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Physical & Interfacial Electrochemistry 2013

Lecture 2.Electrolyte Solutions :

Ion/solvent interactions: ionic solvation.

Module JS CH3304 MolecularThermodynamics and Kinetics

Ionics: the physical chemistryof ionic solutions.

In previous lecture courses at theFreshman level we presented arather general introduction toelectrochemical systems andconcepts. In this lecture course webegin the study of physicalelectrochemistry in earnest and setthe ball rolling by discussing Ionics,the physical chemistry ofelectrolyte solutions.

We recall that electrolytesolutions are solutions which canconduct electricity. We recallfrom basic general chemistry thatin solutions solutes can exist in anumber of possible forms:Molecular units: solution is non-conducting, normal colligativeproperties exhibited, XRDanalysis indicates discretemolecular units in the solid.Molecular units plus ions: solution is weakly conducting, colligativeproperties indicate slightly more than expected numbers of particles present, XRD analysis indicates discrete molecular units in the solid.

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We now begin the study ofphysical electrochemistry inearnest and consider thesolvation of ions in electrolyticsolutions . In particular in thislecture we shall focus on ion -solvent interactions , anddiscuss the progress that hasbeen made in the understandingof the nature and structure ofsolvated ions . Our mainemphasis will be on aqueoussolutions .

The term electrolyte is of particular importancehere. Now electrolytes may be defined as a class ofcompounds , which, upon dissolution in a polar solvent, dissociate at least partially into ions . If we zero inon the concept of dissociation , or the splittingapart to form ions , and concentrate on the extentto which dissociation takes place, then we candistinguish between two types of electrolytes .These two designations are termed true andpotential . A true electrolyte implies completedissociation into ions, whereas for a potentialelectrolyte only limited dissociation takes place .Thus sodium chloride NaCl is a true electrolytewhereas acetic acid CH3COOH is a potentialelectrolyte . These statements are well known frombasic freshman chemistry. The mechanism of ionformation is different for both of these electrolytetypes.A slightly older designation which is still in current

use are the terms strong and weak electrolytes .Note that solutes giving in solution molecular unitsare termed non-electrolytes, those generating bothmolecular units plus ions, weak electrolytes andthose forming ions only, strong electrolytes.

In recent years many advances have been made both in experimental techniques (spectroscopic and diffraction methods) and in computational simulation protocols (such as molecular dynamics MD, especially as applied to biologically important molecules) which may be used as tools for the study of ionic solvation processes .

NaCl is an ionic compound consisting of Na+

and Cl- ions joined together via electrovalent bonds in a crystal lattice . When solid NaCl is put in contact with a solvent such as water , the solvent molecules rapidly attack the lattice , disrupt and break the ionic bonds and thereby generate hydrated ions Na+(aq) and Cl-(aq) as outlined schematically below. Here we see ion/solvent interactions very much at work.

Dissolution of crystalby action of solvent.

Significant (complete)Dissociation/ionization.Ion/solvent interactionStrong.

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A different situation pertains foracetic acid. The latter is a neutralorganic compound. In this case aspecific reaction between the soluteand solvent, a proton transferreaction, results in the generationof acetate CH3CO2

- and hydroniumions H3O+ ions. A proton istransferred from the organic acid (aproton donor) to a water molecule (aproton acceptor) outlined below. Wehave a Bronsted-Lowry acid/basereaction . Now the latter processonly proceeds to a limited extent .Typically the degree of dissociationis ca. 10-3 . Chemical method: proton transfer reaction.

Degree of dissociationsmall, 0.1 %.

Weak acidKA small.

2 3 ( )HA H O H O aq A

Refer to Bronsted-Lowry Acid/BaseBehaviour discussed in JF CH1101.

3

[ ]A

H O AK

HA

The structure of water , the ubiquitous medium .

Water is of course the most generally used solvent system in electrochemistry . However for some applications non aqueous solvents must be used (e.g. in high density lithium battery systems). As a general rule , an electrochemically useful solvent must be able to dissolve a reasonable amount of an inorganic electrolyte (ca. at least 10-3 mol dm-3) .

We see that when compared with other solvents, liquid water exhibits several distinctive properties . For instance , water has a high relative permittivity (er = 78) , high surface tension (72 mN m-1), high thermal capacity (75 J mol-1K-1) and thermal conductivity (0.6 J s-1m-1K-1). Also , its molar volume is the smallest of all the common solvents and so it will exhibit the highest particle density . A further point to note is that the enthalpy of vaporization is greater that the enthalpy of fusion in contrast to the situation observed for non polar solvents where the latter quantities are of similar magnitudes . The surface tension of water is also considerably larger than other solvents. The same can be said for the molar heat capacity . Finally , considering the magnitude of the molar mass of water , we note that its viscosity is anomalously high. Hence these bulk properties of liquid water suggest that the latter must be quite highly ordered and that considerable molecular association is present . Furthermore we can conclude that intermolecular attractive interactions are stronger in liquid water than in other solvents .

Bockris & Reddy, Modern Electrochemistry2nd edition, Kluwer, New York, 1998, Vol.1 Ionics, Ch.2.

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The structure of water and icedepend on the geometry exhibitedby a single water molecule .Molecular structure determinesmacroscopic properties . We knowfrom Freshman Chemistry that asingle H2O molecule has eightvalence electrons (six from oxygenand one each from the two hydrogenatoms). The demands of individualatomic valencies dictate that theoxygen atom is covalently bonded toeach of the hydrogen atoms andthat two lone pairs of electronsreside on the oxygen atom . Hence inthe language of quantum mechanics ,the oxygen atom is sp3 hybridised .Water exhibits a tetrahedralstereochemistry . The HOH bondangle is 104.5 o . The molecule actsas a dipole and the lp electrons mayenter into hydrogen bonding withhydrogens on neighboring watermolecules . In this way a threedimensional network structure maybe generated .

Methods of structural analysis such asXRD, ND, NMR, IR and Ramanspectroscopy applied to liquid waterhave indicated that under normalconditions ca. 70% of water moleculesexist in "ice floes" (clusters of ca. 50molecules exhibiting a structure similarto that of ice) . The mean lifetime ofsuch networked structures is ca. 10-11 s.

And what of the structure of ice ? The best experimental evidence suggests that the latter material consists of a network of open puckered hexagonal rings joining oxygen atoms together . Each oxygen atom is tetrahedrally surrounded by four other oxygen atoms . Also, in between any two oxygen atoms is located a hydrogen atom which is associated to the adjacent oxygen atoms via hydrogen bonding . Hence each oxygen atom has two hydrogen atoms near it at an estimated distance of ca. 0.175 nm . Now such a network structure of associated water molecules contains interstitial regions located between the tetrahedra , which typically are larger than the dimension of a water molecule . Hence it appears that a free non-associated water molecule can enter the interstitial regions without generating any disruption of the network structure .

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We can therefore say that liquid waterunder most conditions is described in termsof a somewhat broken down, slightlyexpanded form of the ice lattice . In short,liquid water partly retains the tetrahedralbonding and resultant network structurecharacteristic of the crystalline structureof ice . Hence by analogy with the icestructure one has associated network waterand structurally free non associated waterlocated in the interstitial regions in thenetwork .Note that the mean oxygen-oxygen bond distance in ice is 0.276 nm whereas in liquid water it is 0.292 nm . Furthermore, the number of oxygen nearest neighbours in ice is 4 and in water is between 4.4 and 4.6 .

It is important to note that this distinction between network and free water is not a static one . Networks can break down : more free water molecules can be formed . Free water molecules may combine to form clusters ( typically ca. 50 water molecules joined together) . The situation is therefore quite dynamic . Free water can be transformed into network water and vice versa . This picture (icelike labile clusters and monomeric water molecules) was initially proposed some time ago by Frank and Wen .

One expects that solvents such asMeOH and formic acid should alsoexhibit considerable association viahydrogen bonding and that a certaindegree of order should be present .However as noted by Desnoyers andJolicoeur 2 the details of solvent-solvent interactions in polar liquids isgenerally unknown at the presenttime , water being the only exception. Solvent-solvent interactions canand do play an important part in ionicsolvation . However their effect isvery difficult to establish in non-aqueous solutions . In this course wewill concentrate mainly on water ,since the latter has been wellcharacterised experimentally and isbest understood .Solvent / solvent interactions can

involve dipole / dipole , dipole /induced dipole , induced dipole /induced dipole (dispersion) anddipole/ quadrupole interactions .

How many water molecules must be joined togetherbefore the bulk properties of water are observed ?Put another way , we ask how many water moleculesdoes it take before the cluster shows theproperties of wetness or before the dielectricproperties approach that of bulk water ? This is nota philosophical question : the answer leads one tothe forefront of research . A single water moleculewill not suffice : this much we know . A recenttextoook author has quoted a value of ca. 6molecules . This is clearly far too low .

Simonson has recently calculated usingMolecular Dynamics (MD) techniques a value for thestatic dielectric constant of water using a simplemodel system consisting of a microscopic droplet ofwater in a vacuum . The idea behind this approach isas follows .

Almost all computational studies to date have been based on simulations of bulk water either using so called periodic boundary conditions (more about these later on) or by combining a finite spherical microscopic region with an outer bulk region modelled as a dielectric continuum. Such computations are difficult, time consuming and hence very expensive to do . Lots of long range interactions have to be taken into account and this causes convergence problems when the computation is done . Optimal computational strategies are still being pursued .

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On the other hand using a modelsuch as a spherical water dropletimbedded in a vacuum , one candispense with long rangeinteractions and one can readilycompute all the interactionswithin the droplet whose size canbe chosen . Simonson has shownthat very good agreement (er inthe range 77 - 87) with theexperimental bulk dielectricconstant of water is obtained fora droplet of 1963 TIP3P watermolecules (sphere 2.4 nm radius)at T = 292 K after a simulationtime of 1000 ps . Hence from thiswork it is clear that a cluster ofsome 2000 water molecules canmimic the behavior of the bulkliquid .

.

We now move on and consider somesimple approaches used for quantifyingwhat happens when an ion enters fromthe gas phase into a polar solvent such aswater and becomes solvated (orhydrated). In particular we shall considerthe energetics of the interactionbetween an ion and the solvent .

We shall initially adopt a verysimple first order approach and assumethat the solvent medium does not have amolecular structure , but instead can beregarded as a dielectric continuum . Inthis approach we assume that theinteractions between the ion and thesolvent are electrostatic in origin and weview the ion as a charged sphere ofradius r and the solvent as a dielectriccontinuum of dielectric constant er

The solvent is the medium in which the solute exists. It is often calledA dielectric. A dielectric can be thought of in terms of an insulator, which isA substance which stops or tends to stop the flow of charge., in other words to stop a current passing through it.If a substance which acts as an insulator is placed between two charges itReduces

• the field strength• the force acting between the charges• the electrostatic potential energy of interaction between the charges

And the factor by which it reduces these quantities is called the relative permittivity r.It is important to note that this definition of relative permittivity is independent of any assumption as to what the dielectric is made of. In particular it is independent of any assumption that the dielectric is composed of atoms and molecules, and so requires no discussion of the medium at a microscopic level.In effect the dielectric constant is just a proportionality constant characteristic of the medium.

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Solvation: Solute-solvent interactions.

Introduction.

We focus attention on chargedsolutes (ions) in polar solvents.Much work has been done in thisarea of Physical Electrochemistryboth from a theoretical and anexperimental viewpoint. The topic isalso of considerable importance inbiophysical chemistry when thesolvation of macromolecular proteinmolecules is addressed.Firstly with respect to thetheoretical analysis we considertwo distinct approaches tosolute/solvent interactions.

Non-structural treatment.This is based on viewing thesolvent as a structurelessdielectric continuum into whichthe ion is embedded as a chargedsphere. The methodology ofclassical electrostatics is used inthe detailed analysis of theenergetics of the problem.The energetics of salvation isexpressed in terms ofthermodynamic cycles and thepertinent thermodynamicquantities such as GS, HS andSS are determined in thecontext of simple electrostaticmodels.

The non-structural approach isbased on the original modeldeveloped by Born (1920). Thistype of analysis is veryapproximate but is still being usedwith some success.Good agreement between theoryand experiment is obtained via theintroduction of empirical fittingparameters which serve toquantify the ionic radius in thesolvent.The non-structural Born approachhas been generalised in recentyears to examine the salvationenergetics of macromolecules, andis so a topic of considerablecurrent interest in biophysicalchemistry.

Structural treatment.This defines the modern approach.Here we view the solvent as amolecular entity with a well definedstructure (as indeed it has). We needa model of the solvent in theimmediate vicinity of the ion. Againthe solvation energetics arequantified in terms of thethermodynamic entities GS, HS andSS. The theoretical analysis is againbased on electrostatic models butthese are more elaborate and includeion/dipole and ion/quadrupoleinteractions. The structuraltreatments based on the early workof Bernal and Fowler (1933). Theresults obtained are encouraging buta fully satisfactory theoretical modelhas yet to be developed.

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In contrast the experimentalanalysis of solvation is quite welldeveloped.Structural information on theenvironment of a solvated ion isobtained from:diffraction methods: X-ray,neutron and electron diffraction,EXAFS etc.spectroscopic studies : uv/vis, ir, nmr, raman spectroscopies etc.Diffraction methods provide directinformation about the environmentsurrounding the ion in solution,whereas indirect structuralinformation is provided fromspectroscopic methods.

One difficulty is thatdilute solutions need to be examinedin order to focus in on ion/solventinteractions. If more concentratedsolutions are used then ion/ioninteractions come into play and cloudthe issue.

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In this section we discuss an approach developed by Born which can be used to model the energetics of ion/solvent interactions in a convenient manner . This approach may have its origins over sixty years ago and is certainly limited, but it can , and has yielded a large number of qualitative and semi-quantitative insights . Now this approach is limited and has had its detractors . However this continuum approach represents a simple, easily interpretable, and computationally inexpensive physical model that does not involve many adjustable parameters . The major problem with this approach is that the concept of a macroscopic dielectric response is applied to a molecular system. Max Born was one of the founding

fathers of modern quantum mechanics having proposed the probability interpretation of the wavefunction for which he ultimately obtained a belated Nobel Prize .

In the Born approach a blend ofclassical electrostatics andthermodynamic cycles is adopted . Theobject of the exercise is to evaluate ,from first principles, the Gibbsenergy change GS associated withthe process of ionic solvation , whereone considers the transfer of an ionfrom a gaseous vapour at low partialpressure to the desired solvent (i.e.water) in the absence of any ion/ioninteractions . In simple terms thework of transfer of an ion fromvacuum to a solvent is equated withthe Gibbs energy of solvation . Hencethe formidable task of quantifying theenergetics of ionic solvation in astructured solvent is reduced usingthis approximate approach toanswering the question : what is thework done in transferring a chargedsphere from vacuum into a dielectriccontinuum ?

Energetics of ionic solvation : Non-structural Treatment.Born Model (1920)

We consider the following situation. Initial state: no ion/solvent interactions. This corresponds to ion in vacuum.Final State: ion/solvent interactions operative.This corresponds to ion in solution.Thermodynamics states that for reversibleProcess taking place at const. T & P, Free Energy change G equals net work (other than PV work) done on system.We need to calculate the work done inTransferring an ion from vacuum into a solvent.

Vacuum

Solvent

Free energy changeGS

Ion

A solvent such as water has a complex structure.To simplify our analysis we dispense with structure and regard the solvent as a structurless dielectric continuum. This medium has a dielectric constant given by S The ion also has structure (ion + associated hydration sheath). We represent the ion as a charged sphere of net charge qi = zie and radius ri.

Charged sphere

nucleus

Electroncloud

i iq z e

irirNet charge

Structured solvent StructurlessDielectriccontinuum

Dielectric constant S

Gibbs energy of solvation computed viaa thermodynamic cycle.

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Born Model : Thermodynamic cycle to calculate the work done in transferring a charged sphere from vacuum into the solvent.

Main assumption:Only charge on ion isResponsible for ion/solventinteraction.Hence ion/solventinteractions are solelyelectrostatic in nature.

Remove charge

Charging process

SG

Work of discharging

Work of charging

Charged sphere in vacuum

Uncharged sphere in vacuum

Uncharged sphere in solvent

Charged sphere in solvent

Zero electrostatic work of transfer

Solvent

Vacuum

WD

WC

WT = 0

The following thermodynamic cycle isdeveloped. The ion i , represented as acharged sphere of radius R,isconsidered in to be initially located in avacuum, and the work Wd required tostrip the ion of its charge q = zie isdetermined . Then this unchargedsphere is transferred into the solventdielectric medium . We assume thatthis transfer process involves no work .Then the charge on the sphere insidethe solvent is restored to its full valueq = zie and the work done in chargingWc is determined . Finally, the ion istransferred from the solvent back intovacuum . The work done in thistransfer process is - Gs .

dcs

sctrd

WWG

GWWW

0

In a thermodynamic cycle the algebraicSum of all the work terms is zero.

We now evaluate the work done in charging a sphere of radius R from q = 0 to q = zie , where zi denotes the valence of ion i and e is the fundamental electronic charge . To do this we start off with an uncharged sphere and add infinitesimal amounts dq to the sphere until the final charge level is attained .The product of electrostatic potential and the charge increment dq gives the work increment dw.

dw dq

The total work done in charging the sphere is obtained by summing over all the little increments dw.

ezez

c

ii

dqdwW00

From fundamental electrostatics we recall that the electrostatic potential at a certain point in space is defined as the work done to transport a unit positive charge from infinity to that specific point . Now the electrostatic potential at a point located a distance r from a charged sphere (r) is given by the product of the electric field strength E(r) (which defines the electric force per unit charge) acting on the moving test charge and the distance r through which the charge is carried .

R

qdr

r

qdrrEr

RR

02

0 44

1

Direction ofIncreasingpotential

Direction of Electric fieldDirection of

transport of charge

PositivelyCharged sphere Unit positive

Test charge

R

ez

dqR

qdqRrW

i

ezez

c

ii

24

1

4

1

2

0

000

0 = permittivity of vacuum = 8.854 x 10-12 C2N-1m-2

Total chargingwork Wc

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Work done in discharging sphereis negative of charging work, Wd = - Wc

R

ezW i

d 24

1 2

0

The recharging work done when the sphere is present in the solvent dielectric is modified by the macroscopic dielectric constant of the medium r .

R

ezW

r

ic 24

1 2

0

r

idielc

vacds R

ezWWG

1

124

1 2

0

Born expression for Gibbs energy of Solvation.

2 2

0 0

1 11 1

4 2 4 2iA A

sr r

z eN N qG

R R

Molar GibbsEnergy ofsolvation

Note that in the expression for Gs there is anegative sign heading the rhs . Now since thedielectric constant r > 1 always and so 1/r < 1 thenGs will always be negative . The solvation proceedsspontaneously : ions prefer to be solvated than to bein vacuum . Hence solvation is thermodynamicallyfavourable. The Born equation also predicts that thesmaller the ion (smaller R) and the larger thedielectric constant , the greater will be the solvationfree energy .

TR

ezN

T

GS r

r

iA

P

ss

2

22

0

1

24

T

T

R

ezNH r

rr

iAs

2

22

0

11

24

The Entropy and enthalpy of solvation are readily evaluated. Note that HS is a quantity readily compared with experiment.

sss STGH

A clear prediction is that thesolvation free energy shouldvary inversely with ion size R

We note that the Born modelmakes predictions whichsuggest that the ion solventinteraction is much strongerthan it actually is . We readilysee this if we compareexperimentally determined Hsvalues for the alkali metal ionsand the halide ions with thatevaluated via the Bornexpression . If one plots Hsversus the inverse of thePauling crystal radius we notethat the linear relationshipbetween Hs and 1/R predictedby the Born expression is notobserved . The experimentaldata points are all much lowerthan the theoretical Bornequation line . In general theuse of ionic radii leads to anoverestimate of the solvationenergies of anions and inparticular the calculatedsolvation energy of cations canbe almost 100 kcal mol-1greater than experimentalvalues . In any event it is quiteremarkable that such a crudeelectrostatic model works atall ! In fact what is measured isthe enthalpy of solvation of asalt not that of an individualion .

(R/)-1

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

-H

S

0

50

100

150

200

250

300

Born TheoryBorn TheoryRegression lineExperiment

Li+Na+K+,F-

Rb+Cs+

Cl-Br-I-

Born Model : Comparing theory and Experiment

Born theory suggests that ion/solvent interactions are much stronger than experiment shows them to be.Need to use effective dielectric constant of solvent region near ion rather than bulk value (ca. 80 for water).

P.W. Atkins, A.J. MacDermott, J. Chem. Ed.59 (1982) 359-360

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Continuum Solvation Models: Recent Developments.

Rashin & Honig Model : J. Phys. Chem., 89 (1985) 5588-5593.In original Born model R was chosen as the ionic radius derived from crystallographic measurements. This leads to quiteconsiderable deviations between experiment and theory. Born Equation estimates (using crystallographic ion radii) of the enthalpy of solvation are considerably higher than values derived from experimental measurements. For example: Li+ :R = 0.60Å, HS,calc = - 277.7 kcal/mol, HS,expt = - 146.3 kcal/mol.Na+: R = 0.95 Å. HS,calc = - 175.5 kcal/mol, HS,expt = - 118.9 kcal/mol.

Rashin & Honig suggested that R should be replaced by the parameter RC termed the cavity radius. Transferring the uncharged ion intothe solvent produces a cavity. The size of this cavity should be used in calculations involving the Born equation. The cavity radius is defined as the distance from the nucleus at which the electron density of the surrounding medium becomes significant.

The cavity radius is a rather arbitary concept, but it has beensuggested that the covalent radii of cations and the ionic radiiof anions provide a useful first approximation to the cavity radius.

B. Honig, K. Sharp, A-S Yang, J. Phys. Chem., 97 (1993) 1101-1109

ric

iAs R

ezNG

1

124

22

0

Predicted and experimental Hs values are in good agreement, with the error typically being in the region of 7 % .

The agreement between theory and experimentis quite good, the calculated values areconsistently larger than the experimental valuesby some 10 - 15 kcal mol-1.

2 2

20

11

4 2iA r

sIC r r

z eN TH

R T

Rashin & Honig Model : J. Phys. Chem., 89 (1985) 5588-5593.

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Bontha & Pintauro Model. J. Phys. Chem., 96 (1992) 7778-7782.Bontha & Pintauro developed a modifiedBorn thermodynamic cycle . It waspresumed that the total free energychange associated with the transfer ofan ion from vacuum to a polarizablesolvent was equal to the sum of the workterms associated with the followingprocesses :• discharge of the ion in vacuum (Wd) ,• transfer of the neutral species fromvacuum into the solvert (Wtr) ,• re-charging the species in the solvent(Wc),• restoring the solvent to its originalstate (WR)• transferring the charged ion back intovacuum .

The WR term is an important addition tothe Born approach . During therecharging of an ion in a polarizablesolvent it is to be expected that thesolvent dipoles are orientated in theelectric field generated by the ion . Thiswill result in a decrease in the potentialenergy of the solvent . Hence, when theBorn cycle is completed an amount ofwork WR must be added to the solvent inorder to restore the rotational andtranslational motion of the solventmolecules i.e. the aligned solvent dipolesmust undergo relaxation .

Discharge Process

Recharge process

SG

Discharge work

Work of charging

Charged sphere in vacuum

Uncharged sphere in vacuum

Uncharged sphere in solvent

Charged sphere in solvent

Solvent

Vacuum

WD

WC

WTTransfer Process

Transfer work

Dipole relaxationprocess

Work to restoreSolvent toOriginal state

WR

Work obtainedfrom solvent dipolerealignment

The net free energy of solvation Gs is given by

s D T c RG W W W W

WD= work done in discharging ion in vacuumWT = work done in transferring neutral species to solventWC = work done in charging neutral species in solventWR = work done in restoring solvent to original state (relaxation of aligned solvent dipoles).It was also assumed that the solventwas a polarizable dielectric medium, which implies that the dielectric constant varies with electric field strength and that dielectric saturation effects must be considered.Hence the dielectric constant near an ion will be much less than the dielectric constant of the bulk solvent

All four work terms can be calculated using Classical electrostatics.Detailed analysis can show that

2

0

0 0

20 0

0

20

8

r

r

r

r

D

T

D

C

V

rE

r r

V V

r

S r

V

qW

a

W ma c

W dV E dD

dV E dE dV E d

W dV E d

0

charge

electric field vector, , electrostatic potential

electric displacement vector

dielectric constant is function of electric field strength.

r

r r

q

E E

D E

E

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The total Gibbs energy of solvation isOutlined across.

2

00 08

E

s T r

V

qG W dV r E dE

a

In order to evaluate the double integral, the interrelationship between the radially dependent electric field and solvent permittivity surrounding an ion must be determined over the entire ion charging process.This is done by simultaneously solving the modified Laplace equation of the electrostatic potential in a polarizabledielectric medium and the Booth equation which describes the variation in the solvent permittivity with electric field strength.These equations (outlined across) are quite complicated and must be solvednumerically using a finite difference technique on a computer.

2

2

2

0

3 1coth

52

2

r

r

r

B

r r

nr n r

r r

nk T

optical refractive index of solvent

= dipole moment of solvent molecule

k Boltzmann constant

r = radial co-ordinate measured from centre of ionB

n

0

0 r r

r

r

dr a

dr r a

= surface charge density C m-2

Boundary Conditions

Bontha & Pintauro Model. J. Phys. Chem., 96 (1992) 7778-7782.

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Firstly, the variation of local dielectric constant with distance forvarious extents of surface charging in the range 40 to 100 % isoutlined. We note that the distance from the ion surface overwhich dielectric saturation is observed gets larger the greater theextent of surface charging. The computed variation in the dielectricconstant of water (at 298 K) as a function of radial distance fromthe ion surface for two univalent ions with different radii (Li+ andCs+) and for cations of similar radius and different charge (Na+,Ca2+ and Y3+) are presented. We also show the computed variationexpected using the one layer dielectric model of Abraham and Lizziin these figures . The Abraham-Lizzi model predicts that thedielectric constant remains at its bulk value until a certain criticaldistance. After that, the value drops to a very low level . Hence theAbraham-Lizzi model does not differentiate between ions ofdifferent charges and size . It does approximate well the computedvariation in dielectric constant with radial distance for univalentions of large radius , but only poorly approximates the variationexpected for small univalent and for all divalent and trivalent ions .

Bontha & Pintauro Model. J. Phys. Chem., 96 (1992) 7778-7782.

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In table above we show the computed and experimental ionic solvationGibbs energies calculated using the Bontha - Pintauro model for a rangeof cations and anions in water at 298 K . Goldschmidt values of the ionicradii were adopted in the calculation . Agreement between theory andexperiment is quite good .

The analysis can also be extended to non aqueous solvents . In table above we show some results for three solvents : methanol, 1,1-dichloroethane and acetonitrile . Again excellent agreement is obtained between theory and experiment . The model predictions are typically within 10 % of the experimental solvation energy values .

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Results from theory agree quiteWell with experiment. TypicallyError is in region of 2=6%. This is agood result given nature of assumptionsmade (solvent is dielectric continuum).

What are the advantages of the Bontha-Pintauro model ? The model contains noadjustable parameters , and specificallyconsiders the effect of dielectric saturation inthe vicinity of the solvated ion . Agreementbetween theory prediction and experimentalmeasurements is quite good, typically within 10%. The Bontha-Pintauro approach is the mostrecent elaboration on the original Born modeldeveloped over 70 years ago , and may be usedas a convenient computational protocol toevaluate Gibbs energies of solvation . Howeverthe model is still firmly based on continuumelectrostatics and does not view the solvent asa structured entity. This is a majordisadvantage , of this and all other Born typecontinuum electrostatic approaches .

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Some elaborations on the structural theory of solvation.

The theoretical models discussedin the previous sections haveneglected the important fact thatwater has a very well definedstructure , and that this structurewill have a very large part to playin the solvation of a particular ion .Details of water structure havebeen presented in a previousslide.We recall that in liquid waterthere are networks of associated(via hydrogen bonding interactions)water molecules and also a certainfraction of free unassociatedwater molecules . We now proceedon and ask the question : whathappens to the structure of waterwhen ions enter the solvent ?

At this point we introduce a centrallyimportant idea . We saw that watermolecules can be represented as dipoles. Hence one might expect that theintruding ions will interact with thedipolar hosts . This interaction is notgentle and causes major changes in thestructure of water . The sphericallysymmetric electric field of the ion cantear water dipoles out of the waterlattice and make them point or orientwith the appropriate charged endtowards the central ion .

Consequently, we arrive at apicture of ion / dipole forces asproviding the principal basis ofionic solvation . Because of theoperation of these ion/dipoleforces, we can imagine that acertain number of watermolecules in the immediatevicinity of the ion may be trappedand orientated in the ionic field .We arrive at a picture of ionsenveloped by a solvent sheath oforientated, immobilised watermolecules. These immobilisedwater molecules move with theion as it moves through thesolvent medium . The fate ofthese immobilised solventmolecules is tied up with that oftheir partner ion . The ion and itssolvent sheath form a singlekinetic entity .

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What of the situation far away fromthe ion ? At a large enough distancefrom the ion, the electric field isnegligible and the solvent structurewill be unaffected by the ionic field .Here the bulk water structure willpertain . The solvent molecules will notrealise that the ion is there at all .

In the region between the solvationsheath (where the electric field of theion determines the water orientation)and the bulk water (where theorientation of the water molecules isuninfluenced by the electric field of theion) , the orientating influences of theion and the water network operate in amicroscopic tug of war . The formertries to align the water dipoles parallelto the spherically symmetric electricfield of the ion, whereas the waternetwork tries to make the watermolecules in this "in between" regioncontinue adopting the tetrahedralbonding arrangement required formembership of the network structure .Hence caught between the two types ofinfluences, the water in this in betweenregion seeks to adopt a compromisestructure that is neither completelyorientated nor disorientated . In thisintermediate region the water structureis said to be partially broken down asoutlined across.

Let us therefore summarise thesituation . We can describe threeregions near an ion : (i) the primaryor structure enhanced regionlocated next to the ion where thewater molecules are immobilised andorientated by the electric field ofthe ion, (ii) the secondary orstructure broken region where thenormal bulk structure of water isbroken down to various degrees , and(iii) the bulk solvent region wherethe water structure is unaffected bythe ion and the tetrahedrally bondednetwork characteristic of bulk wateris exhibited . These regions areillustrated in the previous slide.It should be realised that theseregions differ in their degree ofsharpness . The primary solvationsheath is reasonable well defined .

We have noted that both the central ion and its immobilised layer of water molecules form a single kinetic entity . The number of water molecules associated with the ion during its translational motion is termed the primary solvation (hydration) number nh. We must distinguish the solvation number from the coordination number . The latter defines the number of solvent molecules in contact with the ion and is determined by geometric considerations .

It is important to note that solvation numbers determined via different experimental techniques can give different answers due to the ambiguity inherent in the definition of the solvation number concept . In contrast, the in between region of secondary solvation is not as well defined . The regions of primary and secondary solvation define the region over which ion-solvent interactions operate .

20

We now examine a simple structural model invoking ion-dipole interactions which may be used to provide a quantitative estimate of the energetics of ionic solvation .

We now present an account of a very simple simple model of ionic solvation in which the structural details (although described at a very basic level) of the solvent is considered . This account will again dwell on the energetics of solvation and is based on the work reported many years ago by Bernal and Fowler , Eley and Evans , and Frank and Evans . Again we resort to a simple thoughtexperiment (which is a favouritepursuit of physical electrochemists)to help clarify the situation . Theessentials are presented in across andfollow the essential idea of Bernal andFowler.

We set the ball rolling by assuming that the primary solvation shell occupies a certain volume corresponding to n primary solvent molecules plus one more to make room for the bare ion . As a consequence of this supposition a volume corresponding to n+1 solvent molecules must be made available in the solvent for the immersion of a primary solvated ion . We therefore remove n+1 solvent molecules and transfer them to vacuum . A cavityis left in the solvent as a result of this process . We let WCF denote the work done in cavity formation . We assume of course that the size of an un-solvated ion is the same as that of a solvent molecule . This approximation is reasonable for some ions .

Bernal Fowler Model

21

22

Now before the n+1 solvent moleculescan orient around the ion in vacuum , thewater cluster must firstly dissociateinto n+1 separate molecules . We set WDas representing the work done in thisdissociation process .

Ion-dipole bonds are thenmade between n solvent molecules andthe target ion . A primary solvationsheath is formed . We must determinethe work WID done in this process . Inshort the interaction energy between anion and a dipole must be evaluated .

Then, the ion along with its primary solvation sheath is transferred from vacuum into the cavity within the solvent . The work done in this case is equal to the Born enthalpy of solvation WB . We note that the radius of the solvated ion R + 2rs is used in the Born expression previously derived .

The introduction of the primarysolvated ion into the cavity causesdisturbance of the solvent structurein the immediate neighbourhood ofthe solvated ion . As a result of thisa structure breaking work term WSBmust also be included .

Finally we may ask is thestory complete . Has everythingbeen accounted for ? The answermust be no . What about our extrasolvent molecule left in vacuum ?This species must finally betransferred from vacuum into thesolvent . The work done in thisprocess is Wc , the work ofcondensation .

23

Now this thought experiment or thermodynamic cycle may appear to be quite complicated , but it really is the most simple one that we may devise . More complicated situations may be enumerated as done by Bockris and Saluja . Implicit in our simple model is that the coordination number and solvation number are the same . Also we proposed that only the ion- dipole interaction need be considered . Both of these restrictive assumptions may be removed and more elaborate models developed . We have also not really considered the region of secondary solvation (the second "in between" layer in any specific detail , we merely specify some work WSB . Again this limitation may be dispensed with in a more detailed analysis .

24

We set the ball rolling by assuming that the primary solvation shell occupies a certain volume corresponding to n primary solvent molecules plus one more to make room for the bare ion . As a consequence of this supposition a volume corresponding to n+1 solvent molecules must be made available in the solvent for the immersion of a primary solvated ion . We therefore remove n+1 solvent molecules and transfer them to vacuum . A cavity is left in the solvent as a result of this process . We let WCF denote the work done in cavity formation . We assume of course that the size of an un-solvated ion is the same as that of a solvent molecule . This approximation is reasonable for some ions .

We can calculate the enthalpy ofsolvation by adding together all of thecomponent work terms enumeratedpreviously .

BID

BIDCSBDCF

CSBBIDDCFs

WWW

WWWWWW

WWWWWWH

Here we have differentiated between theelectrostatic terms WID and WB and theother work terms which have all beenlumped together into a composite workterm W . We now must evaluate all ofthese work terms . The electrostatic workterms WB and WID are readily evaluated .The former quantity is obtained directlyfrom the Born theory.

p

r

rrs

iAB T

T

rR

ezNW

20

2 11

22

1

4

Here R denotes the unsolvated ionic radius andrs denotes the radius of a water molecule .Hence the quantity R + 2rs represents theradius of a primary solvated ion . In derivingthis equation we again consider a Born typetransfer process but in this case we are movinga primary solvated ion from vacuum into a cavitywithin the solvent . We also use the bulkdielectric constant of water in this expressionsince we assume for the sake of simplicity thatthe water structure outside the cavity isnormal and undisturbed .

In the R denotes the unsolvatedionic radius and rs denotes theradius of a water molecule . Hencethe quantity R + 2rs represents theradius of a primary solvated ion .In deriving this expression weagain consider a Born typetransfer process but in this casewe are moving a primary solvatedion from vacuum into a cavitywithin the solvent . We also usethe bulk dielectric constant ofwater in this expression since weassume for the sake of simplicitythat the water structure outsidethe cavity is normal andundisturbed .

The other electrostatic term is concerned with the interaction between the ion and the n free solvent dipoles in vacuum to form the primary solvated ion in vacuum . This is a standard problem in electrostatics and the mathematical details are not difficult to follow . Our problem is to calculate the interaction energy between a dipole and an ion placed at a distance r from the dipole center We assume that the dipole is orientated at an angle to the line joining the centers of the ion and the dipole . We assume that the dipole is of length 2L .

r

2L

Ion-dipole work.

25

L

L

rr1

r2

y

x

x + L-

++

P

Now the ion-dipole interaction energyUID is equal to the product of theionic charge qi = zie and theelectrostatic potential y(r) due to thedipole . We recall from basicelectrostatic theory that thepotential due to an assembly ofcharges is given by the sum of thepotentials due to each chargeconstituting the assembly . Since thedipole consists of the charges + q and- q which are located at distances r1and r2 from the point P defining theposition of the ion (see figure onprevious slide) , hence the totalelectrostatic potential is given by :

210

11

4 rr

qr

From basic geometrical considerationswe note that :

xLLyxyLxr

yLxr

2222221

2221

Also we note that

cosand222 rxryx

(1)

(2)

. Hence eqn.2 reduces to

cos2

1cos22

221 r

L

r

LrrLLrr

Inverting this expression we obtain

(3)

2/12

1

cos2

111

r

L

r

L

rr(4)

We now make the point dipole approximationand assume that P is located very far awayfrom the dipole and so r >> 2L . We can thenexpand the rhs of eqn.4 using the Binomialtheorem to obtain :

.....cos2

16

15

cos2

8

3cos

2

2

11

3

2

2

2

2

2

2

2

1

r

L

r

L

r

L

r

L

r

L

r

L

r

r

(5)

26

If we now neglect terms of order higher thanL2/r2 we obtain :

1cos32

cos11 2

3

2

21

r

L

r

L

rr

In a similar manner we may show that :

1cos32

cos1

cos2

111

23

2

2

2/12

2

r

L

r

L

r

r

L

r

L

rr

(6)

(7)

If we substitute eqn.6 and eqn.7 into eqn.1 weobtain the following expression for theelectrostatic potential at the ion due to thepresence of the dipole :

cos4

cos4

2,

20

20 r

p

r

Lqr (8)

In the latter p = 2Lq denotes the dipolemoment . We can replace the distance rby the sum R + rs to obtain thefollowing expression.

cos)(4 2

0 srR

p

(9)

We note from eqn.9 that the potential due to a dipole falls off approximately as the square of the distance r, whereas the potential due to a single point charge varies only as r-1 . This observation is readily explained . The difference in the behavior of the potential at large distances arises from the fact that the charges in a dipole appear close together for an observer located at some distance away from the dipole and their fields cancel more and more as the distance r increases . We also note that the electrostatic potential also depends on the orientation angle .

It is clear for maximum interaction we set = and so cos = 1 and eqn. 9 reduces to

204 s

i

rR

epz

(10)

We assume that n solvent molecules surround the ion in the primary hydration sheath . Hence per mole of ions the ion-dipole work term is given by :

204 s

iAID

rR

epnzNW

(11)

This result may also be obtained in amathematically more rigorous mannerwhich we will not describe here butinvolves Legendre Polynomials which areonly suitable for real black beltelectrochemists at the Graduate level!!!

We can also look at more complex electrostatic interactions. For example we can look at the ion/waterQuadrupole interaction. The latter may be calculated using the methods of electrostatics and the following expression may be derived.

2 3

0 04 2 4A i A i

IQ

i s i s

N nz e N nz epW

r r r r

In latter expression represents theDipole moment and p is the quadrupolemoment and n is the coordination number.For water. The positive sign in second term onrhs corresponds to cations andthe negative sign to anions. Hence in a more Sophisticated nalaysis we should replaceWID by WIQ.

27

We also may have to account for ion/induced dipole interaction work WIID .This term arises from the fact that whenThe water molecule is in contact with the ion, the electric field of the ion tends todistort the charge distribution (quantified in terms of deformation polarizability in the water molecule and so induces a dipolemoment on the water molecule.

2

2 4

04 2iA

IID

i s

n z eNW

r r

Ion-induced Dipole work