liquid systems

29

J. Electroanal. Chem., 313 (1991) 29-35 Elsevier Sequoia S.A., Lausanne

JEC 01558

Linear Gibbs energy relationships and the activated transport model for ion transfer across liquid/liquid systems Theodros Solomon Department

of Chemistry, Addis Ababa University, P. 0. Box 1176, Addis Ababa (Ethiopia)

(Received 20 December 1991; in revised form 11 March 1991)

Abstract

An alternative explanation, based on linear Gibbs energy relationships, is given for the recent experimental observations, and for the comparison between ion transport and ion transfer, made by Shao and Girault (J. Electroanaf. Chem., 282 (1990) 59).

INTRODUCTION

Shao and Girault [l] recently reported the study of the transfer of acetylcholine across the water - 1,Zdichloroethane interface as a function of the viscosity of the aqueous phase. The viscosity of the aqueous phase was varied by the addition of sucrose (s). They concluded that: (i) the diffusion coefficient of the ion in the aqueous sucrose solution (w + s) is related to the Gibbs energy of transfer of the ion AG;*W+S, (ii) the standard rate constant of the ion transfer is directly proportional to the diffusion coefficient, and (iii) the differences in the Gibbs energies of activation for diffusion in (w) and in (w + s), as well as for ion transfer from w + o and from (w + s) + o, are proportional to the differences in the Gibbs energy of hydration and can be expressed as: (diffusion) :

AGZ;; - AG,w,,,d= 0.55 x 10-3RTAGP-

w+s

(ion transfer) : AG,,,tp * o - AG,T,
(1) (2)

The experimental results were interpreted in terms of the formalism for the analysis of the kinetics of ion transfer across a polarised liquid/liquid interface based on the activated transport model for homogeneous media proposed earlier by Girault and Schiffrin [2]. 0022-0728/91/$03.50

0 1991 - Elsevier Sequoia S.A. All rights reserved

The purpose of this communication is to provide an alternative explanation for the experimental observations of Shao and Girault [l], starting from the linear Gibbs energy relationships (LGER). THEORY

Since the activated complex theory is found to be applicable to a variety of rate processes such as diffusion, flow of liquids, etc. then one may write for the diffusion coefficient of an ion in solvents oi and o2 [3]: Do’ = X2( kT/h) Do2 = X’*( kT/h)

exp( - AG&JRT) exp( - A?&JRT)

where X (or h’) is the distance between two successive equilibrium positions, and AGX, d is the Gibbs energy of activation for diffusion. Similarly, the rate constant for the transfer of an ion from w + or and from w -+ o2 may be written in much the same way as [2]: k w4o1 =X( kT/h)

exp( -AGa:,;“‘/RT)

(5)

k w’o2 = X’( kT/h)

exp( -AGGt702/RT)

(6)

where AG,,, t is the Gibbs energy of activation for ion transfer. If the above sets of equations for the diffusion coefficient and for the rate constant are applicable to ion transfer, then it is easy to show that ln( D”l/D”‘)

= ( AG,O,:,,- AG,Od,,,)/RT

(7)

ln( k w -“/kw

‘02) = ( AG,C;;“’ - AG,;,; “‘)/RT

(8)

provided that A is not too different from X’. Linear Gibbs energy relationships may now be introduced as follows: (1) As is observed for many systems [4,5], there is a linear relationship between Gibbs energies of solvation of a series of ions in one solvent and those in another solvent, this relationship obeying an equation of the form: A%% =AG,O,;,+A

(9)

where A is a constant. In terms of the Gibbs energies of transfer, this is equivalent to AG;-“j

=AG,“‘“z+A

(10)

Fig. 1 shows such a relationship between the Gibbs energy of solvation in 1,2-dichloroethane and the Gibbs energy of hydration for the alkali metal ions. The above equation is a special case of a more general LGER of the type log K = x log K ’ + constant where, in this case, x = 1, and K and K’ coefficients.

(11) may be identified

as the partition

31

Fig. 1. AC,%: versus AGhydr for alkali metal ions. (Data

(2) Another

typical

LGER

from ref. 6.)

that may be applied

to the ion transfer

AG,“,,T;O’= cyAG,” - O’+ B

process

is

(12)

where B is a constant; i.e. the Gibbs energy of activation for ion transfer is linearly proportional to the Gibbs energy of ion transfer, where (Yis a proportionality factor. Similarly, for the ion transfer process, w + 02, AG;,I; Oz= (uAG; -02 + B'

(13)

where, in view of Eqn. (lo), it is assumed that there is also a constant difference between the acriua~ion energies for the transfers w -+ o, and w --, oz. (3) If the interface between two immiscible electrolyte solutions is regarded as a non-homogeneous liquid phase where the solvation energy is dependent on the position of the ion [2], and if therefore, the Gibbs energy of activation for diffusion is linearly proportional to the Gibbs energy of solvation, then one may write other LGER’s as AG;;,,, = fiAG& AG;&

+ C

(14

= j3AGS002+ C’

(15)

where C and C’ are constants. Using these relations, the right-hand as

sides of Eqns. (7) and (8) may be rewritten

(AC;&

+ (C’ - C)

- AG:;t9d) = p( AG& - AG&) =PAG;‘-“Z+

(Cl-

C)

(16)

(AG,~,~"'-~G,~,,T;"~)=~(AGP-OZ-~C~-~~)+(B'-B) =~~~AG;‘-“+(B’-B)

(17)

32

Comparison of Eqns. (16) and (17) reveals that the left-hand terms are linear functions of the Gibbs energy of transfer from oi to 02, and hence may be identified as the Gibbs energy of activation for the transfer from o, to oz. In other words, Eqns. (16) and (17) are, in effect, expressing the following LGER: AG,“,‘,T;a= = aAGP’ * O2+ constant

(18)

As a result, Eqns. (16) and (17) and therefore and (8) are the same. Thus, with (Y= p, ln( Do1/Do2)

= CIAGP’+~~/RT + constant

ln( k w+“~/kw~o*)

= CXAG”~‘~~/RT+

If the constant in Eqns. (18-20) C = C’ = 0), then it follows that (DOI/D%)

the right-hand

sides of Eqns.

(7)

(19)

constant

(20)

is equal

to zero,

= (kW+%/kW+%)

(i.e. if B = B’ = 0, and

(21)

This result shows that, since the ratio of the diffusion coefficients of an ion in two solvents is a constant, equal to the inverse ratio of the solvent viscosities, then the ratio of the rate constant for the transfer of an ion from w + o, to that from w + o2 is also a constant, this constant holding for all ions in the series for which the linear Gibbs energy relationships above hold. DISCUSSION

The above formalism may be tested by considering the recent experimental results of Shao and Girault [l]. (a) For the transfer of an ion from (w) + o, and from (w + s) + o Eqn. (20) may be written as ln k”‘”

- ln kwf”‘”

= aAG;” + ,+‘/RT = a(0.40

x W3)AG;

+ w+s

i.e. a plot of In k” -w+s versus AG,w*w+s should yield a straight line with a slope of - ~(0.40 x 10P3). Shao and Girault obtained a slope of -0.45 X 10m3, and hence cu = 1.1. (b) The linear relation between In D and AG, observed by Shao and Girault is a result of the LGER expressed in Eqn. (19) which itself is a consequence of the relations expressed in Eqns. (14) and (15). It is noteworthy that Shao and Girault carefully chose a system in which the viscosity of the medium (and hence the diffusion coefficient of the ion, acetylcholine, in the medium) was varied, while keeping the dielectric constant constant. Thus one could study changes in diffusion coefficient, and how these relate to the Gibbs energy of solvation (or of transfer), without including any effects of the dielectric constant - which would have affected AG,,,, (or AG,). As long as the dielectric constant is kept the same, one can also study changes in diffusion coefficients of

33

Fig. 2. In Dw versus (w) in ref. 8.)

AC,,,,,+ for tetraalkylammonium

ions. (AC,,,,

from ref. 7, D” calculated

from

Xq

different (but related) ions in the same solvent, as they relate to their respective AG,,t, (or AG,). In fact, a consequence of the above formalism is that one should observe a linear relation between In D and the Gibbs energy of solvation for a series of related ions. Thus, substituting Eqns. (14,15) in Eqns. (3,4) yields relations of the type In D = const - /3AG,,,,/RT

(22)

Fig. 2 shows such a relationship. Application of Eqn. (19) to the system studied by Shao and Girault yields In D” - In D”+”

=

~~(0.40 X 10-3)AG~*W+S

The slope of the plot of In D versus AG, obtained by Shao and Girault was -0.55 X 10e3, and hence (Y= 1.4, only slightly larger than the value of CYdetermined from the kinetic result above. The observation made by Shao and Girault, (Eqns. (1) and (2)), that the Gibbs activation energy for diffusion in aqueous solution,’ and ion transfer across a liquid/liquid interface follow almost the same dependence on the variation of Gibbs energy of hydration, therefore follows directly from Eqns. (19) and (20). (c) The linear relationship between diffusion coefficient and the rate constant observed by Shao and Girault follows from E!qn. (21). Thus, for the system studied by the latter authors, Eqn. (21) is D

W+S =

(,W/,W+.),W+S*.

Their experimental results could be expressed by the equation D = 2 X lop4 k, where the slope of 2 x 1O-4 agrees well with the value (D’“/k” ‘“) calculated from their Table 1. The above formalism may therefore be regarded as satisfactory. So, there must also be a similarity between ion transport and ion transfer as maintained by Shao and Girault [l], and Girault and Schiffrin [2]. In conclusion, the observations of Shao and Girault [l] summarized in the Introduction can be explained in terms of the activated transport model and

34

LGER’s. Such LGER’s do exist already, as shown by considering data from the literature on liquid/liquid systems. Apart from Figs. 1 and 2, which illustrate the validity of Eqns. (10) and (22), Samec et al. [9,10] have proposed Brijnsted-type relations~ps between the rate constant and the Gibbs energy of transfer. Such a relationship follows from the substitution of Eqns. (12) or (13) in Eqns. (5) or (6), leading to the result: In k w --)* = constant - CYAG~” + “/RT

(23)

All these LGER’s help to explain the similarity between ion transfer and ion transport. . In a more recent publication, Shao et al. [ll] have studied the kinetics of the transfer of acetylcholine across the water/nitrobenzene + tetrachloromethane interface and provided further experimental proof of the linear relationship between the rate constant for ion transfer and the corresponding Gibbs energy of transfer, thereby also confirming the concept of a linear relationship between the standard Gibbs activation energy and the Gibbs energy of transfer (or of solvation). From Eqns. (12-15), and the conclusion that (Y= p, the conditions 3 = B’ = 0 and C = C’ = 0 lead to an alternative connotation to the proportionality constant 1y, for then 1ymay be defined through the equation (see Eqn. (12)) (r = AGa;,; o’fbGP - O’

(24)

This definition of IY is similar to the “overall chemical transfer coefficient” introduced by Girault and Schiffrin [2]. A similar identification of the Bronsted coefficient for proton transfer reaction, 1y, with the transfer coefficient LY in electrochemical kinetics has, in fact, been made in the theoretical works of Dogonadze and Urushadze [12]. It is therefore tempting to speculate about the significance of the result (Y= 1 (approx.) deduced above from the results of Shao and Girault. If the chemical transfer coefficient signifies the degree of mixed solvation at the transition state, as stated by Girault and Schiffrin [2], then, in view of the similarity of the solvents (w) and (w + s) chosen by Shao and Girault, a value of unity may not be too surprising for the ion transfer process w + w + s. A further test of the above formalism is at present handicapped by the fact that reliable kinetic data for a series of ions in different w/o systems are lacking. It is hoped that future work in this area may shed some light on the attempt to predict and correlate kinetic data in a variety of w/o systems. ACKNOWLEDGEMENTS

Helpful discussions acknowledged.

and comments

by Dr.

B. Hundh~er

REFERENCES 1 Y. Shao and H.H. Girault, J. Eleetroanaf. Chem., 282 (1990) 59. 2 H.H. Girault and D.J. Schiffrin, J. Electroanal. Chem., 195 (1985) 213.

are gratefully

35 3 S. Glasstone, K.J. Laidler and H. Eyring, The Theory of Rate Processes, McGaw-Hill, New York, 1941. 4 T. Solomon, unpublished results. 5 B. Hundhammer, personal communication. 6 M.H. Abraham and J. Liszi, J. Chem. Sot., Faraday Trans. I. 74 (1978) 1604. 7 M.H. Abraham and J. Liszi, J. Inorg. Nucl. Chem., 43 (1981) 143. 8 R.A. Robinson and R.H. Stokes, Electrolyte Solutions, Butterworths, London, 1968, p. 463. 9 2. Samec, V. Marecek, and D. Homolka, J. Electroanal. Chem., 158 (1983) 25. 10 2. Samec and V. Marecek, J. Electroanal. Chem., 200 (1986) 17. 11 Y. Shao, J.A. Campbell and H.H. Girault, J. Electroanal. Chem., 300 (1991) 415. 12 R.R. Dogonadze and Z.D. Urushadze, J. Electroanal. Chem., 32 (1971) 235.