mixed adsorption at mercury/solution interfaces: i. a. thermodynamic analysis

ELECTROANALYTICAL CHEMISTRY AND 1NTERFACIAL ELECTROCHEMISTRY Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

127

MIXED ADSORPTION AT MERCURY/SOLUTION INTERFACES

I. A THERMODYNAMIC ANALYSIS

S. LAKSHMANAN AND S. K. RANGARAJAN

Central Electrochemical Research Institute, Karaikudi-3 (India)

(Received January 7th, 1970; in revised form March 3rd, 1970)

INTRODUCTION

Earlier studies 1 on specific adsorption of a single anion from mixtures containing in addition a non-specifically adsorbed anion at constant total ionic strength were aimed at keeping the Debye thickness constant. This would make possible a comparison of specifically adsorbed charge in such mixtures with those at varying single salt concentrations and would reveal an effect due to imaging conditions in the diffuse layer. Experiments in such mixtures of one specifically adsorbed ion such as iodide with a supposedly non-specifically adsorbed ion such as fluoride at constant total ionic strength were analysed through an elegant thermodynamic analysis by Parsons 1 which gives directly the amount of specifically adsorbed iodide ion. An essentially similar but more general treatment is due to Hurwitz 2 who recognised the role of activity coefficients also in such a derivation. The experimental results subjected to the above analysis in comparison with those for single salts at varying ionic strengths not only point to the significance of imaging in the diffuse layer, at least in solutions of low ionic strength of iodide ~, but also necessitate the use of salt activity rather than ionic activity as the variable determining the amount of iodide ion adsorption. In contrast, similar analyses by Payne 3'4 when applied to fluoride- nitrate and fluoride-perchlorate mixtures at constant ionic strengths could demon- strate the absence of constant imaging in the diffuse layer in these mixtures and it was even suggested how a non-specifically adsorbed ion such as fluoride could influence the specific adsorption of nitrate and perchlorate. This meant recognition of compe- titive adsorption of the anions with each other rather than with the solvent. Moreover the extension of the above treatment to mixtures of ions of nearly similar size, viz. fluoride and chloride, leaves us only in a dilemma as to the choice of salt activity or the anion activity as the proper variable in the isotherm 5. As such, a more generalized study, viz. that of adsorption of two specifically adsorbed anions from solutions containing a common cation at constant total ionic strength, may not be expected to complicate matters further! On the other hand, such a formulation--as is given below--could also be used to test whether a supposedly non-specific adsorption of an ion is really specific.

The thermodynamic analysis given earlier 2 for such systems at constant ionic strength does not yield directly the sum of individual specifically adsorbed charges but only a linear combination of these. This results on the assumption that the surface

J. Electroanal. Chem., 27 (1970) 127-134

128 S. LAKSHAMANAN, S. K. RANGARAJAN

excesses of the anions in the diffuse layer are proportional to their mole fractions in the bulk. Thus we are handicapped by the inability to separate out the total composite specifically adsorbed charge (which is a function of the mole fractions in bulk) without any knowledge of their respective contributions in the diffuse layer. But it is shown here that the sum of the relative surface excesses of the two anions can be evaluated through a knowledge of another differential coefficient such as (07/0 In I)E +_,x or (~+/~ In l)q,~,x where I is the total ionic strength, xI is the mole fraction of one of the anions, that of the other being ( 1 - x)l. From the total surface excess of anions (or cations) thus obtained 6 and the composite surface excesses obtained at a single ionic strength, it is possible to solve for the individual anionic surface excesses in the inner plane. The thermodynamic analysis to be used is given below and this takes into account formally the variation of activity coefficients of the salts.

A general electrocapiUary equation 7 for an ideally polarised electrode at constant temperature and pressure under conditions when the indicator electrode is reversible to the cation j' is :

d T = - q . d E + - Z F , , d u i - Z (rjh,/v~,)dujk, i~i' j~ej,

- Y kCk' hCh'

- 1 r ]Zk,'(V;k,j~j ' jh, Zj ]dpj ,k, (1)

with the symbols having the significance given in ref. 7. For the sake of simplicity, we confine our interest here to the systems containing two anions and a common cation.

For a system containing only one metallic phase and two 1 : I salts with a common cationic species, viz. j' and two anionic species kl and k2, the neutral com- ponent being the solvent h' alone--the above equation reduces to

d T = - q M d E + - Z Fk,a,d/~, (2) i= 1,2

where [Fk,h,]i= 1,2 denote the relative surface excesses of the two anions and [/~i]i= ~.2 the chemical potentials of the two salts. For a mixture of two salts 3~ is a function of the chemical potentials of the two salts

7 = f(/~a,/~2)

or this can also be expressed without losing generality as

= f ( m , I)

where I is the total molal ionic strength. 1. Let the individual molality of salt 1 be m a I so that that of salt 2 is (1 -m~)I

at a total molal ionic strength I. Then it can be shown from eqn. (2) that

( ) /c? In 71+\ 1 { 07 "~ = ml F z - 2 ~ 2 F i t ~ ) , (3) R T k g l n m l / E + J - F1 l - m 1 .= ,

where Fx and/ '2 are surface excesses of anions k~ and k2 respectively and y~_+'s are the mean molal activity coefficients of the salts 1 and 2 in the mixture. This result is essentially similar to that of Hurwitz 2.

J. Electroanal. Chem,, 27 (1970) 127-134

T H E R M O D Y N A M I C S O F M I X E D A D S O R P T I O N 129

2. Let the individual molalities of the salts be m~I and ( 1 - m t ) I at different total molal ionic strengths I, say 1 = 1, 2, 3 . . . . . Then it is possible to write from eqn. (2) that

2RT ~lnn-I +,m~=-(F~+F2)- ~, I ' { gln~i+'] ,=i.2 ~ [ , ~ J , , 1 (4)

The above analysis is strictly true only if molal concentrations are employed. One could however use eqns. (3) and (4) even for experiments under constant molar ionic strength, if I is sufficiently small. When for some reason the variables to be employed are molar instead of molal--transformation of (3) and (4) to suit this need have been made but are not given here for the sake of brevity.

By solving eqns. (3) and (4), it is possible to get individual surface excesses of the anions. But this demands a knowledge of (~ In 7i+/c~ In mt)1 as well as (0 In 7i+_/

In I),,, requiring activity coefficient data in mixed electrolyte systems. Recently, since the advent of cation-sensitive electrodes, numerous such data 8 have been reported at various molal ratios of the salts in the mixture and at constant total molal ionic strengths. In this connection it is worthwhile mentioning that in the earlier analyses of results either the variation of activity coefficients of the salts with the change in the composition of the mixture was neglected or the activity coefficient was assumed to be unity. The former assumption is not unreasonable if the cross terms in the ionic strengths of individual salts are small and especially when the activity coefficients of the salts are closely similar as for K I - K F mixtures. The latter assumption is likely to introduce difficulties while comparing the results obtained for mixed electrolyte systems with those of pure salts.

We illustrate now how the above equations are to be employed. Since y's are functions of ml, I and F j , solution of eqns. (3) and (4) leading to F 1 and F 2 is meaning- ful only when experiments at different total ionic strengths are performed at the same ratios of mole fractions of the salts, viz. mt/1 - m 1. This is better illustrated in the following case where the activity coefficients are neglected. When it is required to evaluate Ft and F2 at mt = 0.5, say at unit molal ionic strength, the derivative

at ( m i = 0 . 5 ) = - 1 l - m 1 - - - - E + , I = 1 ml = 0.5

I = 1 . 0

Similarly

i c3(0_~n~n / ) at ( I = 1 ) = - (Ft+F2),,l=o.5 (6) 2RT e+,ml=O.~ t=a.o

It is easily seen that the solution of the above simultaneous equations will lead to Ft and F2 for the "state" defined by "mr = 0.5, I = 1".

Specifically adsorbed charge Having evaluated the total surface excesses of the two anions, the surface ex-

cess of cations can be found from the identity

- q~a = F ( F + - F - ) (7)

Assuming that all £ + is in the diffuse layer, it is possible to obtain the total diffuse-

J. Electroanal. Chem., 27 (1970) 1 2 7 - 1 3 4

130 s. LAKSHMANAN, S. K. RANGARAJAN

layer charge due to anions as well as their individual contributions F] and F~ with the help of diffuse-layer theory s. The specifically adsorbed charges due to the individual anions F{ and F~ may then be obtained knowing that

r;- = r , ~ + r~ (8)

LIMITING CASES

(i) For a mixture with a non-specifically adsorbed anion, eqn. (3) reduces to the familar Parsons' equation1:

1 (4 ?3' " / ~ l n T i + \ R T . ? l~n~ml)e+,, = - F1- 2 i=~1,2 Fi( ~nn ~l ) I (9)

thus giving directly the specifically adsorbed charge. (ii) For a single salt, eqn. (4) reduces to the well-known form

1 (_~7 "~ _/c'~ln'h+\ 2 R T ~ d l n I / E + = - r - - r

(10)

CONVERSION OF Ere f TO E ÷ SCALE

Equations (3) and (4) imply that the derivatives of ~ should be obtained at constant E ÷ scale. In practice, this can be achieved by following the cation activity with the use of cation-sensitive electrodes. But whenever the potential of the polarised electrode is measured with reference to say an NCE, these potentials must be con- verted to the E + scale before any complete analysis is made. This can be accom- plished by the method due to Frumkin lo. The potential of the reference electrode can be expressed as

Erer = E + + (RT/F) In a+ + "constant" (11)

This "constant" includes liquid junction potential. Neglecting liquid junction potentials, it is possible to write

07 r \ 0 In ml/Erer

For a single 1 : 1 salt

OE ~ _ R T (13) In I/ercf F

since ionic activity is the square root of salt activity. But for a mixture of salts, the individual ionic activities are not simple functions of the salt activity since the individual ionic activity coefficients cannot be identified with the mean activity coefficient of the salt. Hence, to get over this difficulty caused by using a Ere f scale, we have to resort to empirical equations relating the molality of the ions to the individual ionic activity coefficients. We find that the extended Guggenheim relation 11'~3 is best

J. Electroanal. Chem., 27 i1970) 127-134

THERMODYNAMICS OF MIXED ADSORPTION 131

obeyed for NaC1-NaNO3 mixtures*. According to Guggenheim

in 7R = _ a Z 2 x/I + 2 Ex' flRX'mX, (14)

R denotes a particular cation and R' every cation and similarly for the anion X' ; ~ is a coefficient depending only on the solvent and temperature; fl~ are interaction coefficients which depend on the solute, solvent and temperature; m~ are molalities of the ions and Z~ are valencies of the ions.

The variation of cationic activity coefficient 7 + with change in the molality of salt 1 at constant total ionic strength then follows from the extended Guggenheim relation as

/ / ' ~ In 7+ 2 ~k ~ - n D n~l) I = ( / ~ 1 - f 1 2 ) m , I = 2aGm,I (15)

(It is a moot point to ask whether one could substitute for (fl l-f l2) the Harned's slope, 2.303 a12 if the latter is available and is not much different from (ill-f12). It is difficult to support or reject this 'hybrid' approach.)

Rewriting eqn. (11) for a mixture of 1 : 1 salts

Ere f = E + + (RT/F) In m+ 7+ +constant (16)

Hence

eE+ = ( 1 7 )

RT(c~lnT+ ~ c3 In m, JE~o.I F ",alnml}1

From eqn. (14) we have

since

Hence

07 ~7 ~a m x (~ l~l~m~)Eref,l = (~ ln-~l)E + ,I + 2 a T F I (18)

= _ q M = v ( r + - r -)

it follows from eqn. (3) that

1 ( ay ~ = - (F, mt /~ lnT i±~ - ~ \ c~ l--~m l Je.of a l Z--m t Fe ) - 2 i __~l ,2 Fi [ f f ~n m--~l / i

(19)

(20)

2otam 1 I + - - qM F

For different ionic strengths at constant mole fractions of the salts, we may write, similarly to eqn. (12)

* The extended Guggenheim relationship implies "a Harned Slope" for both the salts and is most useful when this can be established 12. Otherwise it can be a good approximation--at least one hopes so. We have also tested the usefulness of this relation for other mixtures at different total ionic strengths and the results will be communicated elsewhere 13.


132 S. LAKSHMANAN, S. K. RANGARAJAN

and it

Hence

-- + t, 0 e + / . , t a In I/E o,

can be shown that

0 ln])+~ = ~x/I ~-ffl~nI/,., 2(1 + ~/I) 2 + 20~Gmi I

it follows from the above relations that

2~RT t ~ ) E ...... = - ( r ,+r~)- ,Z, r, t OlnI )=,

2(1+~/i) 2 + 2~cml

(21)

(22)

(23)

RESULTS CORRESPONDING TO THE E - SCALE

A. The transformation of eqns. (3) and (4) to the E- scale The derivatives of surface tension ? given in eqns. (3) and (4) assume that E +

is held constant. It may be worthwhile to find out the forms of the corresponding equations when the derivatives are needed in the E- scale. They are as follows:

; (O~nml )~ ; , , = ( F+ -2F2\~l-~-~mlJ, (24) - l ~ m l ) {0 In ])21~

0 In ])12

1 d]) (Fx - m x F +) 2F a RT ~ E¢,, = (1 -ml ) Olnml/,

_ 2(r+_r1)(o In 721) g In ml / s

o r

RT 0 In ( l - m 1 ) ee,, = - + - 2r, (25) mj a In ( 1 - m l ) ] i

31n])2~ .~ - 2(r + - r~) ~ In (l--mi~s

1 ( 0 - ~ n / ) E i _ m l = - F + - F 2 - ( F + - F 2 ) ( ~ ) m ' (26, 2RT . t 0 1 n I / m ,

1 f 07 ] = _ F + 0 In ])12~ / . . / 3 I n ])21 \ 2RT\~--i-~nl/~e,m ' - F , ( ?~n] / , . - ( r +- ' ) t ~---~nl ),., (27)

In the equations given above ml is the mole fraction of salt 1, Y12 and 72 t are the mean activity coefficients of salts 1 and 2 in the mixture and [E?] , i= 1,2, denotes potentials measured with respect to a reference electrode reversible to the anions of


THERMODYNAMICS OF MIXED ADSORPTION 133

salts 1 and 2. It may be interesting to observe that (~7/8 In I)s,:m, gives - F + whatever the value o f / ( = 1 or 2) except for the differences in the terms due to the respective salt activity coefficients. The symmetry in eqns. (24) and (25) is obvious.

B. Conversion of E- to Era scale Since

RT Ere f = E;- ~ - In a~_ + constant (i = 1 or 2)

an analysis on the lines made earlier (see: Conversion of Era to E ÷ scale above) reveals (as is shown below) that the forms for the final equations in the Era scale are identical with eqns. (20) and (23) deduced via the E + scale. This is, of course, to be expected and serves as a test on our equations. The basic equations for the transfor- mations are shown below.

R---f( t RT\3lnml /Ec;I

Or 2R-T \~-~In I/~ ... . . , 2RT

I ( lnyl_) ] qM 1 + (28) F \ ~ In m l / l

[ ] qM 1 + (29) 2F \ ~ In I /.,,J

Lastly, we give below the expressions that are similar to those of eqns. (3) and (4) for the derivatives of Parsons' ~ + functions at constant qM-

i ( ) = - (r, m, r2) -2 RT \ 0 In roll,M,, i ~-m, i=~1,2 Fi ~,~ l~-n~mlJ ,

2R T ~ t~ In I /qM.,, ~ i=1,2

(Here, F's are reckoned at the respective qMS.). Similar expressions for the derivatives of [~7]/= ~ or z are easy to obtain.

(30)

(31)

ACKNOWLEDGEMENT

The authors thank Dr. H. V. K. Udupa, Director for his keen interest in the work and Dr. Roger Parsons for his useful comments.

SUMMARY

A thermodynamic analysis is given of the electrocapillary equation suitable for the evaluation of individual surface excesses at the inner planes of two specifically adsorbed anions from a mixture containing these two anions along with a common cation. This analysis takes into account variation of activity coefficients of the salts in the mixture. The problem of interpreting the electrocapillary data obtained in the absence of a cation-sensitive electrode is considered and appropriate expressions are given.


134 S. LAKSHMANAN, S. K. RANGARAJAN

REFERENCES

I E. DUTKIEWICZ AND R. PARSONS, J, Electroanal. Chem., I1 (1966) I00. 2 H. D. HURWITZ, J. Electroanal. Chem., 10 (1965) 35. 3 R. PAYNE, J. Phys. Chem., 69 (1965) 4113. 4 R. PA~'NE, J. Phys. Chem., 70 (1966) 204. 5 R. PAVNE, Trans. Faraday Sot., 64 (1968) 1638. 6 D. C. GRAHAME, Chem. Rev., 41 (1947) 441. 7 D. M. MOmLNER in ALLEN J. BARD (Ed.), Electroanalytical Chemistry, Vol. 1, Marcel Dekker Inc.,

New York, 1966, chap. 4, pp. 275, 401. 8 R. D. LANIER, J. Phys. Chem., 69 (1965) 3992. 9 D. C. GRAnAME, J. Chem. Phys., 21 (1953) 1054.

l0 A. N. FRUMKIN, Zh. Fiz. Khim, 30 (1956) 2066. l l E. A. GUGGENHE~, Phil. Mug., 19 (1935) 588. 12 H. S. HARNED AND B. B.. OWEN, The Physical Chemistry of Electrolyte Solutions, Reinhold Publishing

Corp., New York, 1958, pp. 600-607. 13 S. LAKSnMAr~AN AND S. K. RANGARAJAN, J. Electroanal. Chem., 27 (1970) 170.


mixed adsorption at mercury/solution interfaces: i. a. thermodynamic analysis

Documents