activity and osmotic coefficients from the emf of liquid membrane cells. vi-znso4, mgso4, caso4, and...

Joumal of Solurion Chemistry, Voi. 26, No. 8, 1997

Activity and Osmotic Coefticients from the EMF of Liquid Membrane Cells. VI-ZnS04, MgS04, CaS04, and SrS04 in Water at 25°C

Francesco Malatesta* and Roberto Zamboni

Received March 12, /997; revised July 2, /997

The activity coefficients of ZnSO4, MgSO4 , CaSO4, and SrS04 are measured by means of cells with ion-exchange liquid membranes similar lo those described in the previous papers of this series. Negative deviations from the limiting law are observed in the dilute region. These deviations are, for ZnSO., appreciably more important than recent literature has indicated, and the corresponding activity coefficients need to be corrected by about 12%. Pitzer's theory best-fit coefficients have accordingly been recalculated. The osmotic coefficients are also derived. Accessory infonnation on the hydration state for zinc, magnesium, and sulfate ions, is presented.

KEY WORDS: Activity coefficients; electrolytes: ion hydration: liquid membranes: membrane electrodes: osmotic coefficients; calcium sulfate; magnesium sulfate; strontium sulfate; zinc sulfate.

1. INTRODUCTION

Since the advent of the Debye-Htickel theory (DHT), 2-2 salts have represented a stimulating puzzle far people concemed with electrolytes. These salts, in fact, in the dilute region deviate from the corresponding limiting laws (DHLL) in the directions apposite to those predicted by DHT, i.e., downwards for activity and osmotic coefficients and upwards far partial and apparent molar volumes and enthalpies. Ion pairing was a possible reason far these effects, but La Mer et al. offered convincing proof that the negative deviations they found for CdSO4, O l and even more so for ZnSO4, (2) were in

Dipartimento di Chimica e Chimica Industriale dcli' Università di Pisa, Via del Risorgimento 35, 56126 Pisa, ltaly.

791

()(J95-9782/97/0800-0791Sl2.50/0 () 1997 Plenum Publi,hing Corporalion

792 Malatesta and Zamboni

excellent agreement with the Gronwall-La Mer-Sandved theory.<3> The latter

differs from DHT solely because it retains further tenns, beyond the Iinear one, in the series expansion of the Poisson-Boltzmann equation. Thus, the electrostatic cloudlike interactions were sufficient to explain the inverse deviations, regardless of any ion-association . This finding led researchers to examine other thennodynamic properties, high-charge electrolytes, and ìmproved theoretical treatments carefully. It is now clear that the inverse deviations in the dilute region of 1-4, 2-2, 2-3, 2-4, etc. electrolytes are the regular behavior to be expected as the result of the cumulative interactions of every ion with ali other ions up to Iarge distances, independently of possible, additional, ion-association phenomena.

Meanwhile, however, it happened that Pitzer,<4> and Malatesta and

Rotunno<5> from independent considerations, reached the conclusion that some of the data from Cowperthwaite and La Mer that outlined the impressive negative deviations for ZnSO4,< 2

> were unreliable. Pitzer's new estimates did not reveal any important negative deviations for ZnSO4• <4,

6> Paradoxically,

therefore, the inverse deviations of the high-charge electrolytes were real and theoretically predictable, but the data that most contributed support to the relevant scudies were possibly wrong in origin.

Accurate measurements of activity coefficients are now feasible down to high dilution levels using the liquid membrane cells developed in our laboratory.nl We decided, therefore, to restudy ZnSO4• In addition, the activity coefficients of the alkaline-earth sulfates MgSO4 , CaSO4 , and SrSO4, were examined. The latter are significant exemples of 2-2 salts, free from the hydrolysis problems that may affect the salts of transition metals. Though sparingly soluble, calcium sulfate (ca. 2 g-dm-3) and even strontium sulfate (ca. O. I g-dm-3) still lie inside the application range of the method. As far as we know, no previous thennodynamic infonnation of any sort has been made available for strontium sulfate solutions.

1.1. The Osmotic Coefticients of 2-2 Electrolyes

Osmotic coefficient (<!>) values derived from isopiestic and cryoscopic measurements are generally available for 2-2 electrolytes.<4> At dilution levels around 0.005--0.002m these data usually become imprecise<8l but the Iimitation is not criticai, since <f> (unlike the activity coefficient) is obtained directly as an absolute quantity and does not need extrapolation to infinite dilution. The activity coefficients are then calculated via the Gibbs-Duhem equation. A convenient way consists of detennining the best-fit paramelers of the Pitzer theory<9l from the values of <f> and introducing these parameters into the corresponding equation for ''fa.• As Pitzer and Mayorga have already determined these parameters,<6> it might seem unnecessary to study these salts

EMF of Llquld Membrane Cells 793

again. However, a risk is involved in the procedure which calculates the activity coefficients starting from the osmotic coefficients. Regardless of the use of Pitzer's equation or of any other method, the Gibbs-Duhem equation relates <t> and -y ± through a relation of the kinct< 1 o, 11 l

(J; In -Y± = (cp - I) + 2 Jo (cp - 1)[1 dx (l)

where x is the current value of Jm. In the unexplored region beyond the lowest experimental concentration down to zero, the values of ( <t> - I )x- 1

need a sort of Jong-shot interpolation, 1 or quasiextrapolation (when utilizing the Pitzer equation, one tends no longer to perceive this problem, which, however, still subsists implicitly). The long-shot interpolation does not present any problem for a Jow-charge electrolyte, which approaches the limiting law with the monotone trend of the Debye-Hiickel theory. On the contrary, whcn high-charge electrolytes are involved and inverse deviations occur, the function 4> - l crosses the trace of the limiting law in the dilute region, then reverts, to approach the limiting slope from the wrong (in tenns of DHT) side. Between the breakdown limit m1;m of the ice-point method and the extreme dilution leve! where the slope becomes indistinguishable from DHLL, no accurate prediction of ( <I> - l )[ 1 is possible, and this fact may cause a bias in In 'Y±· With Pitzer's equation,<6

l a 1, a 2, 130, !31, and 132,2 risk being

matched in a wrong way, yielding incorrect values for "Y± even though the osmotic coefficients (and, for m > m1;m, the relative activity coefficients) are still correct.

As direct measurements of activity coefficients are now capable of the requisite precision at appreciably higher dilution levels, it seems more convenient that the Pitzer theory parameters be derived from the activity coefficients and then used to back-calculate the osmotic coefficients, rather than vice-versa.

2. EXPERIMENTAL

ZnSO4, MgSO4, CaSO4, and SrSO4 were from stocks of best purity grade commerciai salts and were employed without any further purification. The salts were handled in the form of stock solutions, and these were standardized, for Zn, Mg, and Ca, by classic EDTA titration (rsd = 0.05, 0.15, and 0.27%, for ZnSO4, MgSO4, and CaSO4, respectively). The strontium sulfate

1 The procedure is to be considered an interpolation rather than an extrapolation because at x = O the function ( cj> - I )x- 1 has the known value of the DHLL slope of ( cj> - I) vs. jm.

2To avoid any possible confusion with the labels of the bibliographic references, Pitzer's originai notations W0l, W11, and ~(2]. have been changed to ~o, ~ 1, and ~2•

794 Malatesta and Zambonl

stock solution was directly prepared by dissolving a weighed amount of the salt (previously dried in a fornace at 250°C) in a weighed arnount of water. Tue solutions to be examined were all prepared from these stock solutions by weighing, using PTFE bottles. Auxiliary solutions of the salts Zn(ClO4h, KC1O4, KCI, MgCl2, CaCl2, and SrCl2 were also prepared, whose concentrations were determined either from the direct weights of the salts (KC1O4

dried at I00°C at reduced pressure; KCI and SrCl2 dried at 250°C) or via EDTA titration (Zn(ClO4h, rsd = 0.15; MgC12, rsd = 0.13; CaC)i, rsd = 0.09%). The technique and instrumentation (in particular, the design of the cell and electrodes) have been described previously,m except for the zinc salt. For M = Mg, Ca, and Sr, the following cells were utilized,

Ag, AgCI IKCI (0.0lm) IK+I K2SO4 (0.007m) ISOj-1 MSO4 (test soln.)

IM2+I MCl2 (0.007m)I AgCl, Ag (celi l)

Ag, AgCI IKCI (0.0lm) IK+I K2SO4 (0.007m) ISOà-1 K2SO4 (test soln.)

IK+I KCI (0.0lm)I AgCI, Ag (cell 2)

Ag, AgCI IKCI (0.0lm) ICI-I MCii (test soln.) IM2+1 MCl2 (0.007m)I

AgCI, Ag (celi 3)

Ag, AgCl IKCl (0.0lm) ICi-1 KCl (test soln.) 1K+1 KCI (0.0lm)I

AgCI, Ag (cell 4)

In accordance with the symbols used already,<7c> IK+I, 1sO1-1, IM2+1, and 1c1-1 indicate the liquid membranes which are permeable to the corresponding ions. These membranes were made of the ion-exchanger salts KTFPB, (TDAhSO4, M(TFPBh, and TDACl, respectively; TFPB- stands for the rerrakis[3,5-bis(trifluoromethyl)phenyl]borate anion and TDA + for the tetradodecylamrnonium cation. Both TON and TFPB- are lipophilic and very hydrophobic, their salts are insoluble in water but soluble, and appreciably ionized, in organic media. <7l The solvent was 2-nitrophenyl octyl ether (NPOE) in the liquid membranes IK+I and IM2+1. For 1C1-1 we selected 3,4-dichlorotoluene, found preferable to NPOE,3 1-chloronaphthalene, 1,2,4-trichlorobenzene, and to ali other solvents previously tested for tetradodecylammonium salts. Also, the ISOi-1 membranes were improved compared with those

3 NPOE, used as a solvent for tetradodecylammonium salts (chloride, nitrate, perchlorate), undergoes some interference effects from the potassium ions; appreciable drifts, the direction of which varies at random for several days without any apparent trend, are observed in potassium chloride, potassium perchlorate, etc., solutions. These effects are not observed with other solvents such as 1-chloronaphthalene, 1.2,4-lrichlorobenzene, 3,4-dichlorotoluene (or also using NPOE, provided no potassium ions come into contact with the membrane).

EMF or Llquid Membrane Cells 795

described in the previous paper, which were made of (TDA)2SO4 dissolved in NPOE or in l-chloronaphthalene,l7b> We found, in fact, that (TDAhSO4 (a white powder, if prepared as already described)<7

b> reacts slowly with water, and forms a viscous liquid insoluble in water. This viscous liquid is a sort of fused salt, whose anion is some hydrated form ofthe sul fate ion, [SOiH2O).]2-. As a result of the increased size of the anions, the lattice weakens, and the salt melts (as described below in Section 3.1, we have an indication that x = 8). The liquid salt (TDAhSOiH2O)x, like (TDAhFe(CN)6 previously noted,<7d>

is directly suitable for preparing high-performance liquid membranes without added solvent (the salts (TDA)3Co(CN)6 and (TDA)4Fe(CN)6 which we pian to employ in future work are also liquid at room temperature).

For zinc, a different assemblage was needed. Preliminary assays showed that a cell like celi 3 does not work properly for ZnCh, since the emf drifts.4

Both zinc-responsive and chloride-responsive electrodes are responsible for these effects, since both keep a memory for many hours of any previous use in zinc chloride solutions when reused separately, the zinc electrode in celi 1, and the chloride electrode in celi 4. It is probable that the zinc-chloride complexes enter the membranes and prevent the cells from working correctly. Thus, ZnCI2 needs to be avoided even in the inner sections of the celi for ZnSO4. We solved the problem by using, as the zinc-responsive electrode, the threemembrane chain whose diagram is shown in Fig. l. The entire cell for ZnSO4,

Ag, AgCl IKCl (0.0lm) IK+I K2SO4 (0.007m) ISOtl ZnSO4 (test soln.)

IZn2+1 ZnSO4 (0.0lm) 1sO~-1 K2SO4 (0.007m)

IK+I KCI (0.0lm)I AgCI, Ag (modified cell l)

contains five liquid membranes; nevertheless, its electrical resistance is reasonably Iow (R < 105 s- 1).

Auxiliary cells for zinc perchlorate and for potassium perchlorate were also prepared,

Ag, AgCI IKCI (0.0lm) IK+I KC1O4 (0.0lm) ICIOil Zn(ClO4h (test soln.)

1zn2+1 ZnSO4 (0.0lm) ISOtl K2SO4 (0.007m)

IK+I KCI (0.0lm)I AgCI, Ag (modified celi 3)

Ag, AgCI IKCI (0.0lm) IK+I KC1O4 (0.0lm) IClO41 KCIO4 (test soln.)

IK+I KCI (0.0lm)I AgCl, Ag (modified cell 4)

4 1n 0.01566m ZnCl2, the emf drifted from +27.70 to +34.50 mV in 2 h: thereafter, an apposite drift (from -3.97 to -6.52 mV in 2 h reaching -8.59 mv in two days) was observed in 0.00592m ZnC)i. But in the same 0.00592m solution the emf was found to drift instead from - IO. 13 to -8.28 mV in 4 h when the electrodes were previously allowed to stand fora day in a more dilute solution, 0.000355m ZnCI~.


b

b b,d

e e,f

g h,f

k j,f

b,d k,l Fig. 1. Diagram of the zinc selective electrode: a = silver: b = P'TFE; e = silver chloride: d = pressing ring; e = O.Olm potassium chloride solution; f = fritted glass; g = potassiumexchange membrane (17% KTFPB in NPOE); h = 0.0067m potassium sulfate solution; i = sulfate-exchangc membrane (tetradodecyl ammonium sulfate in thc hydrated fonn, without solvent); j = 0.0lm zinc sulfate solution; k = zinc-cxchange membrane (35% Zn(TFPBh in NPOE); I = onc of thc six sheets of porous PTFE which bracket the three liquid membranes.

where the IClO4 I membrane was a saturated solution of TDAClO4 in 3,4-dichlorotoluene.

Preparation and purification of the membrane salts KTFPB, M(TFPB)2,

TDACI, and (TDA)iSO4, are described in previous papers;<7a-cl Zn(TFPBh and TDACIO4 do not need any different handling.

3. RESULTS AND DISCUSSION

As long as the transference numbers are indistinguishable from unity for the explored cation across the cation membrane interfaces and for the explored anion across the anion membrane interfaces, the emf of the cells bere considered (i = I to 4) will bave the exact form

(2)

where V+ and v_ are the numbers of cations and of anions, respectively, per molecule of the electrolyte, Z+ the cbarge of tbe cation, and m tbe molality of the test solution.(7) The constant tenn E~ contains the constant activities of the inner-side solutions, plus a constant contribution that arises from the stoichiometric coefficients [the expanded form of Ef, not reported bere for

EMF of Liquid Membrane Cells 797

brevity, leads us to derive useful relationships between the Et of cells that have one electrode in common; an example is Eq. (3)]. The right conditions for Eq. (2) to hold, are usually fulfilled over a wide concentration range, 4-5 orders of magnitude of molality in favorable cases.Oh-el At extreme dilution Jevels, tt1· and OH- will compete with the ions under study, while at high concentrations, the ions may associate, and the associated species may enter the membranes and interfere. Both failures cause the transference number of the right ion through the relevant membrane to be neither equal to unity nor exactly predictable; thus, the celi responses tose significance. Correspondingly, equilibrium conditions at and near the interfaces are no longer reached, and the emfs drift (conversely, when the latter are stable, their values should be significant).

Unlike ZnC12 cited in the experimental section, the MS04 salts yielded stable emfs even in solutions as concentrated as 2.5m MgS04 and 1.5m ZnS04, or in the neighborhood of the saturation limits (SrS04 and CaS04).

In clilute solutions, some drifts (always in the direction negative towards positive, as if the concentration of the electrolyte increased) began to be observed in a quiet solution in the neighborhood of 3 X I 0- 4

- I X I o-4m

depending on the salt, while on mild stirring,5 stable values were read down to about I X 10-4m (magnesium sulfate), 4X 10- 5m (strontium sulfate), 8X 10-5m (zinc sulfate), and 2X 10-4m (calcium sulfate). On further increasing the dilution, the drifts became ineliminable and the readings became sensitive to the rotation rate, thus losing any significance.

Tables I, Il, III, and IV give the observed emfs vs. molal concentrations, divided into different series and the corresponding activity coefficients 'Y-:!:.• Each series refers to a period not exceeding 24 h and includes one measurement in the solution chosen as reference, to minimize the effects of possible changes of E*.6 In the very dilute solutions where no definite equilibrium was obtained, multiple values for E are reported, corresponding to extreme readings made under different, specified, conditions. Though scarcely significant, these readings provide an idea of the possible errar involved and help to identify the range of reliability of the method.

5The stirring was realized using a magnetic bar (5 mm Iength, 3 mm cross section) on the bottom of the cell, about 3-4 cm away from the active surfaces of the membrane electrodes, usually al a rotation rate of about 1-2 rps, up to about 25 rps maximum in a few cases.

6 E* is often found to differ slightly from series to series. This is probably due (in part) to a slow osmotic tlow of water, carried along by the hydrated ions through the membranes, which causes the concentrati on of the nearer aqueous layers lo change slightly. A stepwise variation, and a subsequent very slow relaxation, were also observed in some cases. and were probably due to occasionai contamination of the membrane surfaces, e.g .• from the materials released from the other membrane, when disassembling the celi.


1àble I. Experimental EMF and Activity Coefficients of ZnSO4

Series

l '

2• 2 2 2 2 2 3' 3 4 4 4 4 4 4' 4 4 4 4 4

m•

l .050XJ0- 2

1.516 9.326XI0- 1

5.556x10- 1

3.268XI0- 1

l.050XJ0-2

l.890XI0- 1

1.128XIO-I 6.528X 10-2

3.614XI0-2

2.0)6 XI0-2

l .050Xl0-2

4.993 XJO-l 2.960X 10-3

l.656XI0- 3

9.339X 10-4

5.ISJXI0-4

2.835X 10-4

l.050Xl0- 2

J.450xJ0- 4

8.285X 10-~ 8.285X 10-5

4.691X 10- 5

4.69) X 10-5

"Units: m, mol-kg - •; E, mv.

E•.b In 'Y :!:(hydri"

-0.95 -1.005±0.005 58.66 -3.657±0.006 54.85 -3.320±0.006 50.48 -2.972:':0.006 45.42 -2.638:': 0.006 -0.85 -1.005±0.005 39.35 -2.331 ±0.006 33.86 -2.028:::':0.006 26.85 - 1.745±0.006 18.41 -1 .491 ±0.007 9.63 -1 .249±0.006

-1.25 -1 .005±0.005 -14.17 -0.764±0.006 -24.051 -0.624±0.006 -35.4Jf -0.486±0.007 - 47.391 -0.379±0.006 -60.191 -0.282±0.006 - 73 .561 -0.205±0.007 -1.31' -1.005±0.005

-89.0QI -0.136±0.007 -102.371 -0.096±0.006 -102.29R -0.093±0.006 -116.21 -0.066±? -1 )6.7h -0.085±?

b Unless otherwise specified, values were ali read in unstirred solutions.

In 'Y :!:d

-l.002X0.005 -3.333±0.010 -3.140±0.007 -2.864:=0.007 -2.571 :':0.007 - 1.002±0.005 -2.290:':0.006 -2.002 :t0.006 -1 .738±0.006 - 1.482 ±0.007 - 1.243±0.006 -1.002±0.005 -0.762±0.006 -0.623 ±0.006 -0.485 ±0.007 - 0.379±0.006 -0.282±0.006 -0.205±0.007 -1.002±0.005 -0.136±0.007 -0.096±0.006 -0.093±0.006 -0.066±? -0.085±?

'Values referring to ZnSOiH2O)w- Errors are calculated from the measured or estimated variance of E in the solution and in the corresponding reference solution (::':O. I mV, estimated by excess), with the addition of the possible bias (±0.005, estimated) of In -y",,.

dValues referring to ZnSO4• Errors also include the effect of 0.5 uncertainty in the hydration number w.

'Internal reference point of this series. 1Measured in solution stirred at ca. I rps. 8 Value reached after 4 h at ca. I rps. h Measured in solution stirred at ca. 25 rps.

To derive the activity coefficients from the relative activity coefficients, two independent methods are considered: (a) a theory-assisted extrapolation using (as discussed in Ref. 12) different theoretical approaches together for a mutuai check; (b) an indirect determination of Ef from the Ef values (easily found) of cells 2, 3, and 4, using the therrnodynamic relation

Ef - El - E'f + Et = -2(RTIF) In 2

which also applies to the cells modified for zinc.

(3)


Table Il. Experimental EMF and Activity Coefficients of MgSO4

Series m• f;',b In 'Y ~1hy<1r,'" In -y,/

l.488 X I0-2 -6.30 -1.032:!::0.006 - 1.028 :!::0.006 8.050XI0-3 -17.04 -0.836:!::0.007 -0.834:!::0.007 4.475X 10-3 -27.72 -0.664:!::0.006 -0.663:::0.006 2.sosx10-3 -38.95 -0.521 :t 0.006 -0.520:+:.0.006 l.283X 10-3 -52.45 -0.378:!::0.006 -0.378:!:0.006

I' 5.978xJ0-2 +14.97 - l .595:!::0.005 - J.58/ :+:.0.005 2' 5.978X/0-2 +14.97 - 1.595:!:0.005 - 1.581 ±0.005 2 6.3l3X 10-• -67.50 -0.254:!:? -0.254:':? 2 3.298 X 10-• -82.26 -0.179::t: ? -0.179:':? 2 1.63 I X 10-• -98.56 -O. I IO:':? -O.I IO:':? 2 1.631 X 10-4 -98 .28' -0.099:':? -0.099:':? 2 8.628X 10-5 -113.33 - 0.048:':? -0.048::t:? 2 4.391 Xl0-5 -129.64' -0.007::t:? -0.007::t:? 2 4.391 X 10-5 -128,07/ +0.054:!:?1 +0.054::t:?1

3 5.979x10- 1 43.56h -2.804 ± 0.006 - 2.689::t:0.007 3 3.266X 10- 1 37.08 -2.452±0.006 - 2.387::t:0.007 3' 5.978X 10- 1 /5.47 - l .595:t::0.005 -1.58/ ±0.005 3 l.647XlQ- I 29.08 -2.079:':0.006 - 2.045 :':0.007 3 8.453X 10-2 20.46 - 1.748::t:0.006 -1.729:!::0.006 3 4.538X 10-2 11 .62 - 1.469::t:0.006 -1.458:+:.0.006 3 2.388 X 10-2 1.72 -1.213±0.007 - 1.207 ± 0.007 3 l.240 X 10-2 -9.12 -0.979:!::0.007 -0.976±0.007 3 6.543x10-3 -20.38 -0.778::t:0.007 -0.776±0.007 4' 5.978XJO-Z 15.42 -1.595±0.005 -l.58/:!::0.005 4 6.543X )0-l -20.22 -0.770::t:0.007 -0.768:':0.007 4 3.346X 10-3 -32.99 -0.596::t:0.007 -0.595±0.007 4 l.727X 10-3 -46.36 -0.455 :':0.007 -0.454 ± 0.007 4 9.l70Xl0-4 -59.63 -0.339:':0.007 -0.339:':0.007 4 4.913Xl0-4 -73.50 -0.254::t:0.007 -0.254±0.007 4 2.523XIO-• -88.65 -0.178±0.008 -0.178::t:0.008 5' 5.978XJO-Z 15.06 -J.595±0.005 -1.581±0.005 5 2.495 55.65 -3.747:':0.006 -3.035:!::0.0l9 5 1.338 51.04 -3.303:':0.006 -3.029:':0.009 5 5.772 X I0- 1 42.74 -2.785::t:0.006 -2.674±0.007 5 2.572x10- 1 34.09 -2.314:':0.006 -2.262::t:0.007 6' 5.978XJO-Z 15.25 -1.595:'::0.005 -1.581±0.005 6 I. I 87 X IO- I 24.61 -1.917±0.007 -1.891 ±0.007 6 l.746 x w-1 -3.70 -1.102:!:0.006 -1 .098±0.006 6 7.150X 10-3 -19.11 -0.809:!:0.006 -0.807±0.006 6 3.205x10-3 -34.16 -0.592±0.006 -0.591 ±0.007 'l' 5.97B x 10-z 15. 14 -1.595±0.005 -l.58/:!:0.005 7 l.446 x 10-J -50.22 -0.417±0.006 -0.417±0.006 7 5.9l0X 10-• -69.58 -0.276:!::0.006 -0.276±0.006 7 2.141 x10-4 -92.73 -0.161±0.006 -0.161 :!:0.006 7 I .075 X 10-4 -108.80 -0.098±0.007 -0.098±0.007 7 5.327X 10-5 -125.&Ji -0.058:+:.0.007 -0.058:+:.0.007


Table II. Continued.

Series m" f;",b In 'Y'!'.(hyctri"

7 5.327X 10-5 -125.781 -0.057::!:0.007 7 2.282X 10-5 -140.7' +0.210::!:?1

7 2.282X 10-5 - 141.2" +0.191::!:?1

• Units as in Table I. hUnless otherwise specified, values were ali read in unstirred solutions. cvalues referring to MgSO4(H2O)w (errors calculated as in Table I). dValues referring to MgSO4 (errors calculated as in Table I).

'Internal reference point for this series. fVaJue reached after 2 h in the unstirred solution. ~Tentative extrapolation to zero time. hThe magnesium-responsive electrode was refilled. iMeasured in solution stirred at ca. I rps for ca. I h. 1 Measured after stirring 2h at ca. I rps. • Measured at stirring rate of ca. 25 rps. 1 Absurd value.

( a) Theory-Assisted Extrapolation to Infinite Dilution

In -y,/

-0.057::!:0.007 +0.210::!:?1

+0.19(:t:?1

For 2-2 and other high-charge electrolytes, the Debye-HUckel theory is unsuitable for extrapolation because the experimental slopes are, in the most dilute regions accessible, steeper than the DHLL requires. Pitzer's equation, too, should be considered as theoretically inadvisable for extrapolation with these electrolytes, since it embodies a number of parameters (in particular, o:2 and ~2, but actually also o: 1, ~o, and ~ 1) of not strictly theoretical derivation, whose vaJues may be critical.7 DHLL + B2 approximation of the Mayer theory,< 13J numerica! integration of the Poisson-Boltzmann equation (IPBE),04l and Bjerrum's theory,0 5•16l ali permit treating the data in terms of a single adjustable parameter, the distance of closest approach (or the hardcore mean diameter) a of the ions, as if they were a population of charged spheres in a homogeneous dielectric fluid (primitive model) as in the DebyeHtickel theory.8 One keeps the experimental values of the logarithms of the relative activity coefficients In -y' (In -y' = In 'Y± - In -yi, where -y't to be detennined, is the value in the reference solution) and subtracts them from the values In -y~ calculated from a given theory and a given value of a. The

7 In spi te of these theoretical limitations, once the parameters a 1, a 2, '30, '3 1, '32, and cix are ali chosen with best-fit criteria, the extrapolation values lhat Pitzer's equation suggests are in very good agreement with those calculated by the other methods, as shown in part (e) of this section.

8 Mayer's theory allows the closest approach distances a++, a __ , and a+- to be considered as distinguishable, but the results do noi depend appreciably on the choice of a ... and a __ ; thus, a single a, a.-, is practically involved.


Table lii. Experimental EMF and Activity Coefficients of CaSO4

Series mu E°•b In 'Y =lhydri'"

I' / .273X 10-i -12.31 -1.038±0.005 I 7.109X 10-3 - 22.01 -0.833:!:0.006 I 3.549X 10- 3 -34.64 -0.630:!:0.007 I l.!!27X 10-3 -47 .75 -0.476±0.006 I 9.759X 10-4 - 60.68 -0.352:!:0.006 2 5.126X 10-4 -74.6(}/ -0.250::t::0 007 2 2.512X 10-• -90.88/ -0.180:!:0.007 2' l .273 X 10-i -12.071 - /.038:!:0.005 2 l .285 X 10- • -105.88/.R -0.093:!:0.009 3' J.273 X JO-l -12.17' - /.038:!:0.005 3 4.9(6Xl0- 3 - 28.551 - 0.724:!:0.006 3 2.671 X 10-3 -40.ISI -0.565:!:0.006 4 2.671 X 10-3 -40.171 -0.568:!:0.006 4 l.396 X 10- 3 -53.01 1 -0.4 l 9:!:0.007 4 6.912 X 10-• -67.9()1 -0.298:!:0.007 4 3.614X 10- • -82.411 -0.212±0.007 4 1.991 X 10-4 -96.31 1 -O. I 56:!:0.007 4' l .273 X JO-l -12.121 - /.038:!:0.005 4 9.860X IO_, -16.361 -0.948±0.006 4 l.295 X 10-4 -106.04' ... -0.105±?

"Units as in Tahle I. hUnless otherwise specified, values were all read in unstirred solutions. 'Values referring to CaSO,<H2O)w (errors calculated as in Table I). "Values referring to CaSO4 (errors calculated as in Table I) . 'Internal reference point of this series. IMeasured in solution stirred at ca. I rps). 8 Values estimated in the presence of drifts that could not be eliminated.

In 'Y~ J

-1 .035±0.005 -0.831 :!:0.006 -0.629:!:0.007 -0.476:!:0.006 -0.352:!:0.006 -0.250:!:0.007 -0.180:!:0.007 -1.035:!:0.005 -0.093 ::t::0.009 -1.035:!:0.005 - 0.722:!:0.006 -0.565 :!:0.006 -0.567:!:0.006 -0.419:!:0.007 - 0.298:!:0.007 -0.212:!:0.007 -O. I 56 ±0.007 -l.035:!:0.005 - 0.946±0.006 -0.105±?

differences, In 'Y~ - In -y', are plotted vs. log m, and the procedure is repeated, at different values of a, for the three theories considered. Provided the concentrations are sufficiently low for the validity of the theory, the values In 'Y~ - In -y' will all converge. on dilution, towards the value of In -y'1. An example of the procedure is shown in Fig. 2 for ZnSO4 ; ignoring the points below I X I o-4m, of uncertain precision, the value of In 'Y~ is suggested to lie around - l.01 0 (m0 = l.050X 10-2m.). We do not have any definite criterion to evaluate the range of uncertainty, but approximately speak.ing, it should not exceed ±O.O I (i.e., l % of 'Y-z.). For the other salts, the same treatment suggests In -y1 = -1.590 for 5.978X 10-2m MgS04 • ignoring the points below 2X 10-4m; In 'Y~ = - l.040 for l.273X 10-2m CaSO4, ignoring the points below 1.991 X 10-4m; In 'Y1 = -0.250 or -0.245 for 5.051 X 10-4m SrSO4, ignoring the points below 7X 10-5m.


Table IV. Experimental EMF And Activity Coefficients of SrSO,

Series ma E",b

l'" 5.05/XI0-4 -72.72 1 2.742X (0-4 -86.70 1 l.424X 10-4 -102.11 I 7.2nx10-5 -118.55' I 7.272Xl0-s -118.451

I 7.272X (0-5 -I 18.35' 1 3.495X 10-< -136.671

I 3.495X 10-5 - 136.6QR 1 3.495X 10-5 -136.42i I l.914X (0- 5 - 150.75' 1 l.914X 10-< -150.151

l.914X 10-5 -151.10" 2' 5.05/X/0-4 -70.321

2 2.924XIO-• -82.82 2 l.561XI0-4 -97.83 2 7.997X 10-5 -113.90 J< 5.05/X[0- 4 -70.20 3 7.997X 10-s -113.72 3 4.383X 10-s -128.60 3 2.321x10-5 -144.00' 3 2.321 X IQ-S -143.90f 4,· 5.05/X[O-• -70.09 4 7.997X IQ-5 -113.26 4 4.065X 10-s -130.70" 4 4.065X 10-s -130.86" 4 l.604X 10-s -153.251

4 l.604X 10-5 -153.17h 4 5.051 X 10-5 -70.09

•Units as in Table I. bUnless otherwise specified, values were ali read in unstirred solutions. 'Internal reference point of this serie&.

In·'/±

-0.250:::0.005 -0.183:::0.006 -0.128:::0.007 -0.096±:0.007 -0.092:::0.007 -0.088::1:0.007 -0.068:!:0.007 -0.066:::0.007 -0.059:!:0.007 -0.014:::? + O .009 :!: ?"' -0.028:!:? -0.250:::0.005 -0.190::0.006 -0.147:!:0.006 -O. I 03 :!:0.006 0.250::0.005

-0.101±0.006 -0.079::0.006 -0.042:!:? -0.038:!:? 0.250:!:0.005

-0.087 :!:0.009 -0.089:::? -0.095±? -0.037±? -0.034:::? -0.250±0.006

dcìVaJues read in unstirred solution al ca. 0.5, I, 2, 3, 4, 5, 6 h, respectively, after the experiment was started.

kVaJue read in solution stirred 5 min. at ca. 5 rps, after 4 h standing. 1The strontium-selective electrode was refilled after series I was terrninated. "'Absurd value.

Owing to the Iow concentrations involved, any differences between molar, molai, and rational activity coefficients, and between the values pertaining to the Gibbs or MacMillan-Mayer reference systems, were ignored in these calculations. We did not consider, either, the moderate corrections needed to transform the molai concentrations of the experimental points into the volume scale concentrations utilized by ion solution theories; as a consequence, the values of In 'Y~ and those of In -y' inside the differences

EMF of Llquld Membrane Cells 803

o ····o······ ·o····· ·o··· i----..... ... • .. . ................ . . . .ci. o. • •.

o o -1 .1 ____ _._ _________ _._ __ ~---

.. .Cl..... - 1 ...... o

-, . ,

o -1.1 o

o

0.0001 0.001 m Fig. 2, The extrapolation function In 'Y~ - In -y' for ZnSO4, using the IPBE algorithm (top section), the Bjerrum theory (middle section), and the DHLL + B2 approximation of the Mayer theory (bottom section). for the indicative values of a of 0.38 nm (open circles), 0.35 nm (filled circles), and 0.33 nm (squares). The dashed horizontal lines indicate the value of In -y~ resulting from Eq (3) with its range of uncertainty.

In 'Y~ - In -y', actually refer to slightly different concentrations. However, the empirical value of a eliminates the residuai effects of these imprecise settings.

(b) lndirect Determination of Ef The activity coefficients of Zn(CIO4)z,(17l MgC12,0 8l CaCI2,09> sre1z,< 18

>

K2SO4,<20l KC1,<7a> and KCIOP1l are all available in the literature; hence, E!, Et, and Et can ali be obtained immediately from the E2, E3, and E4

values read at any known molality of the corresponding salts. Afterwards, Ef can be calculated from Eq. (3), thus circumventing the extrapolation problems peculiar to 2-2 electrolytes. A suitable procedure to minimize the errors has already been described.r1c.22l Since the electrodes tend to change slowly during their lifetime, the procedure is used to find once and for ali In -yi in the reference solution, rather than Ef. The values found are In 'Y~ = -1.003 ±:0.028 in 1.050 X 1 o-2m ZnSO4; In 'Y~ = - 1.598 ±:0.034 in 5.978 X I o-2m MgSO4; In 'Y~ = -1.038 ±:0.024 in 1.273 X I o-2m CaSO4;

ln -yi = -0.252±0.029 in 5.051 X w-4m SrSO4• The broad range of uncertainty does not arise from the readings for E1, E2, E3, and E4, which contribute ±0.004 or Iess, but it reflects the previous uncertain values of the activity coefficients of the bivalent salts used to deduce E! from E2 and Et from E3•

For MgC)i, SrCI2, and Zn(ClO4)z different interpolating equations have in fact been proposed,<' 7•18·23

> whose results disagree appreciably from one


another (up to 0.03 for MgCh) at the concentrations at which we measured E3• Average values were selected: In 'Y-z. = -0.368 for l.648X 10-2m MgC)i, ln 'Y:1:. = -0.313 for 9.54xl0- 3m SrC12, and ln 'Y± = -0.132 for l.188X 10-3m Zn(ClO4h. For CaCl2, two interpolating equations are available,<19·23> and both agree on In 'Y-:: = -0.121 (~0.001) at l.049X 10-3m. To deduce Ei from E2, the activity coefficients of K2S04 are needed; we used Pitzer's equation with our own interpolation parameters,<7b> yielding In 'Y-z. = -0.507 at 2.325X 10-2m. (Unfortunately, the measurements for K2S04 were not particularly precise, and thus the standard deviation between our experimental and interpolated values of In 'Y± was 0.016.17b> Note that Pitzer's originai parameters would yield -0.482 instead of -0.507,l23 l and the three smoothing equations from Goldbergr20> would yield -0.518, -0.455, and -0.610, respectively). For6.151 X 10-2m KCl, In 'Y-z. = -0.084 is assumed.(7aJ For 8.69X 10-4m KCIO4, finally, we obtained ln ''fa. = -0.032 from the values reported by Harned and Shropshire,<21 i corrected for the different reference scale and interpolated by means of the Pitzer equation.9

The values for KCI and KC1O4 are accurate within a few tenths of one percent at the worst. and do not contribute any appreciable uncertainty. On the contrary, it is advisable to reconsider in future the activity coefficients of the 1-2 and 2-1 salts; should more accurate values become available at the concentrations at which we measured E2 and E3, the corrections for the MSO4 salts are easily calculated (a variation a in ln 'Y-::. for K2SO4 or MX2

yields a variation 1.5 Ll for MSO4, in the same direction).

(e) Establishing the Reference Values

The values obtained via Eq. (3) are in good agreement with those assigned via theory-assisted extrapolation to infinite dilution, with their maximum discrepancy (0.8% of the value of ''fa, for MgSO4) well inside the range of uncertainty of both direct and indirect methods. The extrapolation method avoids any cumulation of errors from other salts; moreover it is, in part, selfcorrecting for the possible systematic error introduced in the standardization of the stock solutions. By contrast, it is sensitive to any kind of errar able to modify the trend ofthe emfs in the very dilute regions, e.g., the interference from H+, OH- HCO3, and other foreign ions (for this reason we did not take into consideration the more dilute solutions). We considered also the values suggested by the Pitzer equation using a 1, a2, 130, 131, 132, cix, and In 'Y~,

9The originai data, deduced from the diffusion coefficients, were for the molar concentration scale. The translation into the molai scale was based on the approximate density values of the Debye-HUckel limiting law for volumes, assuming 54 cm3-mo1- 1 as the partial molar volume of KCIO4 to infinite dilution.124

> The resulting values of ·'h were processed with the Pitzer theory, thus correcting a minor extrapolation errar of the originai data. Actually, these corrections were practically negligible.

EMF of Liquld Membrane Cells 805

as adjustable parameters, to be varied unti! the sum of the squared differences between experimental and calculation values reached its minimum. For zinc sulfate, discarding the two points for m < 5X 10-5, Pitzer's equation yields the best-fit value In -y1 = -1.002±0.005, which agrees with the extrapolation - I .O 10 of the direct method and the result -1.003 of the indirect method (the corresponding best-fit values of the other adjustable pararneters are a 1 = 1.02±0.33, a 2 = 17.3::!::l.O, !30 = -0.06±0.34, 13 1 = 2.36±0.15, 132 = -121 ± 14, cix = 0.09±0.17). Far magnesium sulfate, discarding the points for m < 2X 10-4

, Pitzer's equation yields In -y1 = -1.588±0.006, which agrees with -1.590 of the direct method and - 1.598 of the indirect method (the best-fit values of the other parameters are a 1 = 1.56:!::0.10, a 2

= 15.3± 1.1, 130 = 0.258±0.037, 13 1 = 3.31 ±0.18, 132 = -65.1 ±9.5, CMx = -0.08±0.02). Far calcium sulfate, the value of CMx was set at zero, owing to the low concentration levels concemed; even so, we were not able to obtain convergence of the computation prograrn if the parameters were treated ali together. Thus, we fixed a1 at 1.4 and optimized a2, 130, 13 1, 132, and In -yi. The result far In -y~ was -1.036±0.006 (the best-fit parameter values were a 2 = 16.4±5.8, 130 = -13.7±31.9, 13, = 20.4±47.6, /32 = -94.4± 10.2, with <J = 0.003; the points far m < l .9X 10-4 were discarded). The originai Pitzer and Mayorga parameters yield a quite similar value,'6l

ln -y~ = -1.040 (actually, this value should not be compared directly with ours, which stili includes the ion hydration contribution discussed below in Section 3.1; but this contribution is negligible, -0.003 logarithmic units, at the corresponding concentration). As regards strontium sulfate, the concentration range explored was tao limited far the best-fit program to converge, unless the values for a 1, a 2, and 130 were previously fixed, and then the corresponding best-fit values of 13 1, 132, and In -y1 were sought. By imposing a 1 between I and 2 (i.e., around the value l.4 recommended for 2-2 salts),<6> a 2 between IO and 30 (Pitzer and Mayorga select 12 forali 2-2 salts),(6> and 130 between 0.2 and 0.4 (typical of 2-2 salts),<6> we find that 131 and 132 vary considerably, but In -y~ and the standard deviation do not change appreciably. The value far In -y1 was around -0.25 l if the points for m < 5 X I o-s were discarded, or -0.246 if only the points for m < 2X 10-s were discarded.

The following reference values, In -yi = - 1.005 for l.050X 10- 2m ZnSO4, In -yi = - l.595 for 5.978X 10-2m MgSO4, In -y~ = - 1.038 for 1.273 X I o-2m CaSO4, and In -y~ = -0.250 for 5.05 Ix I o-4m SrSO4, are intermediate between the different estimates of the theory-assisted extrapolations and the indirect method. and also take into some consideration the suggestions of the Pitzer equation. We took these figures as our best evaluations (they are presumably accurate within ±0.005 or better) and used them to turn the relative activity coefficients into activity coefficients in Tables I-IV. However, these values stili refer to hydrated forms MSO4(H2O)w of the


relevant electrolytes and need correction (Section 3.1) to be expressed in terrns of nonhydrated MSO4• The reference values corrected for the hydration effects, i.e., referring to the actual MS04 salts, become In -y~ = -1.002

(instead of - 1.005) for ZnSO4, - 1.58 1 (instead of - l.595) for MgSO4,

-1.035 (instead of -1.038) for CaSO4• As the reference solution was very dilute, SrS04 does not need a correction.

3.1. The Concentrated Region; Ion Hydration Effects

The activity coefficients that the liquid membrane cells identify in the concentrated region are not equivalent to those derived via cj,.<7H> This is visible in Fig. 3, where the curve from Pitzer and Mayorga for MgSO4 (shifted down by O.O 15 units, to correct the minor systematic difference of the different extrapolations to zero) diverges from our values in the fourth column of Table 11. The same thing happens for ZnSO4 (Fig. 4). The membrane cells identify the activity coefficients -Y:<::chydrJ of a somewhat different electrolyte, MSOiH2O)w instead of MSO4, where w is the tota! number of ion-hydration molecules of water which enter the membranes together with one M2+ ion and one sor ion. For a 2-2 salt, the theoretical relationship between these two kinds of activity coefficients is easily deduced

-1

-2

-3

- 4

In y i

1 o-•

In -Y:<::(hydr) = In -Y:<:: + (w/2) In a 1 (4)

MgSO 4

• m •

,0·3 ,o·2 10"1 1 o0

Fig. 3. Values of In "/,e for MgSO4• Filled circles. celi I without the hydration correction; bold line, Pitzer and Mayorga values (from osmotic coefficients) shifted by -0.015; open circles. celi I corrected by 20.4 moles of water per mole of the sali; narrow line, Pitzer's curve recalculated (coefficients from Table V). The double cross indicates the reference point derived via Eq. (3) end its uncertainty. Dashed line. DHLL.

EMF or Llquid Membrane Cells

-1

-2

-3

In y ±

807

ZnSO 4

• m

Fig. 4. Values of In 'Y~ for ZnS04• Filled circles, celi I without the hydration correction; small open circles, Bray (lowered by -0.121); small crosses, Cowperthwaite and La Mer (lowered by -0.037); bold line, Pitzer and Mayorga (lowered -0.112): large open circles, celi I corrected by 22.6 moles of water per mole of the sali. Ali other symbols are as in Fig. 3.

(ai, the water activity); In 'Y:':(hyctr> and 'Y='= become identica} in dilute solutions. In the concentrated regi on, accurate values are available for <j>, <6

•27

> thus In a1 and (apart from a possible shift) In 'Y='= are correspondingly known, and w can be deduced from the experimental values of In 'Y=<hydrJ· To eliminate the shifts, we rewrite Eq. (4) in the equivalent, relative fonn

w = l0OO[illn (m-y='=) - (RT/F)- 1ilE]/[18.015 il(m<!>)] (5)

(dq, the difference of q between two different solutions), and calculate the needed dln 'Y='= and d(m<!>) directly, using the originai Pitzer and Mayorga parameters.<6> Repeated determinations of dE between the more concentrated solution (m = 2.495 mol-kg- 1, the stock solution; a high concentration enhances lhe precision for w) and the reference solution, lead to w = 20 (20.45:::!:0.44) for MgSO4• The splitting of w into a W+ = 12 and w_ = 8 is strongly suggested, since the previous analysis for water and the metal ion, in a water-saturated solution of Mg(TFPBh in NPOE. seemed to indicate twelve molecules of water per magnesium.(7eJ (The analytical measurements were actually not definitive, and 11 molecules per magnesium was also a possible interpretation of the results, though less likely. Twelve plus eight appeases chemical intuition much better than 11 plus nine, but an elegant hypothesis does not necessarily mean a right one).


Table V. Pitzer Theory Best-Fit Coefficients for Activity and Osmotic Coefticients of MS04 Salts"

Salt Cl1 Cl2 130 131 132 cix b (Jc 104m1• m/

ZnS04 0.902 17.8 -0.170 2.54 -128.3 0.225 0.008 ),

±0.327 ±0.7 ±0.448 ±0.26 ± 8.8 ±0.208 MgSO, 1.396 16.6 0.226 3.27 -77.8 0.035 0.007 4,

±0.082 ±0.7 ±0.042 ±0.12 ±5.1 ±0.021 CaS04 1.4 18.2 -6.75 9.91 -99.3 0.003 2,

±2.8 ±16.71 ±24.06 :!:4.6 SrS04 1.4 17 0.2 50. ) -173.1 0.005 0.7,

:!:27.9 :::47.2

• Values in itaJics are estimated instead of being derived by a best-fit procedure. h CZ.x = 3Ctixl2. "Standard deviation for In"'{,:;; in the region between m1 and m2.

dRange of molality in which the best-tit procedure was applied.

1.52

2.50

0.013

5 X I0-4

For ZnSO4, the emfs measured at m = l.516 and m = l.050X w- 2 (the reference solution) yield w = 22.6±0.5; the hydration state for zinc in the water-saturated NPOE phase should be two or three water molecules higher than for magnesium, thus 14 or 15.

The values of ln ''fa shown in the last columns of the Table I, Table Il, and Table III have been recalculated from the corresponding In 'Y:!:Chydr> in the last column but one of the corresponding tables, assuming w = 22.6 and 20.45 for ZnSO4 and MgSO4, respectively, according to the crude experimental results rather than to our own opinion about the probable integer values of w. For CaSO4, we assumed w = 20 (however, the correction tenns are very small).

Pitzer parameters able to reproduce our final values are given in Table V. The residuai uncertainty of w causes our values of In 'Y:!: referred to the anhydrous salt to be possibly incorrect by 0.017 in 2.5m MgSO4, and by 0.007 in 1.5m ZnSO4, for a 0.5 mole errar in w. On dilution, this possible inaccuracy decreases rapidly, to become for both salts ca. 0.005 at l m, ca. 0.002 at 0.5m, and ca. 0.0005 at 0. lm. For m > I, it should be preferable to use the originai parameters of Ref. 6 rather than those of Table V, provided the values of In 'Y = are then decreased, by -0.112 for ZnSO4 and -0.015 for MgSO4, to correct for the shift of the reference points.

3.2. Comparison with Other Results

The originai evaluations of Pitzer and Mayorga of the activity coefficients of MS04 salts(4·6l are in very good agreement with the present ones for CaSO4 (their 'Y :!: are only 0.5% lower, practically indistinguishable within

EMF of Llquid Membrane Cells 809

the respective uncertainty), and in reasonable agreement for MgSO4 (their 'Y~ are 1.5% higher). On the contrary, for ZnSO4 the corresponding 'Y~ are 11.9% higher than the present ones.(6l This discrepancy is much larger than the most unfavorable estimate for the range of uncertainty of 'Y1 (3% maximum, even if the results frorn Eq. (3) are the only ones considered; but the different methods compared with one another indicate 0.5% or rather less), and arises from the reasons considered in Section 1.1. Our values agree with those from Cowperthwaite and La Mer<2l rather than with the previous and subsequent ones. To summarize the different values assigned to the activity coefficients of this controversia! salt during the years, we consider that at O.O! mol-kg- 1 Bray (1927) reported "Y=. = 0.421,(25

> Cowperthwaite and La Mer (193 I) proposed 0.387,(2

> which became 0.376 once re-extrapolated by Malatesta and Rotunno ( I 984),<5l and Pitzer calculated 0.412 ( 1971) or 0.418 (l974).(4·6l According to the present data, the value is, instead, "Y:1:. = 0.373±0.002. Bray's,l25l Cowperthwaite and La Mer's,m and Pitzer's values(4·61 of In 'Y:1:. only need to be shifted by -0.12 1, -0.037, and -0.l00(4l or -0.11 2/ 6> respectively, to reach very good agreement with our own data (Fig. 4). The problem of the activity coefficients of ZnSO4 has recently concemed Hamer and Wu, who re-exarnined the literature data and obtained an average value, 'Y:1:. = 0.0523, as the probable one for the saturated solution of ZnSO4 (m = 3.5856 mol-kg- 1).(261 This value is not far different from those from e.g., Bray (0.0532)<25·26> or Pitzer (0.0531 from the parameters of Ref. 6, slightly extended outside their range). But, once shifted, Bray's and Pitzer's values becorne 0.047 1 and 0.0475, respectively, still close to one another but appreciably far from 0.0523. The corresponding correction for the systematic error should be introduced in Ref. 26. (Similar problems could also affect the saturated solution of CdSO4, in the same paper. We pian to examine this salt in the future).

The parameters of Table V can also be used to calculate the osmotic coefficients (Eq. (5) of Ref. 6; be sure to use C;t,x = 2Ctxl3 in the calculation), even though the results are not recommended for m > 1, owing to the possibly inexact correction far the ion hydration. Far magnesium sulfate, the new values far <I> are in reasonable agreement with those reported in Ref. 4, those from Robinson and Stokes,127l and those calculated by means of the originai parameters of Pitzer and Mayorga in the respective concentration ranges.l6l We expected that also for ZnSO4 the osmotic coefficients would be practically unchanged except for a slightly different trend in the more dilute solutions, just as needed to alter "Y°zc by the necessary amount. lt was not so; quite surprisingly, our values for <I> look different from those of Pitzer and Mayorga(6> (Fig. 5). The results do not change significantly even if other parameter sets derived from our emf data are used, e.g., a set in which In ì'~ has been treated as an additional adjustable pararneter instead of as an


0.9 ZnSO

4

o.a

f2J 0.7

IPBE

0.6 DHLL

0.5

, o·• 10·3

Fig. 5. Smoothed osmotic coefficients for ZnSO4 according to Pitzer and Mayorga (narrow line) or to Table V (bold fine), compared with isopiestic data from Robinson and Jones (diamonds) and those from ice-points from Brown and Prue (circles). The limiting law for <I> (dash-and-dot line) and an IPBE curve (a = 0.36 nm, dashed line) are shown for comparison.

external datum, or cx I and a 2 have been set at a particular value ( 1.4 to 0.9 for a 1, 16 to 20 for cx2) different from the respective best-fit values, or if some or all of the points below 5 X 10-4m have been excluded from the leastsquares processing. A comparison with the originai experimental data sources reveals that no significant discrepancies are actually involved, except for the two most dilute isopiestic points from Robinson and Jones,<28> and the two most dilute freezing-points from Brown and Prue.<29> It is likely that the relevant series of measurements were affected by higher errors at the lowest concentrations examined. The values of Brown and Prue referring to the freezing temperature were set at 298.15 K by shifting them by -0.008 in accordance with Pitzer (<!>25-c - <f>o•c = -0.008±:0.008),<4> even though -0.011 was in better agreement with our curve. Once the two points for m = 0.00450 and m = 0.00612 which fall off significantly have been discarded, the remaining twenty-two points from Brown and Prue agree very well with our 4> curve, and also seem to outline the slow hump our parameters draw between 0.01 and 0.lm (Fig. 5).

3.3. Association or Nonassociation?

Figure 6 shows the trend in the dilute region, compared with the limiting law. for the activity coefficients of the salts considered. The negative devia-

EMF of Llquid Membrane Cells 811

0.6 In y •• In y t(DHLL)

0.4

... ~

0.2 ~/ 6 ....

<) . .. 8 o :

o 1--.i;i..,.'1lt~~~=o---;;-------..i~,·r=0;..:·-.. ----+

--:~Ol-~ -i~i ,.· -. . .. • __ .a . . .• -· m

Fig. 6. The activity coefficients of 2-2 salts compared with the limiting law in the dilute region. Experimental values, ZnSO4 (filled circles), CaSO4 (squares), SrSO, (open circles). and MgSO4 (diamonds). Calculated values, IPBE, a = 0.36 nm (lower), a = 0.41 nm (upper).

tions typical of the high-charge systems,<7Hl are clearly visible on the left. Some indicative curves that represent (within the lPBE approximation) the behavior of a hypothetical population of + 2 and -2 charged hard spheres. in a homogeneous dielectric fluid having the same physical characteristics as bulk water, are reported for comparison. The curves of the Bjerrum theory (not shown) are able to reproduce the experimental results Lo the same good leve! of approximation as IPBE, while the Mayer theory is slightly less satisfactory.

Bjenum's theory introduces afictitious association constant. whose value ranges between 230 and 410 for the salts bere considered. The choice for the distance of fonnation of the ion-pairs in the originai theory, d = q (q denoting Bjerrum's criticai distance) is in part arbitrary,<30> and a different selection for d (e.g., 2q, 3q/2, 2q/3, etc.) causes the value of the association constant and the subsidiary quantities (ion-pair concentrations, acti vity coefficients of the free ions, beat of formation of the ion-pairs, etc.) to vary markedly.<31

> However, these quantities cannot be measured, and Guggenheim and others bave pointed out that the thennodynamic quantities that actually can be measured (mean activity coefficients, osmotic coefficients, relative apparent molar enthalpies or volumes, etc.) depend on a, but are largely independent of d.<30-31

> In other words, the ion pair of the Bjerrum theory is a short-cut to solve the problem (so far unresolved in exact terms) of the calculation of the radiai functions of the primitive model for a high-charge


electrolyte, nota way to verify whether ion-associations occur physically or not; in fact, IPBE and other sophisticated treatments of the primitive model do not need an association hypothesis to reproduce the same curves_Pb-c.30-33

>

Unfortunately, a real association would lead to the same results as regards the shape of the experimental curves. An association reaction with a constant of the same arder of magnitude as those of the Bjerrum theory for the same solution, cannot emerge from the sea of electrostatic diffuse interactions.

Therefore, no clear conclusion about possible associations in ZnS04, MgS04, CaSO4, and SrSO4 solutions can be reached from the shape of the curves of In 'Y ±· Our own point of view, however. is that these salts are really strong electrolytes. Contrary to zinc chloride, the metal sulfates considered here, as well as lanthanum chloride, tris(ethylenediamine) cobalt (III) chloride, tris(ethylenediamine)cobalt (Ili) sulfate, potassium and alkaline-eanh ferricyanides, etc., previously examined, have not been found to yield any appreciable drift or memory effect, even in the most concentrated solutions explored. An ion-pair, once formed, has a Iower charge and a higher hindrance than a free ion and by consequence has to be less hydrophilic. Thus, an ionpair bearing an electric charge should react with the relevant membrane, viz.

2 MX+ (aq)+M2 + (org) ➔ 2 MX+ (org) + M2 + (aq) (cation exchange membranes)

2 AY-(aq) + Y2-(org) ➔ 2 AY-(org) + Y1 -(aq) (anion exchange membranes)

hence causing the surf ace compositions and the emfs to drift. Stable potentials should therefore exclude any presence of paired ions, in asymmetrical electrolyte solutions. Por symmetrical electrolytes, this principle need not apply and, therefore, no definite conclusion can be drawn from the lack of drifts and memory effects in the solutions of the alkaline earth and zinc sulfates. However, Malatesta and Carrara also observed drifts and memory effects in concentrated solutions of 1-1 electrolytes, specifically for HCl, using the cell'34l

Ag, AgCIIHCl(m)ITDACl in C2H5Br or C2HCl31HCl(m0 )1AgCl, Ag

and far KCI, using the analogous celi with a membrane of TDACI dissolved in NPOE.<7al If these 1-1 electrolytes that have no particular tendency to associate in water, are able to enter the organic phase where they are forced to exist principally as ion pairs, why should the MSO4 electrolytes not do so if they are also associated in water? It would seem to be a really improbable occurrence. Of course, one might assume that the M2+sol- pairs fonned in water are just unable to enter the NPOE + M(TFPBh and (TDA)iSO4CH2O)x phases; but that is an additional conjecture, and does not seem to be particularly convincing.

EMF or Llquld Membrane Cells 813

By way of contrast, we are also obliged to admit that the sinuous trend of the osmotic coefficient of ZnS04 in Fig. 5 does not seem to be likely to arise from the mere coulombic and hard-sphere interactions of the primitive model. The relevant questions are to be considered as stili open at the present stage of research, although with a bit more infonnation available.

4. CONCLUSION

The activity coefficients of the 2-2 salts examined agree, in the dilute region, with the predictions of IPBE and Mayer's theory fora population of rigid spheres bearing charges +2 and -2 in a medium having the same bulk properties as water, for hard-core diameters of 0.35-0.37 nm (ZnSQ4), 0.40-0.42 nm (MgS04), and 0.38--0.40 nm (CaS04, SrS04). For ZnS04 the deviation from the limiting law, in the direction opposite to the Debye-Huckel predictions, is actually as great as Cowpenhwaite and La Mer reponed, or even slightly greater. Thus, Cowperthwaite and La Mer were fundamentally right, in spi te of the anomalies Pitzer<4> and Malatesta and Rotunno<S> noticed in their data. All subsequent revisions of the activity coefficients of this salt, except the work of Malatesta and Rotunno, <5l worsened instead of ameliorating the situation.

There is no clear evidence that the ions of the 2-2 salts examined associate, even though Bjerrum's ion-pair theory, or any free matching of a Debye-Hi.ickel-like formula with an ad hoc best-fit association constant, may be used in the dilute region to reproduce the data with a lower computational effort than using Mayer's theory or IPBE. Occasionai values of these bestfit association constants have been reported in the literature as the formation constants of the corresponding metal-ion complexes, e.g., log K = 2.25 for MgS04 and log K = 2.38 for ZnS04.t35·36l Their physical meaning is, however, very indefinite.

The concentrated regions reveal that zinc, magnesium, and sulfate ions are appreciably hydrated when they enter the membrane materials. It seems likely that magnesium exists in the water-saturated NPOE as rMg(H20)12]

2•,

and zinc as [Zn(H20)14 )2

• or perhaps [Zn(H20)15)2+, while in the liquid salt (TDA)iS04CH20), the sulfate should be an octahydrated ion. These figures, however, still require some corroboration from improved chemical analysis for the composition ofthe membranes. Moreover, it is likely that the hydration numbers of the ions in the membranes have to change when the membranes are allowed to equilibrate with very concentrated solutions of ZnS04 or MgS04, made of rolling hydrated ions without any appreciable residual amount of free water. In fa.et the hydration state of the ions in these aqueous phases does not reach the high values found in the membranes, since around 25°C a solution saturated by MgS04 • 7H20, or by ZnS04 · 7H20, contains


approximately 19, or 16 (15.5),'26l moles of water per mole of the respective anhydrous salt.

ACKNOWLEDGMENTS

This work was supported by the MURST and the ltalian National Council of Research (CNR).

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EMF of Liquid Membrane Cells 8IS

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activity and osmotic coefficients from the emf of liquid membrane cells. vi-znso4, mgso4, caso4, and...

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