Hindawi Publishing CorporationInternational Journal of ElectrochemistryVolume 2013 Article ID 610310 12 pageshttpdxdoiorg1011552013610310

Research ArticleBuoyancy Effect of Ionic Vacancy on the Change ofthe Partial Molar Volume in Ferricyanide-Ferrocyanide RedoxReaction under a Vertical Gravity Field

Yoshinobu Oshikiri1 Makoto Miura2 and Ryoichi Aogaki3

1 Yamagata College of Industry amp Technology Department of Environmental Engineering 2-2-1 Matsuei YamagataYamagata 990-2473 Japan

2 Polytechnic College Akita 6-1 Ohgida-michishita Ohdate Akita 017-0805 Japan3 Polytechnic University 2-20-12-1304 Ryogoku Sumida-ku Tokyo 130-0026 Japan

Correspondence should be addressed to Yoshinobu Oshikiri oshikiriastroyamagata-citacjp

Received 12 October 2012 Revised 4 January 2013 Accepted 15 January 2013

Academic Editor Shen-Ming Chen

Copyright copy 2013 Yoshinobu Oshikiri et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

With a gravity electrode (GE) in a vertical gravity field the buoyancy effect of ionic vacancy on the change of the partial molarvolume in the redox reaction between ferricyanide (FERRI) and ferrocyanide (FERRO) ions was examined The buoyancy force ofionic vacancy takes a positive or negative value depending on whether the rate-determining step is the production or extinctionof the vacancy Though the upward convection over an upward electrode in the FERRO ion oxidation suggests the contributionof the positive buoyancy force arising from the vacancy production the partial molar volume of the vacancy was not measuredOn the other hand for the downward convection under a downward electrode in the FERRI ion reduction it was not completelybut partly measured by the contribution of the negative buoyancy force from the vacancy extinction Since the lifetime of thevacancy is decreased by the collision between ionic vacancies during the convection the former result was ascribed to the shortenedlifetime due to the increasing collision efficiency in the enhanced upward convection over an upward electrode whereas the latterwas thought to arise from the elongated lifetime due to the decreasing collision efficiency by the stagnation under the downwardelectrode

1 Introduction

In electrochemistry the density change in the solution duringan electrode reaction gives rise to a gravitational convection[1] Constant currents observed for hanging electrodes oftencome from the steady-state mass transfer by the gravitationalconvection under a parallel gravity field As a force similarto gravitational force centrifugal force often affects ionictransportation processes for example leading to thermody-namic emf generation [2] and bringing about hydrodynamicconvection

Gravity electrode (GE) is as shown in Figure 1 operatedin a high gravity field arising from a centrifugal force whichprovides a great buoyancy force in the solution promoting agravitational convection As shown in the theoretical analysis

in Appendix it was clarified that the buoyancy force comesfrom the change in the partialmolar volume between productand reactant ions [3] First the effective density coefficientin the limiting diffusion (120574)lim in an electrode reaction ismeasured Then from (A33) in Appendix the change in thepartial molar volume Δ119881

119866is calculated On the other hand

in (A30a) in Appendix Δ119881119866is defined by the partial molar

volumes of the product ion119881119875and reactant ion119881

119877in the form

of 119881119875minus119881119877 Reflecting themeasurement of the diffusion cur-

rent is expressed by 119911119877119863119877(119911119875119863119875) in (A30c) in Appendix

where 119911119877and 119911

119875are the electron numbers without signs

transferring between the reactant and product in the reactionrespectively 119863

119877and 119863

119875are the diffusion coefficients of the

reactant and product respectivelyTherefore if 119881119875and119881119877are

consistent with the equilibriumvalues119881eq119875

and119881eq119877

measured

2 International Journal of Electrochemistry

a b

c

de

Figure 1Gravity electrode of verticalmode a the electrolysis cell bthe electrodes c the counterbalance identical to the electrolysis celld the rotor e metal brushes and lead wires electrically connectedwith a potentiostat

by pycnometer (PM) mentioned later the calculated value119881

eq119875

minus 119881eq119877

will agree with Δ119881119866 This is the blank test for thevalidity of the GEmethod FERRO ion oxidation was actuallyperformed so that it was concluded that the data obtainedby this method do not contain any other extra partial molarvolumes and agree with thermodynamic data [4]

On the other hand by using the GE method as an extracomponent the partial molar volume of ionic vacancy hasbeen newly measured in copper electrodeposition [5 6]Figure 2 represents a schematic figure of the ionic vacancythe center of an ionic vacancy is a charged vacuum voidsurrounded by an ionic cloud and it has a size of the order of01 nm In Figure 3 it is shown that themeasured radii of ionicvacancies agree well with the theoretical values As shown inFigure 3 the vacancies created in cupric sulfate and cupricchloride solutions have minus2 and minus1 unit charges respectively

Recently with a new electrode system called cyclotronmagnetohydrodynamic (MHD) electrode (cyclotronMHDE)the lifetimes of ionic vacancies in some reactions have beenmeasured where the electrolyte solution together with ionicvacancies circulated along a pair of concentric cylinderelectrodes in a vertical magnetic field [7] The lifetime wasmeasured from the circulating velocity As a characteristicfeature the lifetime decreased with an increasing collisionefficiency of ionic vacancies copper depositions in magneticfields from 1 T to 18 T lead to the vacancy lifetimes varyingfrom 10 s to 01 s [7] Though 10 times smaller than in thecase of copper deposition that is 1 s to 001 s in FERRI-FERRO redox reaction the lifetime of ionic vacancy wasalso measured [8] From these results it was thought thatthere are two processes to determine the lifetime of ionicvacancy one is the decay to the initial state and the otheris the conversion to nanobubble [7] Here nanobubble isthe smallest type of bubble containing a gas Recently theformation process from ionic vacancies has been theoreticallyclarified [6] Actually in Figure 4 the plots of the lifetime 120591

against the cell constant 120574cell in copper deposition andFERRI-FERRO reaction are represented [7 8] where 120574cell impliesthe collision efficiency of ionic vacancy [7] For 120574cell = 0 thelifetime thus corresponds to the decay rate of ionic vacancywithout collision whereas for 120574cell = 1 it implies the formationtime of nanobubble by collision of each vacancy As will bediscussed in the following these experimental data elucidatethe reason why by means of the GE method ionic vacancywas observed in copper deposition but not in FERRO ionoxidation that is the lifetimes in FERRO ion oxidation aretoo short to measure by the GE method Since the time scaleof the convection in GE is estimated of the order of 1 s or sothe GE method has the upper limit of the same order for themeasurement

In a large amount of supporting electrolyte the ther-modynamic measurement such as pycnometer (PM) candetermine the partial molar volumes of the individual ions inequilibrium state [3] As a result the validity and accuracy ofthe GE method to determine the change of the partial molarvolume in an electrode reaction were ascertained by the PMmethod The oxidation of FERRO ion to FERRI ion was firsttaken up as a test reaction

[Fe(CN)6]4minus997888rarr [Fe(CN)6]

3minus+ eminus (1)

If there is no effect of ionic vacancy as mentioned above thechange in the partial molar volume Δ119881

119866measured by the GE

method is equal to the equilibriumdataΔ119881pyc obtained by thePMmethod [3] as follows

Δ119881119866= Δ119881pyc (2)

where

Δ119881pyc equiv 119881eq119875

minus 119881eq119877 (3)

where is defined by (A30c) in Appendix 119881eq119875

and 119881eq119877

are the partial molar volumes of the product (FERRI ion)and reactant (FERRO ion) in equilibrium respectively Sinceat the same concentration the density of the FERRO ionsolution is higher than that of the FERRI ion solution FERROion oxidation decreases the solution density In a verticalgravity field for a convection flow to occur an upwardelectrode was thus chosen as the working electrode [4] Thedata obtained by the GE method agreed with those of thePM method within a root mean square (rms) error of 148times 10minus5m3molminus1 [4] Since as will be mentioned later thepartial molar volume of ionic vacancy is estimated as theorder of 10minus4m3molminus1 this result assures that the partialmolar volume of ionic vacancy cannot be observed in FERROion oxidation However an ionic vacancy is as mentionedabove a kind of bubble made of a free vacuum space sothat when ionic vacancies are introduced to a large amountof solution the total volume is expanded with a constantmass From (A14) in Appendix thismeans that the buoyancycoefficient of ionic vacancy 120573

119881is positive that is the ionic

vacancy production generates an upward buoyancy forcewhich accelerates the upward convection and decelerates thedownward convection resulting in the change in the partialmolar volume

International Journal of Electrochemistry 3

Vacuum

minus minusminusminusminusminusminusminusminusminus

minusminusminusminusminus

+ ++++++++++ +++++++ + +

+++ +++

+ +++++

+

(a)

Electrode

IHP OHP IHP OHP

minusminusminusminusminusminus

minusminus

119911119898119890minus

H2O H2O

H2O H2O

H2O

H2OH2O

++

+

+++

+

+

MM119911119898

A119911 A119911

A119911

119911minus

(b)

Figure 2 Ionic vacancy [5 6] (a) structure (b) formation process A119911minus counter anion M119911119898 metallic ion e electron IHP inner Helmholtzplane OHP outer Helmholtz plane

0

05

1

15

119877lowast

(nm

)

10minus2

119898119877 (mol kgminus1)

119911 = minus2

119911 = minus1

119911 = 0

Figure 3 Comparison of experimental data on the radius ofionic vacancy with theoretical calculations in copper electrodepo-sition [5 6] ∙ CuSO

4+ 100molmminus3H

2SO4solution ∘ CuCl

2+

100molmminus3 KCl solution 119911minus the charge number of ionic vacancy

broken line the theoretical value of the radius R lowast the radius ofionic vacancy119898

119877 the molality of the cupric salt

In the present paper in view of the lubricant effect of ionicvacancy the convective-diffusion process in a vertical gravityfield is first reexamined Then the change of the partialmolar volume including ionic vacancy in a redox reactionis theoretically derived Finally the buoyancy effect of ionicvacancy in the redox reactions between FERRI and FERROions is experimentally examined

2 Theoretical

21 Diffusion Current Equation including the Lubricant Effectof Ionic Vacancy In the analysis of the chirality appearing inelectrodeposition under a vertical magnetic field [10 11] ithas been clarified that ionic vacancy plays an important roleof a lubricant [9 12] This is because ionic vacancy is easy

0 02 04 06 08 1001

01

1

10

Cell constant 120574cell

Life

time120591119904

Figure 4 Lifetime of ionic vacancy in the redox reaction betweenFERRI andFERRO ions [7 8] 120591 the lifetime 120574cell the cell constant ofthe cyclotron MHD electrode ∙ FERRI-FERRO ion redox reactionin a 100molmminus3 KCl solution ∘ copper deposition in CuSO

4+

100molmminus3H2SO4solution

to coalesce at electrode surface so that the friction betweenthe solution and the electrode is greatly decreased As aresult two different kinds of surfaces emerge on an electrodesurface that is surfaces covered without and with ionicvacancies which are classified as rigid and free surfaces withand without friction respectively In the previous analysis ofthe convective-diffusion current under a high gravity fieldhowever we considered only a conventional rigid surfacewithout ionic vacancies [13] Figure 5 exhibits the formationprocesses of the free and rigid surfaces by ionic vacancieson an upward electrode under the upward flow followingthe stream lines ionic vacancies are gathered to the centercovering the surface whereas under the downward flow theyare swept away from the center exposing the bare surfaceNamely the free and rigid surfaces arise from upward anddownward flows respectively In Figure 6 a set of convection

4 International Journal of Electrochemistry

Rigid surface

(a)

Free surface

(b)

Figure 5 Rigid and free surfaces formed by ionic vacancy [9] (a) the rigid bare surface under a downward flow (b) the free surface coveredwith ionic vacancies under an upward flow ∘ ionic vacancy

Free surfaceRigid surface

Figure 6 A pair of upward and downward flows in a convection cellaccompanied by ionic vacancy production ∘ the ionic vacancy

flows formed on the free and rigid surfaces is exhibited theflow ascending from the free surface with ionic vacanciesdescends to the rigid surface without them As discussedelsewhere [13] the convection in a vertical gravity field takesplace between the electrode surface and the outer boundaryof the convective-diffusion layer Since the outer boundaryprovides the free surface as shown in Figure 7 for therigid electrode surface without ionic vacancies the boundaryconditions of the convection are consistent with those ofrigid and free boundaries (Figure 7(a)) whereas for the freeelectrode surface completely covered with ionic vacanciesthe conditions are given by two free boundaries (Figure 7(b))

According to the discussion in the previous paper [13]it is apparent that the onset of a vertical convection flowrequires as the necessary and sufficient conditions not onlythe top-heavy distribution of fluid density but also a Rayleighnumber 119877 larger than or equal to the critical value 119877119888 Wemust strongly emphasize that these conditions are valid onlywhen the thickness 120575 of the convective-diffusion layer is apriori fixed in the same way as the conventional thermalconvection (Benard cell convection) [14] that is under thefixed thickness 120575 the convection occurs only when the valueof the adjustable parameter R increases beyond the critical

value 119877119888 However in the present case the situation is quite

different under the top-heavy distribution the convective-diffusion layer is self-organized on the electrode surface thatis the thickness 120575 is not fixed but automatically determinedtogether with the formation of the convection cells The elec-trode system by itself seeks the most stable nonequilibriumstate and determines the thickness 120575 In this case Rayleighnumber R always keeps the critical value 119877

119888 whereas the

thickness 120575 changes as an adjustable parameter From the self-organized convection cells the value is finally given by [13]

120575 = 1205841198631198771198771198881003816100381610038161003816Δ119862119877120573

1003816100381610038161003816 120572

13

(4)

where 120572 is the gravitational acceleration (msminus2) 120584 is thekinematic viscosity (m2 sminus1) and119863119877 is the reactant diffusioncoefficient (m2 sminus1)Δ119862

119877is the concentration difference of the

reactant between the bulk and the surface (molmminus3)

Δ119862119877equiv 119862119877 (119904) minus 119862119877 (119908) (5)

where 119862119877(119904) and 119862

119877(119908) are the bulk and surface molar

concentrations of the reactant (molmminus3) respectively120573 is thebuoyancy coefficient (m3molminus1) defined by

120573 equiv1

120588(120597120588

120597119862119877

)1205831015840

(6)

where 120588 is the density (kgmminus3)119862119877 is themolar concentrationof the reactant (molmminus3) and the subscript 1205831015840 means thatthe composition of the solution except for the reactant is keptconstant In many cases under a gravity field of the order of103msminus2 the value of 120575 is of the order of 10 120583m

In accordance with the way in the previous paper [3]the Rayleigh numbers corresponding to the two kinds ofmathematical solutions for the convection on rigid and freeelectrode surfaces are separately calculated against the nondi-mensional wave number of the convection flow Figure 8represents the result of the calculation the critical Rayleighnumber corresponding to the rigid electrode surface119877crigid =11007 for the critical dimensionless wave number 119886crigid =

International Journal of Electrochemistry 5

Free

RigidElectrode surface

120575rigid

120572

(a)

Free

Electrode surface

120575free

120572

Free

(b)

Figure 7 Boundary conditions for the rigid and free surfaces onthe upward electrode (a) the case of a rigid surface without ionicvacancies (b) the case of a free surface completely covered withionic vacancies 120572 the gravity acceleration 120575rigid the convective-diffusion layer thickness in the case of the rigid surface 120575free theconvective-diffusion layer thickness in the case of the free surface ∘the ionic vacancy

268 is larger than that to the free electrode surface 119877cfree =65751 for 119886cfree = 2221 so that under the condition 119877 =

119877cfree the convection cells are possible for the upward flowson the free surface but are impossible for the downward flowon the rigid surface that is the convection cells shown inFigure 6 are as a whole not completed To compensate thisinsufficient condition it is necessary that the R increasesup to the 119877crigid for the downward flow to start whichtakes the same value of 120575 as that of the preceding case of arigid electrode surface [13] so that it is concluded that thediffusion current density equation obtained is consistent withthe previous one

In the previous papers the diffusion currents of activespecies in parallel and vertical gravity fields were theoreticallyformulated and experimentally validated [13 15ndash19] For thevertical field according to above discussion we obtain [13]

119894 = 1198601199071205741312057213Δ119862119877 (7a)

119860119907 = 119911119877119865119863119877(120584119863119877119877crigid)minus13

= 0969119911119877119865119863119877(120584

119863119877

)

13

120584minus23

(7b)

119911119877is the transferring electron number 119865 is the Faraday

constant and 120574 is the effective density coefficient discussedlater and defined by

120574 equiv minus120573Δ119862119877 (8)

0

1000

2000

3000

A

B

0 1 2 3 4119886 119886119888free 119886119888rigid

119877119888free

119877119888rigidRayl

eigh

num

ber119877

Figure 8 Plots of Rayleigh number versus nondimensional wavenumber A the case of the rigid surface (solid line) 119877crigid = 11007

for 119886crigid = 268 B the case of the free surface (dotted line)119877cfree =

65751 for 119886cfree = 2221

22 Rate-Determining Process of the Convection In FERROion oxidation as mentioned above ionic vacancy has notbeen detected by GE However as shown in Figure 4 inthe redox reactions including the same reaction the lifetimeof ionic vacancy has been measured In the FERRI-FERROredox reaction as will be shown in Figures 11 and 12 thechange in the partial molar volume between the production and the reactant ion is of the order of 10minus5m3molminus1whereas the estimated partial molar volume of ionic vacancyis of the order of 10minus4m3molminus1 Therefore if the lifetime wassufficiently long GE could detect the partial molar volumeof the ionic vacancy However in Figure 4 the lifetimemeasurement by the cyclotron MHDE suggests that theincreasing flow velocity promotes the collision between ionicvacancies [7] assisting the rapid conversion to nanobubbleswhich due to much larger buoyancy forces quickly escapefrom the electrode surface If the buoyancy force of ionicvacancy enhances the convection the lifetime will thusbecome shorter than that in a stationary solution In theFERRI-FERRO redox reaction as shown in Figure 4 thelifetime in the case of perfect collision is about 001 s whichis only 1100th of the intrinsic lifetime

Based on these discussions in Figures 9(a) and 9(b) wecan elucidate the different contributions of ionic vacancies tothe convections on upward and downward electrodes for theFERRO ion oxidation due to upward convection as shownin Figure 9(a) the upward electrode is used as the workingelectrode The vacancy production with upward buoyancyforce thus accelerates the convection promoting the collisionbetween ionic vacancies which decreases the lifetime atmost down to 001 s This is the reason why the vacancy isnot detected in the FERRO ion oxidation by the GE method

6 International Journal of Electrochemistry

120572

119865redox119865119907

Electrode surface

(a)

Electrode surface

120572

119865redox119865119907

(b)

Figure 9 Effect of the buoyancy force of ionic vacancy on the total buoyancy force (a) the case of upward electrode (b) the case of downwardelectrode 119865redox thick black arrow the buoyancy force by the product and reactant ions upward by the density decrease (a) and downwardby the density increase (b) 119865

119907white arrow the buoyancy force by the ionic vacancy upward by the production (a) and downward by the

extinction (b) 120572 thin black arrow the gravity acceleration

RECE

WE

120572

OO

(a)

REWE

CE

120572

OO

(b)

Figure 10 Upward and downward working electrode configurations [9] (a) upward working electrode for the oxidation of FERRO ion (b)downward working electrode for the reduction of FERRI ion o o-ring

As have been discussed in Section 1 since the GE methodrequires the measurement time more than 1 s it is too shortto measure the lifetime In the case of FERRI ion reductiondue to downward convection as shown in Figure 9(b) thedownward electrode is usedThe production of ionic vacancywith upward buoyancy force thus decelerates the convection

forming a stagnation area under the downward electrodethe created vacancies with upward buoyancy forces are firstaccumulated at the stagnation area lightening the upperpart of the solution which results in the suppression ofthe convection The extinction of the ionic vacancies thengradually takes place inducing the downward convection

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal ofPhotoenergy

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Theoretical ChemistryJournal of

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Journal of

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Quantum Chemistry

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Organic Chemistry International

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CatalystsJournal of

2 International Journal of Electrochemistry

a b

c

de

Figure 1Gravity electrode of verticalmode a the electrolysis cell bthe electrodes c the counterbalance identical to the electrolysis celld the rotor e metal brushes and lead wires electrically connectedwith a potentiostat

by pycnometer (PM) mentioned later the calculated value119881

eq119875

minus 119881eq119877

will agree with Δ119881119866 This is the blank test for thevalidity of the GEmethod FERRO ion oxidation was actuallyperformed so that it was concluded that the data obtainedby this method do not contain any other extra partial molarvolumes and agree with thermodynamic data [4]

On the other hand by using the GE method as an extracomponent the partial molar volume of ionic vacancy hasbeen newly measured in copper electrodeposition [5 6]Figure 2 represents a schematic figure of the ionic vacancythe center of an ionic vacancy is a charged vacuum voidsurrounded by an ionic cloud and it has a size of the order of01 nm In Figure 3 it is shown that themeasured radii of ionicvacancies agree well with the theoretical values As shown inFigure 3 the vacancies created in cupric sulfate and cupricchloride solutions have minus2 and minus1 unit charges respectively

Recently with a new electrode system called cyclotronmagnetohydrodynamic (MHD) electrode (cyclotronMHDE)the lifetimes of ionic vacancies in some reactions have beenmeasured where the electrolyte solution together with ionicvacancies circulated along a pair of concentric cylinderelectrodes in a vertical magnetic field [7] The lifetime wasmeasured from the circulating velocity As a characteristicfeature the lifetime decreased with an increasing collisionefficiency of ionic vacancies copper depositions in magneticfields from 1 T to 18 T lead to the vacancy lifetimes varyingfrom 10 s to 01 s [7] Though 10 times smaller than in thecase of copper deposition that is 1 s to 001 s in FERRI-FERRO redox reaction the lifetime of ionic vacancy wasalso measured [8] From these results it was thought thatthere are two processes to determine the lifetime of ionicvacancy one is the decay to the initial state and the otheris the conversion to nanobubble [7] Here nanobubble isthe smallest type of bubble containing a gas Recently theformation process from ionic vacancies has been theoreticallyclarified [6] Actually in Figure 4 the plots of the lifetime 120591

against the cell constant 120574cell in copper deposition andFERRI-FERRO reaction are represented [7 8] where 120574cell impliesthe collision efficiency of ionic vacancy [7] For 120574cell = 0 thelifetime thus corresponds to the decay rate of ionic vacancywithout collision whereas for 120574cell = 1 it implies the formationtime of nanobubble by collision of each vacancy As will bediscussed in the following these experimental data elucidatethe reason why by means of the GE method ionic vacancywas observed in copper deposition but not in FERRO ionoxidation that is the lifetimes in FERRO ion oxidation aretoo short to measure by the GE method Since the time scaleof the convection in GE is estimated of the order of 1 s or sothe GE method has the upper limit of the same order for themeasurement

In a large amount of supporting electrolyte the ther-modynamic measurement such as pycnometer (PM) candetermine the partial molar volumes of the individual ions inequilibrium state [3] As a result the validity and accuracy ofthe GE method to determine the change of the partial molarvolume in an electrode reaction were ascertained by the PMmethod The oxidation of FERRO ion to FERRI ion was firsttaken up as a test reaction

[Fe(CN)6]4minus997888rarr [Fe(CN)6]

3minus+ eminus (1)

If there is no effect of ionic vacancy as mentioned above thechange in the partial molar volume Δ119881

119866measured by the GE

method is equal to the equilibriumdataΔ119881pyc obtained by thePMmethod [3] as follows

Δ119881119866= Δ119881pyc (2)

where

Δ119881pyc equiv 119881eq119875

minus 119881eq119877 (3)

where is defined by (A30c) in Appendix 119881eq119875

and 119881eq119877

are the partial molar volumes of the product (FERRI ion)and reactant (FERRO ion) in equilibrium respectively Sinceat the same concentration the density of the FERRO ionsolution is higher than that of the FERRI ion solution FERROion oxidation decreases the solution density In a verticalgravity field for a convection flow to occur an upwardelectrode was thus chosen as the working electrode [4] Thedata obtained by the GE method agreed with those of thePM method within a root mean square (rms) error of 148times 10minus5m3molminus1 [4] Since as will be mentioned later thepartial molar volume of ionic vacancy is estimated as theorder of 10minus4m3molminus1 this result assures that the partialmolar volume of ionic vacancy cannot be observed in FERROion oxidation However an ionic vacancy is as mentionedabove a kind of bubble made of a free vacuum space sothat when ionic vacancies are introduced to a large amountof solution the total volume is expanded with a constantmass From (A14) in Appendix thismeans that the buoyancycoefficient of ionic vacancy 120573

119881is positive that is the ionic

vacancy production generates an upward buoyancy forcewhich accelerates the upward convection and decelerates thedownward convection resulting in the change in the partialmolar volume

International Journal of Electrochemistry 3

Vacuum

minus minusminusminusminusminusminusminusminusminus

minusminusminusminusminus

+ ++++++++++ +++++++ + +

+++ +++

+ +++++

+

(a)

Electrode

IHP OHP IHP OHP

minusminusminusminusminusminus

minusminus

119911119898119890minus

H2O H2O

H2O H2O

H2O

H2OH2O

++

+

+++

+

+

MM119911119898

A119911 A119911

A119911

119911minus

(b)

Figure 2 Ionic vacancy [5 6] (a) structure (b) formation process A119911minus counter anion M119911119898 metallic ion e electron IHP inner Helmholtzplane OHP outer Helmholtz plane

0

05

1

15

119877lowast

(nm

)

10minus2

119898119877 (mol kgminus1)

119911 = minus2

119911 = minus1

119911 = 0

Figure 3 Comparison of experimental data on the radius ofionic vacancy with theoretical calculations in copper electrodepo-sition [5 6] ∙ CuSO

4+ 100molmminus3H

2SO4solution ∘ CuCl

2+

100molmminus3 KCl solution 119911minus the charge number of ionic vacancy

broken line the theoretical value of the radius R lowast the radius ofionic vacancy119898

119877 the molality of the cupric salt

In the present paper in view of the lubricant effect of ionicvacancy the convective-diffusion process in a vertical gravityfield is first reexamined Then the change of the partialmolar volume including ionic vacancy in a redox reactionis theoretically derived Finally the buoyancy effect of ionicvacancy in the redox reactions between FERRI and FERROions is experimentally examined

2 Theoretical

21 Diffusion Current Equation including the Lubricant Effectof Ionic Vacancy In the analysis of the chirality appearing inelectrodeposition under a vertical magnetic field [10 11] ithas been clarified that ionic vacancy plays an important roleof a lubricant [9 12] This is because ionic vacancy is easy

0 02 04 06 08 1001

01

1

10

Cell constant 120574cell

Life

time120591119904

Figure 4 Lifetime of ionic vacancy in the redox reaction betweenFERRI andFERRO ions [7 8] 120591 the lifetime 120574cell the cell constant ofthe cyclotron MHD electrode ∙ FERRI-FERRO ion redox reactionin a 100molmminus3 KCl solution ∘ copper deposition in CuSO

4+

100molmminus3H2SO4solution

to coalesce at electrode surface so that the friction betweenthe solution and the electrode is greatly decreased As aresult two different kinds of surfaces emerge on an electrodesurface that is surfaces covered without and with ionicvacancies which are classified as rigid and free surfaces withand without friction respectively In the previous analysis ofthe convective-diffusion current under a high gravity fieldhowever we considered only a conventional rigid surfacewithout ionic vacancies [13] Figure 5 exhibits the formationprocesses of the free and rigid surfaces by ionic vacancieson an upward electrode under the upward flow followingthe stream lines ionic vacancies are gathered to the centercovering the surface whereas under the downward flow theyare swept away from the center exposing the bare surfaceNamely the free and rigid surfaces arise from upward anddownward flows respectively In Figure 6 a set of convection

4 International Journal of Electrochemistry

Rigid surface

(a)

Free surface

(b)

Figure 5 Rigid and free surfaces formed by ionic vacancy [9] (a) the rigid bare surface under a downward flow (b) the free surface coveredwith ionic vacancies under an upward flow ∘ ionic vacancy

Free surfaceRigid surface

Figure 6 A pair of upward and downward flows in a convection cellaccompanied by ionic vacancy production ∘ the ionic vacancy

flows formed on the free and rigid surfaces is exhibited theflow ascending from the free surface with ionic vacanciesdescends to the rigid surface without them As discussedelsewhere [13] the convection in a vertical gravity field takesplace between the electrode surface and the outer boundaryof the convective-diffusion layer Since the outer boundaryprovides the free surface as shown in Figure 7 for therigid electrode surface without ionic vacancies the boundaryconditions of the convection are consistent with those ofrigid and free boundaries (Figure 7(a)) whereas for the freeelectrode surface completely covered with ionic vacanciesthe conditions are given by two free boundaries (Figure 7(b))

According to the discussion in the previous paper [13]it is apparent that the onset of a vertical convection flowrequires as the necessary and sufficient conditions not onlythe top-heavy distribution of fluid density but also a Rayleighnumber 119877 larger than or equal to the critical value 119877119888 Wemust strongly emphasize that these conditions are valid onlywhen the thickness 120575 of the convective-diffusion layer is apriori fixed in the same way as the conventional thermalconvection (Benard cell convection) [14] that is under thefixed thickness 120575 the convection occurs only when the valueof the adjustable parameter R increases beyond the critical

value 119877119888 However in the present case the situation is quite

different under the top-heavy distribution the convective-diffusion layer is self-organized on the electrode surface thatis the thickness 120575 is not fixed but automatically determinedtogether with the formation of the convection cells The elec-trode system by itself seeks the most stable nonequilibriumstate and determines the thickness 120575 In this case Rayleighnumber R always keeps the critical value 119877

119888 whereas the

thickness 120575 changes as an adjustable parameter From the self-organized convection cells the value is finally given by [13]

120575 = 1205841198631198771198771198881003816100381610038161003816Δ119862119877120573

1003816100381610038161003816 120572

13

(4)

where 120572 is the gravitational acceleration (msminus2) 120584 is thekinematic viscosity (m2 sminus1) and119863119877 is the reactant diffusioncoefficient (m2 sminus1)Δ119862

119877is the concentration difference of the

reactant between the bulk and the surface (molmminus3)

Δ119862119877equiv 119862119877 (119904) minus 119862119877 (119908) (5)

where 119862119877(119904) and 119862

119877(119908) are the bulk and surface molar

concentrations of the reactant (molmminus3) respectively120573 is thebuoyancy coefficient (m3molminus1) defined by

120573 equiv1

120588(120597120588

120597119862119877

)1205831015840

(6)

where 120588 is the density (kgmminus3)119862119877 is themolar concentrationof the reactant (molmminus3) and the subscript 1205831015840 means thatthe composition of the solution except for the reactant is keptconstant In many cases under a gravity field of the order of103msminus2 the value of 120575 is of the order of 10 120583m

In accordance with the way in the previous paper [3]the Rayleigh numbers corresponding to the two kinds ofmathematical solutions for the convection on rigid and freeelectrode surfaces are separately calculated against the nondi-mensional wave number of the convection flow Figure 8represents the result of the calculation the critical Rayleighnumber corresponding to the rigid electrode surface119877crigid =11007 for the critical dimensionless wave number 119886crigid =

International Journal of Electrochemistry 5

Free

RigidElectrode surface

120575rigid

120572

(a)

Free

Electrode surface

120575free

120572

Free

(b)

Figure 7 Boundary conditions for the rigid and free surfaces onthe upward electrode (a) the case of a rigid surface without ionicvacancies (b) the case of a free surface completely covered withionic vacancies 120572 the gravity acceleration 120575rigid the convective-diffusion layer thickness in the case of the rigid surface 120575free theconvective-diffusion layer thickness in the case of the free surface ∘the ionic vacancy

268 is larger than that to the free electrode surface 119877cfree =65751 for 119886cfree = 2221 so that under the condition 119877 =

119877cfree the convection cells are possible for the upward flowson the free surface but are impossible for the downward flowon the rigid surface that is the convection cells shown inFigure 6 are as a whole not completed To compensate thisinsufficient condition it is necessary that the R increasesup to the 119877crigid for the downward flow to start whichtakes the same value of 120575 as that of the preceding case of arigid electrode surface [13] so that it is concluded that thediffusion current density equation obtained is consistent withthe previous one

In the previous papers the diffusion currents of activespecies in parallel and vertical gravity fields were theoreticallyformulated and experimentally validated [13 15ndash19] For thevertical field according to above discussion we obtain [13]

119894 = 1198601199071205741312057213Δ119862119877 (7a)

119860119907 = 119911119877119865119863119877(120584119863119877119877crigid)minus13

= 0969119911119877119865119863119877(120584

119863119877

)

13

120584minus23

(7b)

119911119877is the transferring electron number 119865 is the Faraday

constant and 120574 is the effective density coefficient discussedlater and defined by

120574 equiv minus120573Δ119862119877 (8)

0

1000

2000

3000

A

B

0 1 2 3 4119886 119886119888free 119886119888rigid

119877119888free

119877119888rigidRayl

eigh

num

ber119877

Figure 8 Plots of Rayleigh number versus nondimensional wavenumber A the case of the rigid surface (solid line) 119877crigid = 11007

for 119886crigid = 268 B the case of the free surface (dotted line)119877cfree =

65751 for 119886cfree = 2221

22 Rate-Determining Process of the Convection In FERROion oxidation as mentioned above ionic vacancy has notbeen detected by GE However as shown in Figure 4 inthe redox reactions including the same reaction the lifetimeof ionic vacancy has been measured In the FERRI-FERROredox reaction as will be shown in Figures 11 and 12 thechange in the partial molar volume between the production and the reactant ion is of the order of 10minus5m3molminus1whereas the estimated partial molar volume of ionic vacancyis of the order of 10minus4m3molminus1 Therefore if the lifetime wassufficiently long GE could detect the partial molar volumeof the ionic vacancy However in Figure 4 the lifetimemeasurement by the cyclotron MHDE suggests that theincreasing flow velocity promotes the collision between ionicvacancies [7] assisting the rapid conversion to nanobubbleswhich due to much larger buoyancy forces quickly escapefrom the electrode surface If the buoyancy force of ionicvacancy enhances the convection the lifetime will thusbecome shorter than that in a stationary solution In theFERRI-FERRO redox reaction as shown in Figure 4 thelifetime in the case of perfect collision is about 001 s whichis only 1100th of the intrinsic lifetime

Based on these discussions in Figures 9(a) and 9(b) wecan elucidate the different contributions of ionic vacancies tothe convections on upward and downward electrodes for theFERRO ion oxidation due to upward convection as shownin Figure 9(a) the upward electrode is used as the workingelectrode The vacancy production with upward buoyancyforce thus accelerates the convection promoting the collisionbetween ionic vacancies which decreases the lifetime atmost down to 001 s This is the reason why the vacancy isnot detected in the FERRO ion oxidation by the GE method

6 International Journal of Electrochemistry

120572

119865redox119865119907

Electrode surface

(a)

Electrode surface

120572

119865redox119865119907

(b)

Figure 9 Effect of the buoyancy force of ionic vacancy on the total buoyancy force (a) the case of upward electrode (b) the case of downwardelectrode 119865redox thick black arrow the buoyancy force by the product and reactant ions upward by the density decrease (a) and downwardby the density increase (b) 119865

119907white arrow the buoyancy force by the ionic vacancy upward by the production (a) and downward by the

extinction (b) 120572 thin black arrow the gravity acceleration

RECE

WE

120572

OO

(a)

REWE

CE

120572

OO

(b)

Figure 10 Upward and downward working electrode configurations [9] (a) upward working electrode for the oxidation of FERRO ion (b)downward working electrode for the reduction of FERRI ion o o-ring

As have been discussed in Section 1 since the GE methodrequires the measurement time more than 1 s it is too shortto measure the lifetime In the case of FERRI ion reductiondue to downward convection as shown in Figure 9(b) thedownward electrode is usedThe production of ionic vacancywith upward buoyancy force thus decelerates the convection

forming a stagnation area under the downward electrodethe created vacancies with upward buoyancy forces are firstaccumulated at the stagnation area lightening the upperpart of the solution which results in the suppression ofthe convection The extinction of the ionic vacancies thengradually takes place inducing the downward convection

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

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CatalystsJournal of

International Journal of Electrochemistry 3

Vacuum

minus minusminusminusminusminusminusminusminusminus

minusminusminusminusminus

+ ++++++++++ +++++++ + +

+++ +++

+ +++++

+

(a)

Electrode

IHP OHP IHP OHP

minusminusminusminusminusminus

minusminus

119911119898119890minus

H2O H2O

H2O H2O

H2O

H2OH2O

++

+

+++

+

+

MM119911119898

A119911 A119911

A119911

119911minus

(b)

Figure 2 Ionic vacancy [5 6] (a) structure (b) formation process A119911minus counter anion M119911119898 metallic ion e electron IHP inner Helmholtzplane OHP outer Helmholtz plane

0

05

1

15

119877lowast

(nm

)

10minus2

119898119877 (mol kgminus1)

119911 = minus2

119911 = minus1

119911 = 0

Figure 3 Comparison of experimental data on the radius ofionic vacancy with theoretical calculations in copper electrodepo-sition [5 6] ∙ CuSO

4+ 100molmminus3H

2SO4solution ∘ CuCl

2+

100molmminus3 KCl solution 119911minus the charge number of ionic vacancy

broken line the theoretical value of the radius R lowast the radius ofionic vacancy119898

119877 the molality of the cupric salt

In the present paper in view of the lubricant effect of ionicvacancy the convective-diffusion process in a vertical gravityfield is first reexamined Then the change of the partialmolar volume including ionic vacancy in a redox reactionis theoretically derived Finally the buoyancy effect of ionicvacancy in the redox reactions between FERRI and FERROions is experimentally examined

2 Theoretical

21 Diffusion Current Equation including the Lubricant Effectof Ionic Vacancy In the analysis of the chirality appearing inelectrodeposition under a vertical magnetic field [10 11] ithas been clarified that ionic vacancy plays an important roleof a lubricant [9 12] This is because ionic vacancy is easy

0 02 04 06 08 1001

01

1

10

Cell constant 120574cell

Life

time120591119904

Figure 4 Lifetime of ionic vacancy in the redox reaction betweenFERRI andFERRO ions [7 8] 120591 the lifetime 120574cell the cell constant ofthe cyclotron MHD electrode ∙ FERRI-FERRO ion redox reactionin a 100molmminus3 KCl solution ∘ copper deposition in CuSO

4+

100molmminus3H2SO4solution

to coalesce at electrode surface so that the friction betweenthe solution and the electrode is greatly decreased As aresult two different kinds of surfaces emerge on an electrodesurface that is surfaces covered without and with ionicvacancies which are classified as rigid and free surfaces withand without friction respectively In the previous analysis ofthe convective-diffusion current under a high gravity fieldhowever we considered only a conventional rigid surfacewithout ionic vacancies [13] Figure 5 exhibits the formationprocesses of the free and rigid surfaces by ionic vacancieson an upward electrode under the upward flow followingthe stream lines ionic vacancies are gathered to the centercovering the surface whereas under the downward flow theyare swept away from the center exposing the bare surfaceNamely the free and rigid surfaces arise from upward anddownward flows respectively In Figure 6 a set of convection

4 International Journal of Electrochemistry

Rigid surface

(a)

Free surface

(b)

Figure 5 Rigid and free surfaces formed by ionic vacancy [9] (a) the rigid bare surface under a downward flow (b) the free surface coveredwith ionic vacancies under an upward flow ∘ ionic vacancy

Free surfaceRigid surface

Figure 6 A pair of upward and downward flows in a convection cellaccompanied by ionic vacancy production ∘ the ionic vacancy

flows formed on the free and rigid surfaces is exhibited theflow ascending from the free surface with ionic vacanciesdescends to the rigid surface without them As discussedelsewhere [13] the convection in a vertical gravity field takesplace between the electrode surface and the outer boundaryof the convective-diffusion layer Since the outer boundaryprovides the free surface as shown in Figure 7 for therigid electrode surface without ionic vacancies the boundaryconditions of the convection are consistent with those ofrigid and free boundaries (Figure 7(a)) whereas for the freeelectrode surface completely covered with ionic vacanciesthe conditions are given by two free boundaries (Figure 7(b))

According to the discussion in the previous paper [13]it is apparent that the onset of a vertical convection flowrequires as the necessary and sufficient conditions not onlythe top-heavy distribution of fluid density but also a Rayleighnumber 119877 larger than or equal to the critical value 119877119888 Wemust strongly emphasize that these conditions are valid onlywhen the thickness 120575 of the convective-diffusion layer is apriori fixed in the same way as the conventional thermalconvection (Benard cell convection) [14] that is under thefixed thickness 120575 the convection occurs only when the valueof the adjustable parameter R increases beyond the critical

value 119877119888 However in the present case the situation is quite

different under the top-heavy distribution the convective-diffusion layer is self-organized on the electrode surface thatis the thickness 120575 is not fixed but automatically determinedtogether with the formation of the convection cells The elec-trode system by itself seeks the most stable nonequilibriumstate and determines the thickness 120575 In this case Rayleighnumber R always keeps the critical value 119877

119888 whereas the

thickness 120575 changes as an adjustable parameter From the self-organized convection cells the value is finally given by [13]

120575 = 1205841198631198771198771198881003816100381610038161003816Δ119862119877120573

1003816100381610038161003816 120572

13

(4)

where 120572 is the gravitational acceleration (msminus2) 120584 is thekinematic viscosity (m2 sminus1) and119863119877 is the reactant diffusioncoefficient (m2 sminus1)Δ119862

119877is the concentration difference of the

reactant between the bulk and the surface (molmminus3)

Δ119862119877equiv 119862119877 (119904) minus 119862119877 (119908) (5)

where 119862119877(119904) and 119862

119877(119908) are the bulk and surface molar

concentrations of the reactant (molmminus3) respectively120573 is thebuoyancy coefficient (m3molminus1) defined by

120573 equiv1

120588(120597120588

120597119862119877

)1205831015840

(6)

where 120588 is the density (kgmminus3)119862119877 is themolar concentrationof the reactant (molmminus3) and the subscript 1205831015840 means thatthe composition of the solution except for the reactant is keptconstant In many cases under a gravity field of the order of103msminus2 the value of 120575 is of the order of 10 120583m

In accordance with the way in the previous paper [3]the Rayleigh numbers corresponding to the two kinds ofmathematical solutions for the convection on rigid and freeelectrode surfaces are separately calculated against the nondi-mensional wave number of the convection flow Figure 8represents the result of the calculation the critical Rayleighnumber corresponding to the rigid electrode surface119877crigid =11007 for the critical dimensionless wave number 119886crigid =

International Journal of Electrochemistry 5

Free

RigidElectrode surface

120575rigid

120572

(a)

Free

Electrode surface

120575free

120572

Free

(b)

Figure 7 Boundary conditions for the rigid and free surfaces onthe upward electrode (a) the case of a rigid surface without ionicvacancies (b) the case of a free surface completely covered withionic vacancies 120572 the gravity acceleration 120575rigid the convective-diffusion layer thickness in the case of the rigid surface 120575free theconvective-diffusion layer thickness in the case of the free surface ∘the ionic vacancy

268 is larger than that to the free electrode surface 119877cfree =65751 for 119886cfree = 2221 so that under the condition 119877 =

119877cfree the convection cells are possible for the upward flowson the free surface but are impossible for the downward flowon the rigid surface that is the convection cells shown inFigure 6 are as a whole not completed To compensate thisinsufficient condition it is necessary that the R increasesup to the 119877crigid for the downward flow to start whichtakes the same value of 120575 as that of the preceding case of arigid electrode surface [13] so that it is concluded that thediffusion current density equation obtained is consistent withthe previous one

In the previous papers the diffusion currents of activespecies in parallel and vertical gravity fields were theoreticallyformulated and experimentally validated [13 15ndash19] For thevertical field according to above discussion we obtain [13]

119894 = 1198601199071205741312057213Δ119862119877 (7a)

119860119907 = 119911119877119865119863119877(120584119863119877119877crigid)minus13

= 0969119911119877119865119863119877(120584

119863119877

)

13

120584minus23

(7b)

119911119877is the transferring electron number 119865 is the Faraday

constant and 120574 is the effective density coefficient discussedlater and defined by

120574 equiv minus120573Δ119862119877 (8)

0

1000

2000

3000

A

B

0 1 2 3 4119886 119886119888free 119886119888rigid

119877119888free

119877119888rigidRayl

eigh

num

ber119877

Figure 8 Plots of Rayleigh number versus nondimensional wavenumber A the case of the rigid surface (solid line) 119877crigid = 11007

for 119886crigid = 268 B the case of the free surface (dotted line)119877cfree =

65751 for 119886cfree = 2221

22 Rate-Determining Process of the Convection In FERROion oxidation as mentioned above ionic vacancy has notbeen detected by GE However as shown in Figure 4 inthe redox reactions including the same reaction the lifetimeof ionic vacancy has been measured In the FERRI-FERROredox reaction as will be shown in Figures 11 and 12 thechange in the partial molar volume between the production and the reactant ion is of the order of 10minus5m3molminus1whereas the estimated partial molar volume of ionic vacancyis of the order of 10minus4m3molminus1 Therefore if the lifetime wassufficiently long GE could detect the partial molar volumeof the ionic vacancy However in Figure 4 the lifetimemeasurement by the cyclotron MHDE suggests that theincreasing flow velocity promotes the collision between ionicvacancies [7] assisting the rapid conversion to nanobubbleswhich due to much larger buoyancy forces quickly escapefrom the electrode surface If the buoyancy force of ionicvacancy enhances the convection the lifetime will thusbecome shorter than that in a stationary solution In theFERRI-FERRO redox reaction as shown in Figure 4 thelifetime in the case of perfect collision is about 001 s whichis only 1100th of the intrinsic lifetime

Based on these discussions in Figures 9(a) and 9(b) wecan elucidate the different contributions of ionic vacancies tothe convections on upward and downward electrodes for theFERRO ion oxidation due to upward convection as shownin Figure 9(a) the upward electrode is used as the workingelectrode The vacancy production with upward buoyancyforce thus accelerates the convection promoting the collisionbetween ionic vacancies which decreases the lifetime atmost down to 001 s This is the reason why the vacancy isnot detected in the FERRO ion oxidation by the GE method

6 International Journal of Electrochemistry

120572

119865redox119865119907

Electrode surface

(a)

Electrode surface

120572

119865redox119865119907

(b)

Figure 9 Effect of the buoyancy force of ionic vacancy on the total buoyancy force (a) the case of upward electrode (b) the case of downwardelectrode 119865redox thick black arrow the buoyancy force by the product and reactant ions upward by the density decrease (a) and downwardby the density increase (b) 119865

119907white arrow the buoyancy force by the ionic vacancy upward by the production (a) and downward by the

extinction (b) 120572 thin black arrow the gravity acceleration

RECE

WE

120572

OO

(a)

REWE

CE

120572

OO

(b)

Figure 10 Upward and downward working electrode configurations [9] (a) upward working electrode for the oxidation of FERRO ion (b)downward working electrode for the reduction of FERRI ion o o-ring

As have been discussed in Section 1 since the GE methodrequires the measurement time more than 1 s it is too shortto measure the lifetime In the case of FERRI ion reductiondue to downward convection as shown in Figure 9(b) thedownward electrode is usedThe production of ionic vacancywith upward buoyancy force thus decelerates the convection

forming a stagnation area under the downward electrodethe created vacancies with upward buoyancy forces are firstaccumulated at the stagnation area lightening the upperpart of the solution which results in the suppression ofthe convection The extinction of the ionic vacancies thengradually takes place inducing the downward convection

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Journal of

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Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

4 International Journal of Electrochemistry

Rigid surface

(a)

Free surface

(b)

Figure 5 Rigid and free surfaces formed by ionic vacancy [9] (a) the rigid bare surface under a downward flow (b) the free surface coveredwith ionic vacancies under an upward flow ∘ ionic vacancy

Free surfaceRigid surface

Figure 6 A pair of upward and downward flows in a convection cellaccompanied by ionic vacancy production ∘ the ionic vacancy

flows formed on the free and rigid surfaces is exhibited theflow ascending from the free surface with ionic vacanciesdescends to the rigid surface without them As discussedelsewhere [13] the convection in a vertical gravity field takesplace between the electrode surface and the outer boundaryof the convective-diffusion layer Since the outer boundaryprovides the free surface as shown in Figure 7 for therigid electrode surface without ionic vacancies the boundaryconditions of the convection are consistent with those ofrigid and free boundaries (Figure 7(a)) whereas for the freeelectrode surface completely covered with ionic vacanciesthe conditions are given by two free boundaries (Figure 7(b))

According to the discussion in the previous paper [13]it is apparent that the onset of a vertical convection flowrequires as the necessary and sufficient conditions not onlythe top-heavy distribution of fluid density but also a Rayleighnumber 119877 larger than or equal to the critical value 119877119888 Wemust strongly emphasize that these conditions are valid onlywhen the thickness 120575 of the convective-diffusion layer is apriori fixed in the same way as the conventional thermalconvection (Benard cell convection) [14] that is under thefixed thickness 120575 the convection occurs only when the valueof the adjustable parameter R increases beyond the critical

value 119877119888 However in the present case the situation is quite

different under the top-heavy distribution the convective-diffusion layer is self-organized on the electrode surface thatis the thickness 120575 is not fixed but automatically determinedtogether with the formation of the convection cells The elec-trode system by itself seeks the most stable nonequilibriumstate and determines the thickness 120575 In this case Rayleighnumber R always keeps the critical value 119877

119888 whereas the

thickness 120575 changes as an adjustable parameter From the self-organized convection cells the value is finally given by [13]

120575 = 1205841198631198771198771198881003816100381610038161003816Δ119862119877120573

1003816100381610038161003816 120572

13

(4)

where 120572 is the gravitational acceleration (msminus2) 120584 is thekinematic viscosity (m2 sminus1) and119863119877 is the reactant diffusioncoefficient (m2 sminus1)Δ119862

119877is the concentration difference of the

reactant between the bulk and the surface (molmminus3)

Δ119862119877equiv 119862119877 (119904) minus 119862119877 (119908) (5)

where 119862119877(119904) and 119862

119877(119908) are the bulk and surface molar

concentrations of the reactant (molmminus3) respectively120573 is thebuoyancy coefficient (m3molminus1) defined by

120573 equiv1

120588(120597120588

120597119862119877

)1205831015840

(6)

where 120588 is the density (kgmminus3)119862119877 is themolar concentrationof the reactant (molmminus3) and the subscript 1205831015840 means thatthe composition of the solution except for the reactant is keptconstant In many cases under a gravity field of the order of103msminus2 the value of 120575 is of the order of 10 120583m

In accordance with the way in the previous paper [3]the Rayleigh numbers corresponding to the two kinds ofmathematical solutions for the convection on rigid and freeelectrode surfaces are separately calculated against the nondi-mensional wave number of the convection flow Figure 8represents the result of the calculation the critical Rayleighnumber corresponding to the rigid electrode surface119877crigid =11007 for the critical dimensionless wave number 119886crigid =

International Journal of Electrochemistry 5

Free

RigidElectrode surface

120575rigid

120572

(a)

Free

Electrode surface

120575free

120572

Free

(b)

Figure 7 Boundary conditions for the rigid and free surfaces onthe upward electrode (a) the case of a rigid surface without ionicvacancies (b) the case of a free surface completely covered withionic vacancies 120572 the gravity acceleration 120575rigid the convective-diffusion layer thickness in the case of the rigid surface 120575free theconvective-diffusion layer thickness in the case of the free surface ∘the ionic vacancy

268 is larger than that to the free electrode surface 119877cfree =65751 for 119886cfree = 2221 so that under the condition 119877 =

119877cfree the convection cells are possible for the upward flowson the free surface but are impossible for the downward flowon the rigid surface that is the convection cells shown inFigure 6 are as a whole not completed To compensate thisinsufficient condition it is necessary that the R increasesup to the 119877crigid for the downward flow to start whichtakes the same value of 120575 as that of the preceding case of arigid electrode surface [13] so that it is concluded that thediffusion current density equation obtained is consistent withthe previous one

In the previous papers the diffusion currents of activespecies in parallel and vertical gravity fields were theoreticallyformulated and experimentally validated [13 15ndash19] For thevertical field according to above discussion we obtain [13]

119894 = 1198601199071205741312057213Δ119862119877 (7a)

119860119907 = 119911119877119865119863119877(120584119863119877119877crigid)minus13

= 0969119911119877119865119863119877(120584

119863119877

)

13

120584minus23

(7b)

119911119877is the transferring electron number 119865 is the Faraday

constant and 120574 is the effective density coefficient discussedlater and defined by

120574 equiv minus120573Δ119862119877 (8)

0

1000

2000

3000

A

B

0 1 2 3 4119886 119886119888free 119886119888rigid

119877119888free

119877119888rigidRayl

eigh

num

ber119877

Figure 8 Plots of Rayleigh number versus nondimensional wavenumber A the case of the rigid surface (solid line) 119877crigid = 11007

for 119886crigid = 268 B the case of the free surface (dotted line)119877cfree =

65751 for 119886cfree = 2221

22 Rate-Determining Process of the Convection In FERROion oxidation as mentioned above ionic vacancy has notbeen detected by GE However as shown in Figure 4 inthe redox reactions including the same reaction the lifetimeof ionic vacancy has been measured In the FERRI-FERROredox reaction as will be shown in Figures 11 and 12 thechange in the partial molar volume between the production and the reactant ion is of the order of 10minus5m3molminus1whereas the estimated partial molar volume of ionic vacancyis of the order of 10minus4m3molminus1 Therefore if the lifetime wassufficiently long GE could detect the partial molar volumeof the ionic vacancy However in Figure 4 the lifetimemeasurement by the cyclotron MHDE suggests that theincreasing flow velocity promotes the collision between ionicvacancies [7] assisting the rapid conversion to nanobubbleswhich due to much larger buoyancy forces quickly escapefrom the electrode surface If the buoyancy force of ionicvacancy enhances the convection the lifetime will thusbecome shorter than that in a stationary solution In theFERRI-FERRO redox reaction as shown in Figure 4 thelifetime in the case of perfect collision is about 001 s whichis only 1100th of the intrinsic lifetime

Based on these discussions in Figures 9(a) and 9(b) wecan elucidate the different contributions of ionic vacancies tothe convections on upward and downward electrodes for theFERRO ion oxidation due to upward convection as shownin Figure 9(a) the upward electrode is used as the workingelectrode The vacancy production with upward buoyancyforce thus accelerates the convection promoting the collisionbetween ionic vacancies which decreases the lifetime atmost down to 001 s This is the reason why the vacancy isnot detected in the FERRO ion oxidation by the GE method

6 International Journal of Electrochemistry

120572

119865redox119865119907

Electrode surface

(a)

Electrode surface

120572

119865redox119865119907

(b)

Figure 9 Effect of the buoyancy force of ionic vacancy on the total buoyancy force (a) the case of upward electrode (b) the case of downwardelectrode 119865redox thick black arrow the buoyancy force by the product and reactant ions upward by the density decrease (a) and downwardby the density increase (b) 119865

119907white arrow the buoyancy force by the ionic vacancy upward by the production (a) and downward by the

extinction (b) 120572 thin black arrow the gravity acceleration

RECE

WE

120572

OO

(a)

REWE

CE

120572

OO

(b)

Figure 10 Upward and downward working electrode configurations [9] (a) upward working electrode for the oxidation of FERRO ion (b)downward working electrode for the reduction of FERRI ion o o-ring

As have been discussed in Section 1 since the GE methodrequires the measurement time more than 1 s it is too shortto measure the lifetime In the case of FERRI ion reductiondue to downward convection as shown in Figure 9(b) thedownward electrode is usedThe production of ionic vacancywith upward buoyancy force thus decelerates the convection

forming a stagnation area under the downward electrodethe created vacancies with upward buoyancy forces are firstaccumulated at the stagnation area lightening the upperpart of the solution which results in the suppression ofthe convection The extinction of the ionic vacancies thengradually takes place inducing the downward convection

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Medicinal ChemistryInternational Journal of

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Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Theoretical ChemistryJournal of

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Quantum Chemistry

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Organic Chemistry International

ElectrochemistryInternational Journal of

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CatalystsJournal of

International Journal of Electrochemistry 5

Free

RigidElectrode surface

120575rigid

120572

(a)

Free

Electrode surface

120575free

120572

Free

(b)

Figure 7 Boundary conditions for the rigid and free surfaces onthe upward electrode (a) the case of a rigid surface without ionicvacancies (b) the case of a free surface completely covered withionic vacancies 120572 the gravity acceleration 120575rigid the convective-diffusion layer thickness in the case of the rigid surface 120575free theconvective-diffusion layer thickness in the case of the free surface ∘the ionic vacancy

268 is larger than that to the free electrode surface 119877cfree =65751 for 119886cfree = 2221 so that under the condition 119877 =

119877cfree the convection cells are possible for the upward flowson the free surface but are impossible for the downward flowon the rigid surface that is the convection cells shown inFigure 6 are as a whole not completed To compensate thisinsufficient condition it is necessary that the R increasesup to the 119877crigid for the downward flow to start whichtakes the same value of 120575 as that of the preceding case of arigid electrode surface [13] so that it is concluded that thediffusion current density equation obtained is consistent withthe previous one

In the previous papers the diffusion currents of activespecies in parallel and vertical gravity fields were theoreticallyformulated and experimentally validated [13 15ndash19] For thevertical field according to above discussion we obtain [13]

119894 = 1198601199071205741312057213Δ119862119877 (7a)

119860119907 = 119911119877119865119863119877(120584119863119877119877crigid)minus13

= 0969119911119877119865119863119877(120584

119863119877

)

13

120584minus23

(7b)

119911119877is the transferring electron number 119865 is the Faraday

constant and 120574 is the effective density coefficient discussedlater and defined by

120574 equiv minus120573Δ119862119877 (8)

0

1000

2000

3000

A

B

0 1 2 3 4119886 119886119888free 119886119888rigid

119877119888free

119877119888rigidRayl

eigh

num

ber119877

Figure 8 Plots of Rayleigh number versus nondimensional wavenumber A the case of the rigid surface (solid line) 119877crigid = 11007

for 119886crigid = 268 B the case of the free surface (dotted line)119877cfree =

65751 for 119886cfree = 2221

22 Rate-Determining Process of the Convection In FERROion oxidation as mentioned above ionic vacancy has notbeen detected by GE However as shown in Figure 4 inthe redox reactions including the same reaction the lifetimeof ionic vacancy has been measured In the FERRI-FERROredox reaction as will be shown in Figures 11 and 12 thechange in the partial molar volume between the production and the reactant ion is of the order of 10minus5m3molminus1whereas the estimated partial molar volume of ionic vacancyis of the order of 10minus4m3molminus1 Therefore if the lifetime wassufficiently long GE could detect the partial molar volumeof the ionic vacancy However in Figure 4 the lifetimemeasurement by the cyclotron MHDE suggests that theincreasing flow velocity promotes the collision between ionicvacancies [7] assisting the rapid conversion to nanobubbleswhich due to much larger buoyancy forces quickly escapefrom the electrode surface If the buoyancy force of ionicvacancy enhances the convection the lifetime will thusbecome shorter than that in a stationary solution In theFERRI-FERRO redox reaction as shown in Figure 4 thelifetime in the case of perfect collision is about 001 s whichis only 1100th of the intrinsic lifetime

Based on these discussions in Figures 9(a) and 9(b) wecan elucidate the different contributions of ionic vacancies tothe convections on upward and downward electrodes for theFERRO ion oxidation due to upward convection as shownin Figure 9(a) the upward electrode is used as the workingelectrode The vacancy production with upward buoyancyforce thus accelerates the convection promoting the collisionbetween ionic vacancies which decreases the lifetime atmost down to 001 s This is the reason why the vacancy isnot detected in the FERRO ion oxidation by the GE method

6 International Journal of Electrochemistry

120572

119865redox119865119907

Electrode surface

(a)

Electrode surface

120572

119865redox119865119907

(b)

Figure 9 Effect of the buoyancy force of ionic vacancy on the total buoyancy force (a) the case of upward electrode (b) the case of downwardelectrode 119865redox thick black arrow the buoyancy force by the product and reactant ions upward by the density decrease (a) and downwardby the density increase (b) 119865

119907white arrow the buoyancy force by the ionic vacancy upward by the production (a) and downward by the

extinction (b) 120572 thin black arrow the gravity acceleration

RECE

WE

120572

OO

(a)

REWE

CE

120572

OO

(b)

Figure 10 Upward and downward working electrode configurations [9] (a) upward working electrode for the oxidation of FERRO ion (b)downward working electrode for the reduction of FERRI ion o o-ring

As have been discussed in Section 1 since the GE methodrequires the measurement time more than 1 s it is too shortto measure the lifetime In the case of FERRI ion reductiondue to downward convection as shown in Figure 9(b) thedownward electrode is usedThe production of ionic vacancywith upward buoyancy force thus decelerates the convection

forming a stagnation area under the downward electrodethe created vacancies with upward buoyancy forces are firstaccumulated at the stagnation area lightening the upperpart of the solution which results in the suppression ofthe convection The extinction of the ionic vacancies thengradually takes place inducing the downward convection

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Medicinal ChemistryInternational Journal of

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Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Theoretical ChemistryJournal of

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Quantum Chemistry

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Organic Chemistry International

ElectrochemistryInternational Journal of

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CatalystsJournal of

6 International Journal of Electrochemistry

120572

119865redox119865119907

Electrode surface

(a)

Electrode surface

120572

119865redox119865119907

(b)

Figure 9 Effect of the buoyancy force of ionic vacancy on the total buoyancy force (a) the case of upward electrode (b) the case of downwardelectrode 119865redox thick black arrow the buoyancy force by the product and reactant ions upward by the density decrease (a) and downwardby the density increase (b) 119865

119907white arrow the buoyancy force by the ionic vacancy upward by the production (a) and downward by the

extinction (b) 120572 thin black arrow the gravity acceleration

RECE

WE

120572

OO

(a)

REWE

CE

120572

OO

(b)

Figure 10 Upward and downward working electrode configurations [9] (a) upward working electrode for the oxidation of FERRO ion (b)downward working electrode for the reduction of FERRI ion o o-ring

As have been discussed in Section 1 since the GE methodrequires the measurement time more than 1 s it is too shortto measure the lifetime In the case of FERRI ion reductiondue to downward convection as shown in Figure 9(b) thedownward electrode is usedThe production of ionic vacancywith upward buoyancy force thus decelerates the convection

forming a stagnation area under the downward electrodethe created vacancies with upward buoyancy forces are firstaccumulated at the stagnation area lightening the upperpart of the solution which results in the suppression ofthe convection The extinction of the ionic vacancies thengradually takes place inducing the downward convection

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

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CatalystsJournal of

International Journal of Electrochemistry 7

0 001 002 0030

5

10

119898119877 (mol kgminus1)

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 11 Comparison of the partial molar volume change betweenthe GE and PM methods in the oxidation of FERRO ion at theupward electrode ∘Δ119881

119866 ∙Δ119881pyc119863119875 (FERRI) = 828times10

minus10m2 sminus1119863119877(FERRO) = 698 times 10minus10m2 sminus1 120584 = 901 times 10minus7m2 sminus1

with an increasing density Such extinction process mayproceed in a rate of the intrinsic lifetime that is 1 s whichis much longer than that of FERRI ion reduction

23 Measurement of the Buoyancy Effect of Ionic VacancyIn accordance with above discussion instead of (A26) inAppendix the effective density coefficient is more correctlyexpressed by adding the effect of ionic vacancy as follows

120574 = 120574redox + 120574119907eff (9)

where 120574redox and 120574119907eff are the effective density coefficients ofthe redox reaction and ionic vacancy respectively As shownin (A33) in Appendix 120574 is used as (120574)lim in the limitingdiffusion which is related to the change of the partial molarvolume in the electrode reaction Δ119881

119866(m3molminus1) explicitly

written as in (A33) where Δ119872119898

is the difference of themolar mass between the product 119872

119898119875and the reactant

119872119898119877

(kgmolminus1) Δ119872119898equiv 119872119898119875

minus119872119898119877

(see (A30b)) 119898119877(119904)

is the molality of the reactant in the bulk solution (mol kgminus1)and 120588

1199040is the density of the bulk solution with supporting

electrolyte (kgmminus3) In view of the effect of ionic vacancy in(9) using the equilibrium data of 119881eq

119875and 119881eq

119877measured by

the PMmethod the partialmolar volumes of the product andthe reactant 119881119875 and 119881119877 in the electrode reaction are definedby

119881119875= 119881

eq119875

+ (119881119881)eff

119881119877= 119881

eq119877

(10)

where (119881119881)eff is the effective partial molar volume of the

vacancy (m3molminus1) Instead of (2) and (3) the change in the

119898119877 (mol kgminus1)0 001 002 003

0

minus5

minus10

minus15

Δ119881119866Δ119881

pyc

(10minus5

m3

molminus1)

Figure 12 Comparison of the partial molar volume change betweenthe GE and PM methods in the reduction of FERRI ion at thedownward electrode ∘ Δ119881

119866 ∙ Δ119881pyc 119863119875 (FERRO) = 698 times

10minus10m2 sminus1 119863119877(FERRI) = 828 times 10minus10m2 sminus1 120584 = 901 times

10minus7m2 sminus1

partial molar volume Δ119881119866measured by the GE method is

rewritten by Δ119881pyc and (119881119881)eff as follows

Δ119881119866 = Δ119881pyc + (119881119881)eff (11)

Therefore (119881119881)eff is calculated by

(119881119881)eff =Δ119881119866minus Δ119881pyc

(12)

As the lifetime becomes longer (119881119881)eff approaches plusmn119881

119881

where 119881119881is the intrinsic partial molar volume of the ionic

vacancy (m3molminus1) and the sign plusmn corresponds to positive(upward) and negative (downward) buoyancies respectivelyOn the other hand (119881119881)eff converges to zero as the lifetimedecreases

3 Experimental

31 GE Method As test reactions redox reactions betweenFERRO and FERRI ions were adopted ConcerningFERRO and FERRI ions each of eighteen samples of100molmminus3 K

2SO4solutions was prepared for the molar

concentration from 10molmminus3 to 30molmminus3 For themeasurement of the density coefficient (120574)lim in a limiting-diffusion current a GE (GE01 Nikko Keisoku Co) wasused in the vertical gravity mode As shown in Figures 10(a)and 10(b) a pair of circular Pt plates with 5mm diametershielded by o-rings was used for working and counterelectrodes where the active areas inside the o-rings were314mm2 Since the oxidation of FERRO ion decreases the

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Carbohydrate Chemistry

International Journal of

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Journal of

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Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

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Quantum Chemistry

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Organic Chemistry International

ElectrochemistryInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

8 International Journal of Electrochemistry

minus50 001 002 003

0

5

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 13 Plot of the effective partialmolar volume of ionic vacancyin the oxidation of FERRO ion against the molality of FERRO ion ∘the measured values broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus877 times 10

minus6plusmn 133 times

10minus5m3molminus1 the expected partial molar volume of ionic vacancywith minus1 unit charge 119881

119907= 213 times 10minus4m3molminus1

solution density the upward electrode configuration bringshydrodynamic instability whereas the reduction of FERRIion increases the solution density so that the convection isexpected for a downward electrode Therefore the upwardelectrode was used as the working electrode for the oxidationof FERRO ion and the downward electrode was used as theworking electrode for the reduction of FERRI ion As thereference electrode a silver wire coated by AgCl film with a1mm diameter was used The reactions were performed atoverpotentials of +200mV and minus200mV for the oxidationand reduction respectively that is at the limiting diffusionranges Prior to the measurement argon bubbling wasperformed to evacuate dissolve oxygen in the solution Tocalculate the constant 119860119907 in (7b) the kinematic viscosity120584 was measured by the Cannon-Fenske viscometer (SibataScientific Technology Ltd) and the diffusion coefficients119863119875 and 119863119877 were determined by the rotating disk electrode(RRDE-1 Nikko Keisoku Co) The solution was also kept at27 plusmn 1∘C After the measurement according to the procedureelucidated in the preceding paper [4] the data obtained werecalculated

32 PM Method Concerning FERRO and FERRI ions thesame samples as those of the GE method were prepared Forthermodynamic data to obtain Hubbard-type PM (SpecificGravity Bottle Sibata Scientific Technology Ltd) was usedThe sample was also kept at 27 plusmn 1∘CThe data obtained weretreated in the same way as that in the preceding paper [4]

minus5

minus100 001 002 003

0

119898119877 (mol kgminus1)

(119881119907) e

ff(10minus5

m3

molminus1)

Figure 14 Plot of the effective partialmolar volume of ionic vacancyin the reduction of FERRI ion against the molality of FERRI ion ∘the measured value broken line the average value The measuredeffective partial molar volume (119881

119907)eff = minus380 times 10

minus5plusmn 109 times

10minus5m3molminus1 the expected negative partial molar volume of ionicvacancy with minus1 unit charge minus119881

119907= minus213 times 10minus4m3molminus1

4 Results and Discussion

Figure 11 shows the plot of the partial molar volume changein the oxidation of FERRO ion measured by the upwardelectrode together with the thermodynamic data by thePM method Both data take positive values consistent witheach other within a relative error of 148 times 10minus5m3molminus1As discussed above such agreement is attributed to theshortened lifetime of the ionic vacancy due to the increasingcollision efficiency in the convection flow driven by theupward buoyancy forces arising from the vacancy productiontogether with the positive partial molar volume changebetween FERRO and FERRI ions

In Figure 12 the partial molar volume change in thereduction of FERRI ion at the downward electrode is exhib-ited Different from the above case all the data are shiftedtoward the negative side which is attributed to the negativebuoyancy forces occurring from the vacancy extinctiontogether with the negative partial molar volume changebetween FERRI and FERRO ions which suggests that thestagnation under the downward electrode keeps the collisionof vacancy away making the lifetime longer

In Figures 13 and 14 the effective partial molar volumes(119881119881)eff extracted by (12) are exhibited For the FERRO ion

oxidation in Figure 13 the average value of (119881119881)eff is nearly

equal to zero that is minus877 times 10minus6m3molminus1 In comparisonwith the expected value of the ionic vacancy with ndash1 unitcharge given by 119881

119881= 213 times 10minus4m3molminus1 it is concluded

that the FERRO ion oxidation at the upward electrode notonly always assures the validity of the diffusion current

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Medicinal ChemistryInternational Journal of

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Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Quantum Chemistry

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Organic Chemistry International

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CatalystsJournal of

International Journal of Electrochemistry 9

equations (7a) and (7b) but also can be used as the blanktest for the GE method On the other hand for the FERRIion reduction in Figure 14 due to the negative buoyancyforce of ionic vacancy the (119881

119881)eff takes a negative average

value minus380 times 10minus5m3molminus1 which in comparison with theintrinsic value minus213 times 10minus4m3molminus1 can be regarded as ameaningful value The difference between the experimentaland the intrinsic values in Figure 14 results from the fact thatthe intrinsic lifetime of the vacancy is not sufficiently long forcomplete observation Accordingly it is concluded that theexact measurement for the size of ionic vacancy by the GEmethod requires at least an intrinsic lifetime longer than 1 sActually for the ionic vacancy with an intrinsic lifetime of86 s in copper deposition from acidic cupric sulfate solutionas shown in Figure 3 in spite of upward electrode observeddata agreed with the intrinsic value

5 Conclusions

In a vertical gravity field ionic vacancies create partly rigidand partly free surfaces on an electrode surface with andwithout friction respectively However since the convectionflow on the rigid surface is rate-determining step the wholeconvection process is controlled by the convection on therigid surface Accordingly it is concluded that in this situa-tion the same equation as that of rigid surface is derived

The gravitational convection for FERRI-FERRO ionredox reaction in the GE method arises from the buoyancyforce occurring in the reaction which takes a positiveor negative value according to the fact whether the rate-determining step is the production or extinction of ionicvacancy together with the positive or negative partial molarvolume change between product and reactant ions Whetherthe total buoyancy force of the reaction is positive ornegative can be discriminated by the upward or downwardelectrode used for measuring diffusion current As a resultit was found that in FERRO ion oxidation and FERRI ionreduction upward and downward convection cells arisefrom the positive and negative buoyancy forces respectivelyHowever in the FERRO ion oxidation the positive partialmolar volume of the vacancy from the vacancy productionwas not observed whereas in the FERRI ion reduction thenegative partial molar volume from the vacancy extinctionwas not completely but partly observed The former resultwas ascribed to the short vacancy lifetime shortened by theaccelerated convection on the upward electrode and the latterwas to the lifetime elongated by the stagnation under thedownward electrode

Appendix

The Partial Molar Volume Change Measuredby GE in a Redox Reaction

In accordance with the foregoing paper [3] the partial molarvolume change measured by GE is formulated as follows

First the density of a solution is defined by the mass119872 andvolume 119881 of the solution

120588 =119872

119881 (A1)

In the solution the volume 119881 is thought to be a physicalquantity defined in the area in local equilibrium Resultantlyinside the volume the temperature T the pressure p and thecomposition119862119894 (molar concentration) are regarded constantThe change in the density during the reaction is assumedto proceed at constant temperature and pressure Since theconcentration of the active ion is quite low (the order of1molmminus3) we can neglect the thermal effect due to Jouleheat as well as exthothermic and endothermic reactions Alarge amount of supporting electrolyte brings large dielectricrelaxation to the bulk solution so that the electric fieldstrength is drastically weakenedThemigration of supportingelectrolyte can be therefore disregarded In the presence ofa large amount of supporting electrolyte sharing the counterion with the active ion except for a narrow region of electricdouble layer the counter ion together with the other ionof the supporting electrolyte is thus thought to distributehomogeneously in the solution so that the active ion isindependently treated from other components

A solvent containing only supporting electrolyte is firstassumed of which mass and volume are denoted as119872

1199040and

1198811199040 respectively The reactant and product of the electrode

reaction are then virtually introduced to the solvent Accord-ing to the increments of the mass and volume d119872 and d119881the solution density also changes by d120588 The infinitesimalvolume d119881 is defined by the scale of length much smallerthan the diffusion layer thickness but sufficiently large incomparison with the double layer thickness Therefore d119872contains a sufficient large number of solution particles Atconstant temperature and pressure (A1) is expanded toobtain the equation of the first expansion of 120588 with regard tod119881and d119872 at119872 = 119872

1199040and 119881 = 119881

1199040

d120588 = (120597120588

120597119881)119872=119872

1199040

119881=1198811199040

d119881 + (120597120588

120597119872)119872=119872

1199040

119881=1198811199040

d119872

(A2)

where

(120597120588

120597119881)119872=119872

1199040

119881=1198811199040

= minus1198721199040

11988121199040

= minus1205881199040

1198811199040

(A3a)

(120597120588

120597119872)119872=119872

1199040

119881=1198811199040

=1

1198811199040

(A3b)

where 1205881199040 denotes the density of the solvent with the support-ing electrolyte

In the presence of a large amount of electrolyte we canexpress the change in the volume by the change in the molarnumber 119899

119896of the active ion k that is reactant or product ion

(119896 = 119877 or P) as follows

d119881 = 119881119896d119899119896 (A4)

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

10 International Journal of Electrochemistry

where 119881119896is the partial molar volume of the ion 119896 exhibited

as

119881119896equiv (

120597119881

120597119899119896)1205831015840

(A5)

Then the molar change of the ion 119896 by the change in its masscan be expressed by

d119899119896=

1

119872119898119896

d119872 (A6)

where119872119898119896

is the molar mass of the active ion 119896 From (A4)and (A6) the following relationship is derived

d119881 =119881119896

119872119898119896d119872 (A7)

Substitution for d119872 from (A7) in (A2) allows us to obtain

d120588 = minus120588lowast119896

1198811199040d119881 (A8)

where

120588lowast

119896equiv 1205881199040 minus

119872119898119896

119881119896

(A9)

The density 120588 of the solution is also a function of the molarconcentration 119862

119896 that is

d120588 = (120597120588

120597119862119896

)1205831015840

d119862119896 (A10)

where subscript 1205831015840 means that other concentrations exceptfor 119862

119896together with other physical parameters are kept

constant The volume variation by the change in the molarconcentration can be expressed by

d119881 = (120597119881

120597119862119896

)1205831015840

d119862119896 (A11)

Substituting (A11) into (A8) and comparing the resultantequation with (A10) we obtain

(120597120588

120597119862119896

)1205831015840

= minus120588lowast

119896

1198811199040

(120597119881

120597119862119896

)1205831015840

(A12)

According to (6) the buoyancy coefficient of a single activeion 119896 is defined as

120573119896equiv minus

1

1205881199040(120597120588

120597119862119896)1205831015840

(A13)

Substitution of (A12) into (A13) leads to

120573119896= (

120588lowast119896

1205881199040

)1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A14)

Practically as have already been shown in (7a) and (7b)the following density coefficient 120574

119896of the single active ion

119896 is rather important for the measurement by GE Thedensity coefficient is the relative density of the ion 119896 for thegravitational convection instead of (8) newly defined for theion k by

120574119896equiv minus120573119896Δ119862119896 (A15)

120574119896 is a dimensionless number intrinsic to the solvated activeion 119896 Δ119862119896 is as shown in (5) the molar concentrationdifference between the bulk and surface

Δ119862119896= 119862119896 (119904) minus 119862119896 (119908) (A16)

where 119862119896(119904) and 119862

119896(119908) are the bulk and surface concentra-

tions respectively 119862119896is the molar concentration of the ion 119896

defined as

119862119896equiv119899119896

119881 (A17)

Differentiating (A17) we obtain the relationship

d119899119896= 119881d119862

119896+ 119862119896d119881 (A18)

The total volume 119881 is a function of the molar number of theactive ion 119896 at constant temperature and pressure that is

d119881 = 119881119896d119899119896 (A19)

Substitution for d119899119896from (A18) in (A19) leads to

d119881 =119881119881119896d119862119896

1 minus 119881119896119862119896

(A20)

Comparing (A20) with (A11) we obtain

119881119896

1 minus 119881119896119862119896

=1

119881(120597119881

120597119862119896

)1205831015840

(A21)

Assuming the following condition where the molar concen-trations of the active ions are sufficiently low and their molarvolumes are small that is

10038161003816100381610038161198811198961198621198961003816100381610038161003816 ≪ 1 (A22)

We rewrite (A21) as

119881119896=

1

1198811199040

(120597119881

120597119862119896

)1205831015840

(A23)

where in viewof the fact that the total volume is approximatedby the volume of the solvent with the supporting electrolyte119881 in (A21) is replaced by119881

1199040 Inserting (A14) into (A15) and

substituting (A9) and (A23) into the resultant equation weobtain the density coefficient of the ion k as follows

120574119896 = minusΔ119862119896 (119881119896 minus119872119898119896

1205881199040

) (A24)

This is the basic equation describing the relationship between120574119896and 119881

119896

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

International Journal of Electrochemistry 11

Based on the above preliminary discussion the densitycoefficient for an electrode reaction can be derived Firstthe effective density coefficient 120574 for an electrode reaction isdefined in the following redox reaction which is generallyexpressed by

reactant plusmn 119911119896e 997888rarr product (A25)

where 119911119896is the number of electron transferring in the reac-

tion Since in the presence of a large amount of supportingelectrolyte the minority active ions are always surroundedby the majority ions of the supporting electrolyte and thecircumstance around each active ion remains the samewhichguarantees the superimposition of the partial molar volumesThe effective density coefficient 120574 in (8) is thus expressed by

120574 = 120574119875+ 120574119877 (A26)

where subscripts 119875 and 119877 denote the product and reactantrespectively As the reactant is consumed during the reac-tion the product is produced As mentioned above due todielectric relaxation a large amount of supporting electrolyteweakens the migration of supporting electrolyte togetherwith the occurrence of electric field As for active ionssince the minority active ions are always surrounded by themajority ions of the supporting electrolyte the circumstancearound each active ion remains the same Namely an activeion does not feel the existence of other active ions so thatwithoutmigration they can diffuse in the sameway as neutralmolecules Therefore at the electrode surface (119911 = 0) thefollowing Fickrsquos first law is satisfied

119911119877119865119863119877(120597119862119877

120597119911)119911=0

= minus119911119875119865119863119875(120597119862119875

120597119911)119911=0

(A27)

where 119911119896 (119896 = 119877 or 119875) is the electron number in the reactantor product transferring in the reaction and119863119896 is the diffusioncoefficient Using the diffusion layer thickness 120575 and theconcentration differences of the reactant and product ionsΔ119862119877and Δ119862

119875between the bulk and surface we can rewrite

(A27) as

119911119877119863119877Δ119862119877= minus119911119875119863119875Δ119862119875 (A28)

From (A24) (A26) and (A28) 120574 is explicitly written as

120574 = Δ119862119877 (Δ119881119866 minus

Δ119872119898

1205881199040

) (A29)

where Δ119862119877 Δ119881119866 and Δ119872

119898are defined in (5) as the follow-

ing

Δ119881119866equiv119881119875minus 119881119877 (A30a)

Δ119872119898equiv119872m119875 minus119872m119877 (A30b)

equiv119911119877119863119877

119911119875119863119875

(A30c)

Under the condition of limiting diffusion where thesurface concentration of the reactant becomes zero that is

119862119877(119908) = 0 the difference of the molar concentration of the

reactant Δ119862119877is thus rewritten as

Δ119862119877= 119862119877 (119904) = 120588

1199040119898119877 (119904) (A31)

where 119862119877(119904) and 119898119877(119904) are the molar concentration and themolality of the reactant in the bulk respectively Substituting(A31) into (A29) we obtain the relationship correspondingto the limiting diffusion as follows

(120574)lim = 119898119877 (119904) (1205881199040Δ119881119866 minus Δ119872119898) (A32)

where (120574)lim denotes the effective density coefficient mea-sured in the limiting diffusion Therefore from the experi-mental data of (120574)lim in an electrochemical reaction the valueof Δ119881119866is calculated by

Δ119881119866=

1

1205881199040

((120574)lim119898119877 (119904)

+ Δ119872119898) (A33)

All the physical quantities on the right-hand side can bedetermined by electrochemical and thermodynamic mea-surements so that (A33) is applicable to nonequilibrium stateas well as equilibrium state

Acknowledgment

Theauthors are thankful to Dr SugiyamaWasedaUniversityfor providing us with the data of the lifetime of ionic vacancyin FERRI-FERRO redox reaction

References

[1] VM Volgin andA D Davydov ldquoNatural-convective instabilityof electrochemical systems a reviewrdquoRussian Journal of Electro-chemistry vol 42 no 6 pp 567ndash608 2006

[2] D A Bograchev and A D Davydov ldquoTime variation of emfgenerated by the centrifugal force in the rotating electrochem-ical cellrdquo Journal of Electroanalytical Chemistry vol 633 no 2pp 279ndash282 2009

[3] R Aogaki Y Oshikiri and M Miura ldquoMeasurement of thechange in the partial molar volume during electrode reaction bygravity electrode -I Theoretical examinationrdquo Russian Journalof Electrochemistry vol 48 no 6 pp 636ndash642 2012

[4] Y Oshikiri M Miura and R Aogaki ldquoMeasurement of thechange in the partial molar volume during electrode reactionby gravity electrode -II Examination of the accuracy of themeasurementrdquo Russian Journal of Electrochemistry vol 48 no6 pp 643ndash649 2012

[5] R Aogaki ldquoTheory of stable formation of ionic vacancy in aliquid solutionrdquo Electrochemistry vol 76 no 7 pp 458ndash4652008

[6] R Aogaki M Miura and Y Oshikiri ldquoOrigin of nanobubble-formation of stable vacancy in electrolyte solutionrdquo ECS Trans-actions vol 16 no 25 pp 181ndash189 2009

[7] R Aogaki K Motomura A Sugiyama et al ldquoMeasureent ofthe lifetime of Ionic vacancy by the cyclotron-MHD electroderdquoMagnetohydrodynamics vol 48 no 2 pp 289ndash297 2012

[8] A Sugiyama R Morimoto T Osaka I Mogi Y Yamauchiand R Aogaki ldquoLifetime measurement of ionic vacancy inferricyanide-ferrocyanide redox reaction by cyclotron MHDelectroderdquo in Proceedings of the 62nd Annual Meeting Interna-tional Society of Electrochemistry Niigata Japan 2011

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

12 International Journal of Electrochemistry

[9] R Aogkai and R Morimoto ldquoNonequilibrium fluctuations inmicro-MHD effects on electrodepositionrdquo in Heat and MassTransfermdashModeling and Simulation M Hossain Ed pp 189ndash216 InTech Croatia 2011

[10] I Mogi and K Watanabe ldquoMagnetoelectrochemical chiralityin Ag electrodepositionrdquo Journal of Chemistry and ChemicalEngineering vol 4 no 11 pp 16ndash22 2010

[11] I Mogi and K Watanabe ldquoChiral recognition of amino acidsby magnetoelectrodeposited Cu film electrodesrdquo InternationalJournal of Electrochemistry vol 2011 Article ID 239637 6 pages2011

[12] R Aogaki R Morimoto A Sugiyama and M Asanuma ldquoOri-gin of chirality in magnetoelectrodepositionrdquo in Proceedings ofthe 6th International Conference Electromagnetic Processing ofMaterials pp 439ndash442 Dresden Germany 2009

[13] M Sato A Yamada and R Aogaki ldquoElectrochemical reactionin a high gravity field vertical to an electrode surface-analysis ofdiffusion process with a gravity electroderdquo Japanese Journal ofApplied Physics vol 42 no 7 pp 4520ndash4528 2003

[14] S Chandrasekhar Hydrodynamic and Hydromagnetic StabilityDover Publication New York NY USA 1981

[15] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field parallel to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 71 no 3 pp 154ndash162 2003

[16] Y Oshikiri M Sato A Yamada and R Aogaki ldquoElectrochem-ical reaction in a high gravity field vertical to electrode surface-analysis of electron transfer process by gravity electroderdquoElectrochemistry vol 72 no 4 pp 246ndash251 2004

[17] Y Oshikiri M Sato A Yamada and R Aogaki ldquoGravityfield effect on copper-electroless plating -comparison with themagnetic field effectrdquo Japanese Journal of Applied Physics vol43 no 6 pp 3596ndash3604 2004

[18] M Sato Y Oshiklri A Yamada and R Aogaki ldquoMorpho-logical change in multi-layered 2D-dendritic growth of silver-substitution plating under vertical gravity fieldrdquo Electrochem-istry vol 73 no 1 pp 44ndash53 2005

[19] M Sato Y Oshikiri A Yamada and R Aogaki ldquoApplication ofgravity electrode to the analysis of iron-pitting corrosion undervertical gravity fieldrdquo Electrochimica Acta vol 50 no 22 pp4477ndash4486 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Inorganic ChemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Carbohydrate Chemistry

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Physical Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2014

Bioinorganic Chemistry and ApplicationsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

SpectroscopyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Medicinal ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chromatography Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Theoretical ChemistryJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Spectroscopy

Analytical ChemistryInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Quantum Chemistry

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Organic Chemistry International

ElectrochemistryInternational Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CatalystsJournal of