Emfs of galvanic cells with liquid junctions

127

J. Electroanal. Chem., 308 (1991) 127-149 Elsevier Sequoia !%A., Lausanne

Emfs of galvanic cells with liquid junctions Jilska M. Perera and Peter T. McTigue School of Chemistty, (Received

29 January

The University of Melbourne, ParkviNe 3052 (Australia) 1991)

Abstract The emfs of the following

cells, which include Wagner

cells, have been both measured

and calculated

at 298 K: (i) Ag,AgCl IMCl(aq) INCl(aq) )AgCl,Ag; with cations M+; N+ = Li+, Na+ or K+ (ii) Ag,AgCl 1NaCl(aq) 1NaBr(aq) 1AgBr,Ag (iii) Ag,AgCl IKCl(aq) INaBr(aq) IAgBr,Ag (iv) Ag,AgCl INaCl(aq) INaCl(aq-methanol) IAgCl,Ag (v) Ag,AgCl 1MCl(aq) 1NCl(aq-methanol) IAgCl,Ag; M +; N + = Na+ or K + (vi) Ag,AgX INaX JNaY(aq-methanol) JAgY,Ag; X-; Y- = Cl- or BrAll experimental data were obtained using a flowing liquid junction with electrolyte concentrations < 0.10 mol kgg’; for all cells the half-cell electrolyte concentration ratio was varied over two powers of ten. Cell emfs were calculated using available transport numbers, Gibbs’ energies of transfer and other thermodynamic information which, in the case of Wagner cells, included the ionic relative solvation intensities. The Henderson method was used to determine the allowed variations of electrolyte and solvent composition in the junction region. Equations are given for calculating the emfs of the various cells, and the conditions needed for successful emf predictions are outlined.

INTRODUCTION

Our interest lies in the use of cells with liquid junctions for the determination of useful thermodynamic information; we therefore treat the junction as an integral part of the cell by combining the liquid junction potential (ljp) with the electrode potential difference, thereby eliminating all reference to single ion activity coefficients. The appropriate theoretical formulations have already been discussed by us at some length [l-3] in the case of Wagner cells containing a single electrolyte. In this work, we extend and generalise the treatment used in our earlier papers to develop a procedure for the extraction of thermodynamic data from emf measurements on various types of cells with liquid junctions and, just as importantly, to identify the limitations of the method. We have already established [l] that flowing liquid junctions have the advantage of combining rapid attainment of a constant 0022-0728/91/$03.50

0 1991 - Elsevier Sequoia

S.A.

128

steady state potential (1-5 s) with high reproducibility and have therefore our experimental studies to cells with junctions formed in this way.

THEORETICAL

restricted

BACKGROUND

The emf of a galvanic cell with a liquid junction is most conveniently expressed as the sum of an ljp and an electrode potential difference. This combination always leads to an expression for the cell emf that contains no single ion activity coefficients. We shall first present the basic equations for the emfs of the various cells studied in this work derived from a quite general expression for the ljp that is obtained using either quasi-static or irreversible thermodynamics [1,4]:

where E, is the Galvani potential difference between the right and left hand sides of the liquid junction (42 - 4,) F is the Faraday constant, t, the Hittorf transport number, z, the valence and ~1, the chemical potential of ion i. rk is the reduced transport number and pk the chemical potential of solvent component k. The second term on the right hand side of (1) takes account of the contribution to the ljp of ion-assisted solvent transport in Wagner cells. However, if the solvent is the same in both half-cells, this term is zero provided the electrolyte concentration is not too high. We shall now develop the equations for the emfs of the various cells used in this work. (1) The salvation potential The ljp of a Wagner E, = Elj + Es

cell can be expressed

in the form: (2)

E, is directly related to the Gibbs energy change of the processes occurring at the liquid junction that are coupled to the reversible flow of charge through the cell. Eij is the component of E, arising solely from the transport of ions, as given by the first term in eqn. (1) while Es, the solvation potential, arises from the ion-coupled transport of solvent through the junction and is given by the second term in eqn. (1). All single ion activity coefficients are contained in the Eij term. The most general equation [l] for Es in a cell in whrch the half-cells contain a binary solvent with components W and Z is:

129

where l;, z, are the electrical

transport

number

and valence

of the ion i, and

d ln fzxz g=

dln

(4

X,

where xz, fit are the mole fraction

and rational activity coefficient of solvent component Z. X, is the mole fraction of Z in the binary mixed solvent as previously defined [2]. Note that the expression for Es is the sum of individual ionic terms one for each ion that is contributing to the transport of solvent through the liquid is the relative solvation intensity (RSI) for ion i [1,2]: junction; I,, Iwz(i)

=1,(i)

--I,(i)

(5)

where

Z,(i) =Nw/Xw

(6)

NW being the dynamic W-solvation number of ion i. An important property of the solvation potential is its variation with electrolyte concentration. Feakins and Lorimer [5] showed that, in a Wagner cell with equal electrolyte concentrations in the half-cells, the solvation potential varied roughly as the square root of the ionic strength. More recently [l], we have shown that the concentration dependence of Es can be expressed in terms of the concentration dependence of the ionic RSIs and one possible form of this dependence was discussed and its use demonstrated. There were no data on the concentration dependence of individual RSIs and it was assumed for convenience that the chloride ion RSI was concentration invariant. Until appropriate concentration dependence data on RSIs become available it seems preferable to use a simple empirical equation, similar to that proposed by Feakins and Lorimer [5], to take account of the variation of Es with concentration, viz: E,=E;(l-~‘rn”~)

(7)

where E,” is the limiting value of Es at infinite dilution, u’ is a parameter that applies to a particular electrolyte and a particular solvent system and m is the electrolyte molality. In cases where there is a difference in molality between the two half-cells an average value of the square root of the molality, m1’2, is taken. As will be shown later, eqn. (7) is moderately successful in fitting the observed concentration dependence of our Es data. It follows from eqn. (7) that we can write:

Es

=

(1 - a’m’/2)Eso

= (1 _ u’m’/2

gZGz(i)dXz )RT F x/;‘$ i

where ZGz is the limiting value of the RSI currently tabulated [2,3] have been obtained

at infinite dilution. The Z values by combining solvent-electrolyte

130

diffusion generate

ratios with the ES0 data from ref. 2. Thus, when these Z values are used to solvation potentials, they give ES0 which must be converted to ES via eqn.

(7). (2) Emfs of cells with different solvents in each half-cell (a) Junctions with two different binary electrolytes and a common ion; AC(1, ) I BC(2, W-Z) Cells (v) and (vi), which are based on this type of junction, are the most complex cells studied in this work. The expression for the ljp obtained by using eqn. (1) is:

o is the standard chemical potential of ion i. where a, is the activity of ion i and EL, is the standard chemical potential difference of ion C across the liquid 4-G junction. ES is the solvation potential as defined in eqn. (3) and given by eqn. (8). When this expression in eqn. (9) is combined with the electrode potential difference to give an expression for the cell emf the first two terms, which contain single ionic quantities, vanish. The emfs of cells (v) and (vi) are given by eqns. (10) and (11): 1 E,=

-f

j2tM 1

d/&.,-F

2 j1

(1

2tMgZ&(M+) 1

‘t,,gZ&(Cl-) j1

-

2RT

t,

G-G,C,

dX,

-

2 (j

7

1 tM

+ j2tNgZ&(N+)

2 d

In

aMCl

+

j1

tN

d

In

aNCl

dX,

1

dX,

(10)

where M and N represent two different singly-charged cations, e.g. alkali metal ions. The final term in the equation is the sum of three separate ionic contributions to the solvation potential from the anion and two different cations.

& 2t~dPNaY +j2'X

E,i = EYO-(aq)lAgy,~g - EXO~~aq)lA~.Ag +

1

+

2RT

Fin-

aNaY NaY

aNaX

(1

2tNagZ&(Na+) 1

-

2tXgZ.&(X-) /1

where X-

dPNaX

and Y-

dX,

dX,

1

dX,

represent

- j2tygZ&(Y-)

NaX

(1 - u’rnl”)

(11)

i different

singly charged

anions,

e.g. Cl-

and Br-.

131

(b) Junctions formed from a single binary electrolyte; AC(I, W) 1AC(2, W-Z) This type of junction appears in cell (iv) and the emf of this cell is given by: Ei, = - $j2tM 1

2RT 2 - 7 / tM

d&C,

d

In

aMCl

1

v

2tMgl$Z(M+)

dXz - /2t,-,gl$z(C1P)

1

1

where M+ represents

a singly-charged

cation,

dXz

(1 - a’m’j2) i e.g an alkali metal ion.

(12)

(3) Cells with aqueous solutions in both half-cells (a) Junctions formed from two different electrolytes with one common ion; AC(l) 1BC(2) This is a well known situation where ES equals zero. Equation (1) then leads to the familiar equation for cell (i): Ei = - F

(/‘tM

d In aMC, + S2tN d In aNc,}

1

(13)

1

where M, N represent Li, Na or K as appropriate and aMC, is the mean activity electrolyte MCI. Similarly, for cell (ii), the appropriate equation is: Eii = E o - yin*

+ 7

(/‘t,,

aN.Cl

d In a,,,,

1

where E o is the standard

+ 12tg, d In aNaBr 1

)

of

(14)

emf of the cell.

(6) Two different electrolytes with no common ion; AC(I) 1BD(2) This is a more complex situation and it is worth noting the expression for the ljp, since we are unaware that it has been recorded elsewhere; again, ES is equal to zero. The general ljp equation for this situation is:

(15) For cell (iii) the resultant

expression

for the emf is:

mK d ln Yxcl + j2tNaBr%

d In

YNaCl

1 --

2yln

YN~B~(Z) YNaCl

t2)

(16)

132

where tKc, = t, + t,, and ykc, is the mean molal activity coefficient of KCl. yNaBr(2) is the mean activity coefficient of NaBr in cell compartment 2 and yNac,(2) is the tracer activity coefficient of NaCl in cell compartment 2. Although the final term in this equation is in principle measurable, it has been assumed in all our calculations to be zero, bearing in mind that at low ionic strengths the activity coefficients of NaCl and NaBr are similar [6] over a wide concentration range. EXPERIMENTAL

Chemicals used AR grade NaCl and KC1 were twice recrystallised from triply-distilled water. LiCl was twice recrystallised from an aqueous methanol solution. Merck ultrapure NaBr was used without further purification. The salts were all dried under vacuum; LiCl was stored in a dry box and all other salts in a vacuum desiccator. Water was either triply-distilled or obtained from a Milli-Q filtration and ion-exchange system. Methanol was twice fractionated over 3A molecular sieves. All solutions were made up by weight. Potentiometric measurements Silver/silver chloride [7] and silver/silver bromide electrodes [3] were prepared as previously reported and electrodes were rejected if their bias potentials were > +0.02 mV. The galvanic cells and flowing liquid junctions were set up as previously described [I]. THEORETICAL

CALCULATIONS

The calculation of emfs of cells (i)-(vi) requires the evaluation of a number of integrals. The allowed compositions through the liquid junction were determined using the Henderson assumption [8] that the solution at every point in the junction can be represented as having been formed by the mixing of the solutions on either side of the boundary in a definite volume ratio. Thus, once this mixing fraction, or ratio, has been decided, both the solvent mole fraction and electrolyte molality are fixed for the particular mixing fraction chosen. The limits of integration then cover the whole range of mixing fractions from 0 to 1 in the transition from one side of the junction to the other. Details of the calculation of the various components of the integrals, over the whole of the methanol-water composition range, are as follows: Ionic activities: Given that ionic molalities are obtained at any mixing fraction from the Henderson assumption, the activities are calculated using extended Debye-Htickel theory [l] with the same distance of closest approach as that in pure water. Where appropriate, ion-pairing may be allowed for using known ion association constants, although this was not necessary in any of the calculations reported in this work. Zonic transport numbers: Limiting molar conductivities of ions over the whole solvent composition range are given in refs. 1 and 3 as polynomials in the methanol

133

cell (iii), Table 5

cell (i) Table 1

q

-2

0

-1 lo&

1

2

[wI~ll

Fig. 1. Differences between calculated and experimental emfs for data in Table 1 -(Cell i), Table 4 (Cell ii) and Table 5 (Cell iii). The abscissa gives the logarithm to base ten of the ratio of the electrolyte concentrations in left hand and right hand half-cells.

mole fraction. These can be used, along with a knowledge of the ionic composition, the ionic strength and the Stokes-Onsager conductivity equation, together with any appropriate ion-pairing constants, to calculate the transport number of the ion at the required solution composition. Standard Gibbs energies of transfer: These are recorded as polynomials in the methanol mole fraction in refs. 1 and 3. Solvent non-ideality term, g: This is given as a polynomial in the methanol mole fraction in ref. 1. It has been assumed that its value is independent of the electrolyte concentration. Relative solvation intensities, I: These quantities are tabulated in refs. 2 and 3. Constants for predicting the concentration dependence of the solvation potential, o’: Recommended values have been calculated using data in refs. 1 and 3 and are given in Table 11 below. In all cases, the various terms comprising an integrand were evaluated as a function of mixing fraction and values of the integrand could then be calculated at any required mixing fraction. The data were then used to carry out the required integration numerically. Equations used for cells (i) to (vi) respectively were eqns. (13) (14), (16) (12), (10) and (11). The results of the calculations are shown in the various Tables, including the average differences between calculated and experimental emfs. For types (i) and (ii) cells the average difference between calculated and experimental emfs of about 0.2 mV appears to be random as it is comparable with the

134 TABLE 1 Emf of cell (i): Ag,AgCI jNaCl(aq) IKCl(ag) IAgCl,Ag, at 298 K at various electrolyte molalities. Ecarc obtained using eqn. (13) with M = Na, N = K. Distance of closest approach used for Ecalc = 0.43 nm. (AE 1ay= 0.21 mV where AE = .Ecalc- E,,,,

mNact/mol kg-’

m,,,/mol

0.00509 0.00509 0.00509 0.00509 0.00509 0.00509 0.00993 0.00993 0.01011 0.01005 0.00993 0.00993 0.00993 0.00993 0.01985 0.01985 0.01985 0.01985 0.01985 0.01985 0.03034 0.03034 0.03034 0.03034 0.03034 0.03034 0.04085 0.04085 0.04085 0.04085 0.04085 0.04085 0.05193 0.05193 0.05193 0.05193 0.05193 0.05193

0.00506 0.01007 0.02037 0.03033 0.04031 0.05030 0.00516 0.01047 0.01003 0.01042 0.02020 0.03041 0.04064 0.05064 0.00513 0.01014 0.02004 0.03046 0.04023 0.05045 0.00513 0.01028 0.02036 0.03022 0.04053 0.05063 0.00511 0.01028 0.02023 0.03020 0.04007 0.05037 0.00510 0.01029 0.02014 0.03022 0.04051 0.05063

kg-’

Eel1 /mV - 4.02 - 19.60 - 35.67 -44.51 - 51.28 - 55.48 9.82 - 5.58 - 4.24 - 5.11 - 20.09 - 29.26 - 35.68 - 40.65 24.42 9.91 - 4.73 -13.79 - 19.77 - 24.10 33.04 18.41 4.02 - 4.41 - 10.77 - 15.53 38.91 24.42 10.35 1.86 - 3.91 - 8.86 43.73 29.28 15.29 6.77 0.64 - 3.99

GkJmV - 4.36 19.64 35.39 44.39 50.76 55.74 9.65 -5.69 - 4.37 -5.33 - 20.09 -29.12 - 35.45 -40.32 24.31 9.75 -4.81 - 13.82 - 19.82 - 24.7s 32.68 18.73 3.79 - 4.57 - 10.82 - 15.56 38.62 24.31 10.05 1.58 - 4.29 9.12 43.26 28.73 15.45 6.56 0.45 - 4.21

-

absolute accuracy of the emf measurements and shows no significant variation with the half-cell electrolyte concentration ratio; agreement with experiment must therefore be rated as very good for most practical purposes. Cell (iii> contains four different ions and Fig. 1 shows the difference as a function of the concentration

135 TABLE 2 Emf of cell (i): Ag,AgCl ILiClfaq) INaCl(aq) jAgCl,Ag, at 298 K at various electrolyte molahties. &tc obtained using eqn. (13) with M = Li, N = Na. D tstance of closest approach used for E_,, = 0.47 nm.

1AhE1Bv= 0.25 mV where A E = Ecalc- E,II m tic, /mol kg-’

m,,t/mof

0.00541 0.00541 0.00541 0.00541 0.00541 0.00541 0.01033 0.01033 0.01219 0.01228 0.01033 0.01033 0.01033 0.01033 0.02145 0.02145 0.02145 0.02145 0.02145 0.02145 0.03053 0.03053 0.03053 0.03053 0.03053 0.03053 0.04358 0.04358 0.04358 0.04358 0.04358 0.04358 0.05209 0.05209 0.05209 0.05209 0.05209 0.05209

0.00528 0.00996 0.01989 0.02972 0.03949 0.04969 0.00549 0.01033 0.01011 0.01oQ5 0.02042 0.03002 0.03976 0.04%5 0.00540 0.01018 0.02060 0.03010 0.04014 0.05041 0.00551 0.01026 0.02013 0.03037 0.03931 0.05094 0.00553 0.01091 0.02160 0.03292 0.04291 0.05429 0.00576 0.01087 0.02237 0.03436 0.04269 0.05419

kg-’

Ecett/mV

Gale /mV

-1.97 - 13.74 - 26.68 - 34.25 - 39.12 - 43.76 8.74 - 2.60 - 0.79 -1.15 - 14.83 - 21.65 -26.81 - 30.73 21.53 10.51 - 1.88 - 8.41 - 13.51 - 17.56 27.16 16.32 4.62 - 2.38 - 6.89 - 11.41 32.85 21.18 9.46 2.33 - 2.28 - 6.31 34.98 24.16 11.83 4.49 0.81 - 3.29

- 2.07 - 13.50 -25.98 -33.19 - 38.36 - 42.42 8.65 - 2.55 - 0.75 - 0.9s - 14.64 -21.39 - 26.57 - 30.36 21.35 10.31 - 1.89 - 8.47 - 13.47 - 17.46 26.73 16.16 5.54 - 2.53 - 6.96 - 11.40 32.53 21.11 9.28 2.11 - 2.39 -6.37 34.61 23.89 11.52 4.35 0.69 - 3.33

ratio of the electrolytes in the half-cells and it is clear that the difference depends strongly on this quantity, so that there is good agreement between calculated and experimental values in the special case where the concentrations in the two half-cells are the same. The essentially random differences for cells of type (i) are shown on

136 TABLE 3 Emf of cell (i): Ag,AgCl ILiCl(aq) }KCl(aq) JAgCl,Ag, at 298 K at various electrolyte molaiities. Ecalc obtained using eon. (13) with M = Li, N = K. Distance of closest approach used for Ecalc= 0.45 nm. I A&w = 0.22 mV where AE = Ecalc- E,,, m,c,/mol 0.00543 0.00543 0.00543 0.00543 0.00543 0.00543 0.01047 0.01047 0.01047 0.01047 0.01047 0.01047 0.02053 0.02053 0.02053 0.02053 0.02053 0.02053 0.03045 0.03045 0.03045 0.0304s 0.03045 0.0304s 0.04074 0.04074 0.04074 0.04074 0.04127 0.04127 0.05097 0.05097 0.05097 0.05097 0.05097 0.05097

kg-’

m,t/mol 0.00588 0.01072 0.02010 0.03006 0.04054 0.05061 0.00519 0.00992 0.02011 0.03015 0.03992 0.04998 0.00520 0.01049 0.02063 0.03314 0.0399s 0.0549 0.00509 0.01017 0.02058 0.03047 0.04010 0.05014 0.00514 0.01025 0.01996 0.03100 0.04001 0.05050 0.00495 0.01037 0.02019 0.03031 0.~84 0.05050

kg-’

Gtt /mV

Gt,/mV

- 8.57 - 21.48 - 35.28 - 44.02 - 50.53 - 55.58 7.12 - 6.02 - 20.69 - 29.19 - 35.31 -40.18 19.81 6.21 -7.28 - 16.95 - 20.80 - 25.65 27.36 14.21 0.60 - 7.09 - 12.53 - 17.09 32.41 19.68 6.75 -1.52 - 6.62 - 11.15 36.97 23.29 10.79 3.01 - 2.7% - 6.75

- 8.67 -21.32 - 34.84 - 43.64 - so.22 - 55.07 7.06 -5.99 - 20.63 - 29.24 - 35.16 - 40.04 19.72 6.16 - 7.30 - 16.94 - 20.79 - 25.92 27.17 14.19 0.49 - 7.29 - 12.80 - 17.32 32.03 19.20 6.60 - 1.98 -6.73 -11.35 36.48 22.93 10.22 2.73 - 3.05 - 7.21

the same diagram with the data from Table 1. Data from Table 3 for cell type (ii) are also shown on the diagram and again there is no systematic variation of the deviation with the half-cell electrolyte concentration ratio; there is, however, a small constant positive component (ca. 0.1 mV) in the error which may be due to a sma51 error in the potential of the silver/silver bromide electrode.

137 TABLE 4 Emf of cell (ii): Ag,AgCl INaCl(aq) INaBr(aq) IAgBr, Ag, at 298 K at various electrolyte molalities. Ecalc obtained using eon. (14). Distance of closest approach used for Ecalc= 0.43 nm. 1A E jav = 0.32 mV where AE = &,I, - &II mNaCl

/“‘ol

0.00599 0.01126 0.02026 0.03341 0.04185 0.05713 0.00560 0.01127 0.02214 0.03259 0.04216 0.05361 0.00580 0.01064 0.02170 0.03163 0.04278 0.04988 0.00600 0.01064 0.02119 0.03177 0.04147 0.04986 0.006262 0.01083 0.02058 0.03122 0.04186 0.05116 0.00474 0.00959 0.02108 0.02607 0.03900 0.05158

k

- ’

mNaBr/mol

0.00498 0.00498 0.00498 0.00498 0.00498 0.00498 0.01037 0.01037 0.01037 0.01037 0.01037 0.01037 0.02015 0.02015 0.02015 0.02015 0.02015 0.02015 0.03035 0.03035 0.03035 0.03035 0.03035 0.03035 0.04046 0.04046 0.04046 0.04046 0.04046 0.04046 0.05126 0.05126 0.05126 0.05126 0.05126 0.05126

kg-’

-%,I, /mV

Ecalc /mV

-

-

147.55 135.27 123.88 114.44 109.79 104.04 163.04 149.51 136.58 129.18 124.52 120.14 174.91 162.27 149.60 142.35 136.85 133.89 181.76 170.56 157.53 149.78 144.97 141.46 186.12 175.43 163.32 155.48 150.41 145.95 195.79 182.02 167.20 163.12 155.71 150.49

147.06 135.10 123.88 114.55 110.36 104.73 162.43 149.04 136.49 128.70 124.40 120.00 174.26 161.91 149.23 142.23 136.72 133.29 181.30 169.88 157.28 149.77 144.87 141.51 185.69 175.16 163.10 155.37 149.99 146.33 195.76 181.80 166.79 163.01 155.49 150.49

The study of the emfs of Wagner cells of type (iv) has been reported in detail in refs. 1 and 3. Additional data for these cells are here reported in Table 6 for a methanol mole fraction of 0.5; the data refer to the situation in which the half-cell electrolyte concentration ratio varies over two powers of 10. Figure 2 shows that the error in the calculated emfs is about -0.5 mV and is not a strong function of the

138 TABLE 5 Emf of cell (iii): Ag,AgCl 1KCl(aq) jNaBr(aq) jAgBr,Ag, at 298 K at various electrolyte molalities. Ecalc obtained using eqn. (16). Distance of closest approach used for Ecalc= 0.42 nm. In all cases the distance of closest approach was taken as the average of the distances of closest approach of the electrolytes concerned. 1AE 1av= 0.93 mV where AE = /& - I&.,, m,,-,/mol 0.005699 0.01057 0.02056 0.03104 0.04199 0.05247 0.005417 0.01177 0.02062 0.03184 0.04071 0.05157 0.005714 0.01249 0.02109 0.03189 0.04184 0.05396 0.005264 0.01287 0.02017 0.03159 0.04192 0.05098 0.005621 0.01258 0.02049 0.03168 0.04194 0.05167 0.005939 0.01234 0.02007 0.03211 0.04131 0.05084

kg-’

mNaer/molkg-’

-%u/mv

&,I, /mV

0.005403 0.005403 0.005403 0.005403 0.005403 0.005403 0.01048 0.01048 0.01048 0.01048 0.01048 0.01048 0.02001 0.02001 0.02001 0.02001 0.02001 0.02001 0.03042 0.03042 0.03042 0.03042 0.03042 0.03042 0.04061 0.04061 0.04061 0.04061 0.04061 0.04061 0.05027 0.05027 0.05027 0.05027 0.05027 0.05027

- 145.39 -131.57 - 116.77 - 107.19 - 100.33 - 95.11 - 160.46 - 143.72 - 131.42 - 121.70 - 116.47 - 111.14 - 172.81 - 156.15 - 145.08 - 136.07 - 130.39 - 124.80 - 182.92 - 164.24 - 154.95 - 145.48 - 139.55 - 135.29 - 187.43 - 170.50 - 160.57 - 151.42 - 145.73 - 141.31 - 190.36 - 175.07 - 165.36 - 155.36 - 150.29 - 145.64

-

144.98 130.79 115.20 105.41 - 98.19 - 92.85 - 160.90 - 143.51 - 130.65 - 120.54 - 114.78 - 109.20 - 173.74 - 156.59 - 144.89 - 135.49 - 129.26 - 123.38 - 184.32 - 165.08 - 155.19 - 145.17 - 138.78 - 134.34 - 188.91 - 171.74 - 161.13 - 151.51 - 145.26 - 140.57 - 192.11 - 176.65 - 166.16 - 155.88 - 150.31 - 145.69

concentration ratio, although it seems to become larger as the electrolyte concentration in the methanol + water mixture increases. The calculated emfs of Wagner cells (v) and (vi) given in Tables 7-10 show average deviations of several millivolts. The errors are largely systematic and are shown as a function of methanol mole fraction in Fig. 3.

139 TABLE

6

Emf of cell (iv): Ag,AgCl INaCl(aq,l) jNaCl(aq-methanol.2) IAgCl,Ag at 298 K. E,,,, obtained (12). Distance of closest approach used for Ecalc= 0.43 nm. X,,, = 0.5

m,/molkg-'

&I /mv

LdmV

0.005274 0.01453 0.02162 0.02978 0.04203 0.05365

0.005030 0.005030 0.005030 0.005030 0.005030 0.005030

-33.15 -33.15 -4.21 2.30 9.29 14.45

-33.52 -33.52 -4.39 2.12 9.09 14.05

0.004984 0.01026 0.01984 0.03077 0.04026 0.04980

0.01043 0.01043 0.01043 0.01043 0.01043 0.01043

-48.72 -34.48 -21.13 -12.26 -6.91 -2.53

-48.49 -34.66 -21.25 -12.43 -7.08 -2.89

0.005409 0.01033 0.02010 0.03013 0.04040 0.04801

0.02189 0.02189 0.02189 0.02189 0.02189 0.02189

-61.47 -48.73 -35.43 -27.45 -21.63 -18.23

-62.42 -49.30 -35.91 -27.85 -22.07 -18.70

0.005788 0.01013 0.02043 0.03026 0.04050 0.05237

0.05228 0.05228 0.05228 0.05228 0.05228 0.05228

-76.15 -65.54 -51.83 -44.12 -38.55 -33.44

-77.82 -66.53 -52.59 -44.87 -39.20 -34.24

M, /mol

kg-’

-1

0

using eqn.

1

log([1l/r211 Fig. 2. Difference between calculated and experimental emfs for data in Table 6. The abscissa gives the logarithm to base ten of the ratio of the electrolyte concentrations in left hand and right hand half-cells.

140

-0

0.00

0.20

0.40

0.80

0.60

1.00

Methanol mole fraction Fig. 3. Average differences between calculated and experimental Table 10 (Cell vi) as a function of methanol mole fraction.

TABLE

emfs for data in Tables

7 (Cell v) and

7

Emf of cell (v): Ag,AgCl IKCl(aq) (NaCl(aq-methanol) IAgCl,Ag, at 298 K at various electrolyte molalities and various solvent compositions in the right hand half-cell. Ecalc obtained using equation (10) with M = K; N = Na. Distance of closest approach used for Ecalc = 0.47 nm. 1AE lay given for each solvent composition, where AE = E,,,, - I&. e’ = 0.65 X MeOH 0.1

mKCl

mNaCl

mol kg-’

mol kg-’

&I

/mv

0.005735 0.01043 0.02134 0.05302 0.005842 0.01108 0.02150 0.05361 0.005594

0.007668 0.007668 0.007668 0.007668 0.01238 0.01238 0.01238 0.01238 0.02042

-1.25 0.23 15.61 36.44 - 22.02 - 7.98 6.21 26.25 - 33.10

- 7.80\ 5.09 20.69 40.78 - 17.38 - 3.83 10.35 30.09 - 28.40 )

0.01049 0.02093 0.05303 0.005467 0.01062 0.02184 0.05382

0.02042 0.02042 0.02042 0.03250 0.03250 0.03250 0.03250

- 19.75 - 5.24 14.95 - 42.86 - 28.78 - 13.97 5.28

- 15.33 - 0.84 18.84 - 37.98 - 24.36 - 9.49 9.26)

kc

/mV

IAE I,,/mV

4.55

141

TABLE 7 (continued) X MeOH

0.2

0.3

0.4

&II

/mv

&A

/mv

m KC1

mNaCl

molkg-'

molkg-'

IW,,/mV

0.005581 0.01093 0.02124 0.05308 0.005982 0.01077 0.02179 0.05248 0.005679 0.01066 0.02109 0.05276 0.005435 0.01062 0.02164 0.05245

0.006736 0.006736 0.006736 0.006736 0.01169 0.01169 0.01169 0.01169 0.02102 0.02102 0.02102 0.02102 0.04994 0.04994 0.04994 0.04994

-19.08 -3.77 10.83 31.47 -28.89 -16.13 -1.05 18.29 -41.95 -28.61 -14.40 5.19 -59.51 -45.85 -31.47 -13.13

-15.08\ -0.77 13.54 33.61 -24.93 -12.69 2.14 20.84 -37.62 -24.78 -10.77 8.24 -54.99 -41.57 -27.37 -9.66)

3.51

0.005412 0.01054 0.02110 0.05250 0.005742 0.01077 0.02126 0.05306 0.005474 0.01099 0.02103 0.05263 0.005402 0.009747 0.01966 0.04760

0.004392 0.004392 0.004392 0.004392 0.01010 0.01010 0.01010 0.01010 0.02108 0.02108 0.02108 0.02108 0.05031 0.05031 0.05031 0.05031

-18.22 -3.21 12.05 32.65 -34.45 -21.02 -6.63 13.28 -50.50 -35.70 -22.36 -3.24 -67.09 -55.29 -41.05 -23.25

-15.42, -1.25 13.72 34.04 -31.17 -18.27 -4.16 15.03 -46.43) -32.49 -19.45 -0.86 -62.98 -51.33 -37.59 -20.311

2.82

0.004964 0.009861 0.01936 0.04804 0.005496 0.009908 0.01976 0.04806 0.004882 0.009937 0.01938 0.04787 0.005462 0.00999 0.01938 0.04834

0.005254 0.005254 0.005254 0.005254 0.01030 0.01030 0.01030 0.01030 0.02180 0.02180 0.02180 0.02180 0.04607 0.04607 0.04607 0.04607

-31.65 -16.52 2.09 17.79 -43.14 -30.64 -16.34 2.49 -60.21 -45.56 -32.06 -13.62 -71.94 -59.77 -46.43 -28.38

-28.91\ -14.72 -0.55 18.98 -40.15 -28.31 -14.29 3.99 -56.71 -42.77 -29.63 -11.71 -68.27 -56.55 -43.79 -26.241

2.47

(to be continued)

142 TABLE 7 (continued) X MrOH 0.5

0.6

0.7

mKCl

mNaCl

molkg-'

molkg-'

0.005411 0.01097 0.02006 0.05046 0.005259 0.01046 0.02209 0.05259 0.005220 0.01058 0.02286 0.05430 0.005310 0.01097 0.02078 0.05282

J%,II /mV

Lk

/mV

lAEl,,/mV

0.005158 0.005158 0.005158 0.005158 0.01047 0.01047 0.01047 0.01047 0.02206 0.02206 0.02206 0.02206 0.05091 0.05091 0.05091 0.05091

-36.29 -21.22 -9.37 10.53 -51.23 -37.28 -21.43 -3.55 -65.55 -51.06 -35.61 -18.33 -80.44 -66.05 -53.53 -35.15

-34.261 -19.76 -7.15 12.67 -48.59 -34.98 -19.96 -2.26 -62.60' -48.95 -34.02 -17.10 -77.42 -63.51 -51.44 -33.901

2.02

0.005137 0.01069 0.02129 0.05377 0.005587 0.01105 0.02176 0.05254 0.005247 0.01067 0.02208 0.05280 0.005564 0.01112 0.02181 0.05365

0.007063 0.007063 0.007063 0.007063 0.01058 0.01058 0.01058 0.01058 0.02104 0.02104 0.02104 0.02104 0.05039 0.05039 0.05039 0.05039

-49.72 -34.45 -21.42 -2.25 -56.09 -41.87 -28.34 -10.59 -70.36 -55.96 -41.63 -24.63 -84.87 -71.11 -58.11 -40.72

-48.45, -33.89 -19.96 -0.85 -54.43 -41.12 -27.73 -10.06 -68.21} -54.69 -40.81 -24.02 -82.72 -69.64 -57.11 -40.51J

1.12

0.005170 0.01073 0.02136 0.05283 0.005637 0.01075 0.02162 0.05327 0.005522 0.01084 0.02129 0.05271 0.005640 0.01066 0.02126 0.05273

0.005750 0.005750 0.005750 0.005750 0.01136 0.01136 0.01136 0.01136 0.02109 0.02109 0.02109 0.02109 0.05242 0.05242 0.05242 0.05242

-51.12 -36.33 -22.04 -3.61 -62.34 -49.51 -35.39 -17.47 -74.11 -60.58 -47.79 -30.24 -89.66 -77.25 -64.14 -46.88

-50.84) -36.46 -22.66 -4.07 -61.83 -49.51 -36.04 -18.46 -73.32 -60.69 -48.06 -31.03 -89.02 -77.15 -64.54 -48.18J

0.50

143 TABLE

7 (continued)

xmon

mKCI

mbk4

mol kg-’

mot kg-’

0.8

0.006046

0.004937

- 49.32

- SO.661

0.01094

0.004937

- 37.35

- 39.09 - 25.74

0.9

TABLE

E,,, /mV

Ecalc /mV

0.02146

0.004937

- 23.86

0.05270

0.004937

- 5.99

- 7.39

0.006085

0.009702

- 62.15

- 63.02

0.01082

0.009702

- 50.77

- 52.18

0.02121

0.009702

- 37.55

- 39.39

0.05326

0.009702

- 19.31

- 21.68

0.005497

0.02039

- 77.69

- 78.07

0.01115

0.02039

-63.71

- 65.06

0.02151

0.02039

-51.56

- 53.03

0.05290

0.02039

- 34.32

- 36.51

0.005547

0.04790

- 92.57

- 92.86

0.01067

0.04790

- 79.56

- 80.86

0.02164

0.04790

- 66.24

- 68.19

0.05292

0.04790

- 49.36

- 52.40)

0.005202

0.006014

- 59.73

-61.57

0.01082

0.006014

- 44.74

- 47.77\

0.02145

0.006014

- 31.80

- 34.66

0.05322

0.006014

- 18.85

- 16.84

0.005579

0.01087

- 69.05

- 70.83

0.01066

0.01087

- 55.99

- 58.97 - 46.69

0.02073

0.01087

- 43.55

0.05125

0.01087

- 26.19

- 29.82

0.005437

0.02158

- 81.83

- 83.24

0.01049

0.02158

- 67.08

- 71.39

0.02054

0.02158

- 56.08

- 59.37

0.04984

0.02158

- 39.42

- 43.56

0.005404

0.05116

- 96.61

- 98.41

0.01068

0.05116

- 83.19

- 86.06

0.02087

0.05116

- 70.61

- 74.29

0.05063

0.05116

- 50.06

- 59.06)

lAEl,,/mV

i

1.55

’

3.23

8

Emf of cell (v): Ag,AgCl

INaCl(aq)

ties and various

compositions

M = Na; AE = &,I, m NaCl /mol

N = K.

solvent

Distance

- I&.,,,. e’= kg - ’

0.65.

of

jKCl(aq-methanol)

closest

X,,,, m,o/mol

IAgCl,Ag,

at 298 K at various electrolyte

in the right hand half-cell. approach

used

for

Ecalc obtained

Ecalc = 0.47

nm.

molali-

using eqn. (10) with

)A.E

lav = 2.19

= 0.5 kg-’

0.006116

0.00667

-49.17

- 50.66

0.01108

0.01260

- 50.15

- 51.61

0.02190

0.02491

- 49.57

- 51.61

0.03317

0.03846

- 49.86

- 52.16

0.04311

0.05072

- 49.89

- 52.62

0.05398

0.06452

- 50.03

- 53.13

mV

where

144 TABLE

9

Emf of cell (vi): Ag,AgCl jNaCl(aq) INaBr(aq-methanol) IAgBr,Ag, at 298 K at various electrolyte molalities and various solvent compositions in the right hand half-cell. E,,,, obtained using eqn. (11). Distance of closest approach used for E,,,, = 0.54 nm. 1AE lav = 1.55 mV where AE = E,,,, - Ecea. 0’=0.60. X MeoH = 0.5. In all cases the distance of closest approach was taken as the average of the distances of closest approach of the electrolytes concerned mNaCl/mol

mNaer/mol

kg-’

0.005709 0.01118 0.02163 0.03250 0.04378 0.05411

TABLE

0.006134 0.01225 0.02416 0.03823 0.05143 0.06454

kg-’

E,,t, /mV

Ecalc /mV

-

-

173.93 174.24 174.28 175.08 174.92 175.06

172.05 172.29 172.57 173.63 173.64 174.00

10

Emf of cell (vi): Ag,AgBr INaBr(aq) INaCl(aq-methanol) (AgCl,Ag, at 298 K at various electrolyte molalities and various solvent compositions in the right hand half-cell. E,,,, obtained using eqn. (11). Distance of closest approach used for E,,,, = 0.54 nm. 1AE IBv given for each solvent composition, where AE = EC,,, - EC,,, o ’ = 0.60 X MeOH

0.1

0.2

0.3

0.4

&II /mV

-bc /mV

IAEl,,/mV

0.005155 0.01028 0.02054 0.03074 0.04085 0.05117

0.005143 0.01041 0.02087 0.03109 0.04134 0.05010

142.73 142.09 142.10 141.93 142.20 142.03

141.17 140.89 140.83 140.90 140.88 141.47 I

1.16

0.005312 0.01024 0.02074 0.03150 0.04078 0.05056

0.005515 0.01049 0.02149 0.03279 0.04251 0.05296

133.63 133.76 133.72 133.52 133.65 133.45

132.13 132.42 132.22 132.13 132.09 131.97 1

1.46

0.005098 0.01031 0.02039 0.03075 0.04115 0.05125

0.005354 0.01069 0.02172 0.03310 0.04436 0.05579

126.11 126.08 125.91 125.59 125.76 124.57

124.35 124.67 124.20 124.00 123.95 123.73 I

1.52

0.005046 0.01009 0.02157 0.03118 0.04069 0.05065

0.005505 0.01094 0.02366 0.03522 0.04644 0.05745

118.24 118.43 118.39 117.89 117.93 118.08

116.57 116.81 116.67 116.12 115.91 115.97 I

1.82

~N~B~/~OI

kg-’

145 TABLE 10 (continued) X MeOH

mNaBr/mol kg-’

mNaCI/mol kg-’

-b

-%,I, /mV

I@l,,/mV

0.5

0.005209 0.005736 0.01011 0.02072 0.04992 0.005736 0.01011 0.01058 0.02072 0.04992 0.005636 0.009955 0.02052 0.02055 0.03029 0.04092 0.005636 0.009955 0.02052 0.04959 0.05061

0.005685 0.005044 0.005044 0.005044 0.005044 0.01024 0.01024 0.01174 0.01024 0.01024 0.02012 0.02012 0.02012 0.02300 0.03552 0.04827 0.05017 0.05017 0.05017 0.05017 0.06045

111.54 115.61 126.48 139.84 156.61 102.05 113.21 111.35 126.79 143.11 88.87 99.59 113.49 111.64 110.91 111.09 72.29 83.50 97.49 113.67 110.89

109.94\ 114.09 124.99 138.68 154.94 100.70 11.50 109.79 124.98 141.19 87.92 i 98.68 112.16 109.72 108.86 108.71 71.58 82.18 95.56 111.43 108.42)

1.62

0.005388 0.01035 0.02040 0.03054 0.04098 0.05078

0.005932 0.01156 0.02388 0.03732 0.05082 0.06406

105.49 105.40 105.14 104.59 104.61 104.35

103.66 103.56 102.79 101.98 101.64 101.24 I

2.45

0.00567 0.01141 0.02070 0.03178 0.04035 0.05061

0.006413 0.01347 0.02532 0.04070 0.05266 0.06772

99.88 99.37 99.44 98.89 98.94 98.59

97.68 97.08 96.49 95.59 95.15 94.55 I

3.10

0.005246 0.01045 0.02182 0.03909 0.04055 0.05040

0.006771 0.01283 0.02849 0.04183 0.05595 0.07407

93.66 95.07 94.87 94.43 94.61 93.45

90.55 91.73 90.63 89.85 89.33 87.95 I

4.34

0.005185 0.01023 0.01988 0.02990 0.03990 0.04916

0.006454 0.01266 0.02648 0.04213 0.05823 0.07510

90.83 92.05 91.73 91.21 91.24 90.50

87.96 88.26 86.84 85.59 84.67 83.55

5.12

0.6

0.7

0.8

0.9

/mv

146 DISCUSSION

The recent development [2] of a method for extracting reliable thermodynamic information from a Wagner cell is based on the ability to account successfully for the liquid junction contribution to the cell emf. The formulation of a suitable single ionic quantity [l], the relative solvation intensity (RSI), greatly simplified the calculation of the solvation potential, an important component of the total cell emf. At the same time, we used procedures for the calculation of the various integrals needed in the evaluation of the cell emf that took account of the effects on cell emfs of the variation of both ionic transport numbers and activity coefficients with solution composition in the junction region. For the sake of completeness in this work, we have used the calculation procedures developed earlier to predict the emfs of some single-solvent cells with liquid junctions. A good deal of work has continued to be carried out on this latter type of cell since the early fundamental studies of MacInnes [9], Taylor [lo] and Guggenheim [ll]. Thus, Spiro [12], Smyrl and Newman [13], Chen and Frank [14], Osterberg et al. [15] and, most recently, Bagg [16] have progressively elaborated the calculation procedures for determining both the ionic distributions and the appropriate thermodynamic properties within the junction so as to evaluate the integrals in eqn. (1). Our extensive data on cells of types (i) and (ii) shows that, in these cases, the cell emfs at low ionic strengths can be calculated correctly to a level of accuracy similar to the reproducibility of the emfs of most galvanic cells. In this work, we have verified the validity of the Henderson method under conditions where the electrolyte concentration ratio in the half-cells was allowed to vary over two powers of ten. The differences between calculated and experimental emfs of ca. +0.2 mV are essentially random. Our experimental and calculated emfs for these cells are compared with the results of earlier workers obtained in the same ionic strength range and are shown in Table 12. Chen and Frank’s value [14] on the KCl/NaCl junction is about 1 mV less than either our result or that given by MacInnes [9]. On the other hand MacInnes recorded a cell emf 2.2 mV higher than ours for the KCl/LiCl junction. Given the wide-ranging agreement between calculated and experimental emf values shown in Tables l-4 that encompass a variation of two powers of ten in the half-cell electrolyte concentration ratio, it seems reasonable to conclude that the discrepancies are experimental in origin. Cell (iii) is a particularly interesting case in which there are four different ions in the junction. The average difference between calculated and experimental emfs of

TABLE

11

Values of l7’ parameter for methanol+ water mixtures at 298 K obtained from refs. 1 and 2 by curve-fitting salvation potential data Electrolyte , (r /(mol kg-‘)-“2

mV_’

LiCl 0.4

NaCl 0.6

KC1 0.7

CsCl 0.1

HCI 2.0

NaBQg 0.7

147 TABLE

12

Comparison determined

between experimental using eqn. (13)

emfs

of other

workers

and

data

from

this work

(cell i). E,,,,

Junction

Composition/M

Ref.

EexPl /mV

&,/mV

AE/mV

KC1 1NaCl

0.010/0.010 0.010/0.010 0.01043/0.00989 0.01000/0.01008 0.01038/0.01002

14 9 this work this work this work

4.496 5.65 5.58 4.24 5.11

5.58 5.58 5.69 4.37 5.33

+ 1.35 - 0.07 +0.11 +0.13 +0.22

KC1 1LiCl

0.010/0.010 0.00989/0.01044

9 this work

8.20 6.02

6.02 5.99

-2.18 - 0.03

NaCl 1LiCl

0.010/0.010 0.01008/0.01224 0.01002/0.01224 0.01030/0.01029

9 this work this work this work

2.63 0.79 1.15 2.60

2.60 0.75 0.98 2.55

- 0.03 - 0.04 -0.17 - 0.05

ca. 1 mV arises mainly from a systematic variation of AE with the ratio of the half-cell electrolyte concentrations. The difference is shown as a function of the logarithm of the half-cell electrolyte concentrations in Fig. 1; similar data for a type (i) cell taken from Table 1 are also included. The calculated cell emf agrees well with experiment provided the half-cell electrolyte concentrations are roughly equal. When the concentration ratio is 10 : 1 the emf difference is around 2 mV and it seems clear that under these conditions there is a need to use more sophisticated modelling to determine the allowed solution compositions in the junction region. We have discussed Wagner cells of type (iv) at length in a previous publication [l] and some additional data are given in Table 6 which confirm that the errors in the calculated emfs are not a strong function of the half-cell electrolyte concentration ratios. Wagner cells of types (v) and (vi), with different electrolytes but with one common ion in the two half-cells, show differences between calculated and experimental emfs that vary systematically with the methanol mole fraction as shown in Fig. 3. There is no significant dependence of the error on the half-cell electrolyte concentration ratio over two orders of magnitude. The maximum differences observed are nearly 5 mV giving a maximum error of ca. f0.5 kJ mol-’ in any thermodynamic quantity derived from such emf measurements. Any errors in the assumed concentration distribution of the ions in the junction at the high water end of the solvent composition range where the ionic mobilities are changing rapidly will, in turn, generate errors in the integrals that make up the first two terms of eqn. (10). This problem does not arise so acutely in cell (vi) where the chloride and bromide ions have very similar mobilities. As with type (iii) cells, there is a clear need to use more sophisticated modelling to determine the allowed solution compositions in the junction region of the flowing junctions. Preliminary calculations on a type (iii) cell suggest that the structure of our flowing junction is no better

NaCl Gibbs’ energy term

ibbs energy term -60

-80 ’

0.0

I

,

0.2

0.4 Mole

Fig. emfs KC1 final

fraction

0.6

0.8

1.0

of methanol

4. Contributions of the terms in eqn. (10) to the total calculated emf of cell (v); the experimental given in Table 6 have been corrected to the same electrolyte activity in the half-cells. The NaCl and lines show the first two terms in eqn. (lo), while the solvation potential contribution is given by the term in eqn. (10).

represented with a free diffusion model than with the Henderson model. No calculations based on a diffusion model have yet been attempted for a Wagner cell. Finally, it is worth noting the relative contributions to the cell emf of the various terms in eqn. (10). Figure 4 shows emf data for a type (v) cell, taken from Table 7, and corrected to equal electrolyte activities. It illustrates that the two standard Gibbs energy of transfer terms for the electrolytes, and the solvation potential term, all have similar magnitudes. CONCLUSION

It is now well established that simple galvanic cells with liquid junctions can be used to obtain reliable thermodynamic data. Our work supports and extends earlier studies by confirming that the junction contributions to the emfs of such cells can be calculated to an accuracy similar to the reproducibility of normal galvanic cells. In particular, we have shown that the Henderson method of modelling the allowed solution compositions in the liquid junction leads to correct emf predictions for single solvent cells with two different electrolytes and a common ion over a wide range of half-cell electrolyte concentration ratios. Wagner cells containing a single electrolyte are also reasonably well-behaved. The Henderson method also appears to work for cells with two different electrolytes and no common ion, provided the half-cell electrolytes are at similar concentrations. The calculation of the emfs of Wagner cells with two different electrolytes with a common ion requires better modelling of the structure of the liquid junction so as to

149

reproduce more correctly the allowed solution compositions in the flowing junction. However, even for the Wagner cell with two different electrolytes and a common ion, the Henderson method leads to predictions correct to within + 5 mV and this is adequate for obtaining thermodynamic data if a level of accuracy of & 0.5 kJ mol-’ is acceptable. ACKNOWLEDGMENT

The authors wish to thank Mr.R.E. Gregory for assistance with the computations and the Australian Research Council for an equipment and maintenance grant. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

A.M. Hodges, N.W. Kilpatrick, P.T. McTigue and J.M. Perera, J. Electroanal. Chem., 215 (1986) 63. A.M. Hodges, J.M. Perera and P.T. McTigue, J. Electroanal. Chem., 306 (1991) 41. A.M. Hodges, P.T. McTigue and D. Abela, J. Electroanal. Chem., 306 (1991) 55. C. Wagner, Advances in Electrochemistry and Electrochemical Engineering, Vol. 4, P. Delahay and C.W. Tobias (Eds.), Wiley, New York, 1966, p. 1. D. Feakins and J.P. Lorimer, J. Chem. Sot. Faraday I, 70 (1974) 1888. R.A. Robinson and R.H. Stokes, Electrolyte Solutions, 2nd ed., Buttenvorths, London, 1965. J.R. Farrell and P.T. McTigue, J. Electroanal. Chem., 139 (1982) 37. P. Henderson, Z. Phys. Chem., 63 (1908) 325. D.A. MacInnes, J. Am. Chem. Sot., 37 (1915) 2301. P.B. Taylor, J. Phys. Chem., 31 (1927) 1478. E.A. Guggenheim, J. Am. Chem. Sot., 52 (1930) 1315. M. Spiro, Electrochim. Acta, 11 (1966) 569. W.H. Smyrl and J. Newman, J. Phys. Chem., 72 (1968) 4660. C.H. Chen and H.S. Frank, J. Phys. Chem., 77 (1973) 1540. N.O. Osterberg, J.B. Jensen, T.S. Sorensen and L.D. Caspersen, Acta Chem. Stand., A34 (1980) 523. J. Bagg, Electrochim. Acta, 35 (1990) 367. D.A. MacInnes and Y.L. Yeh, J. Am. Chem. Sot., 43 (1921) 2563. D.A. MacInnes, The Principles of Electrochemistry, Dover, New York, 1961.