Charge transfer between two immiscible electrolyte solutions

J. Electroanal. Chem., 99 (1979) 197--205 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

197

CHARGE TRANSFER BETWEEN TWO IMMISCIBLE ELECTROLYTE SOLUTIONS PART I. BASIC EQUATION FOR THE RATE OF THE CHARGE TRANSFER ACROSS THE INTERFACE

ZDENI~K SAMEC

J. Heyrovsk~ Institute of Physical Chemistry and Electrochemistry, Czechoslovak Academy of Sciences, U tovSren 254, 102 O0 Prague lO-Hostiva~ (Czechoslovakia) (Received 18th October 1978)

ABSTRACT The theoretical fundamentals of the kinetics of ion and electron transfer reactions at the interface between two immiscible electrolyte solutions have been discussed on the basis of the present theory of chemical reactions in polar media.

INTRODUCTION

In principle there are two types of charge transfer across the interface of two immiscible electrolyte solutions 1 and 2, (a) the transfer of an ion X z, with a charge number z, from the phase 1 to the phase 2 and reverse, X z (1) ~ X z (2)

(1)

or (b) the electron transfer between the particle in the phase 1 and the particle in the phase 2, which can be written as Red1 (1) + Ox: (2) ~ Ox~ (1) + Red: (2)

(2)

At the interface a Galvani potential difference A~ = ~(2) - - ~ ( 1 ) is formed. Under equilibrium conditions the sum of the electrochemical potentials of the reactants in a transfer reaction, which takes place at the phase boundary, must be equal to the sum of the electrochemical potentials of the products. After introducing the quantity go = po + RT In 7, where p0 is the standard chemical potential and ~/is the activity coefficient, we get for the equilibrium (1) A~ = A~ ° + ( R T / z F ) ln[c(1)/c(2)]

(3)

In this expression c(1) and c(2) are the concentrations of the ion X z in solutions 1 and 2, respectively, and the Galvani potential difference A¢ ° at [c(1)/c(2)] = 1 is related to the apparent Gibbs energy of ion transfer AGt°~1-~ 2 from the phase 1 to the phase 2, "A~P0 = --( AV°t;1 -' 2/zF) = --[U°(2) -- U°(1) ]/zF

(4)

198 In a similar way, we obtain for the equilibrium (2) A ~ = A~p° + ( R T / n F ) In[cox,(I)

Cred2(2)/Credl(1) Cox2(2)]

(5)

where n is the number of electrons transferred in the reaction (2) and A~ ° is • related to the apparent reaction Gibbs energy A G o 0 2 ) - - g r0e d l ( 1 ) - - P°ox2(2)]/nF A ~ ° = A V ° / n F = [P°oxl(1) + Pred2(

(6)

When the system is out of equilibrium, the charge transfer between the phases 1 and 2 occurs mostly in one direction. During the last several years the kinetics of the transfer of some ions through the water/nitrobenzene interface was studied extensively using both galvanostatic [1--3] and potentiostatic [4--7] methods. The results of these investigations indicate that the polarization phenomena at the interface of two immiscible electrolyte solutions have the same character as those at the metal/electrolyte interface [6]. Namely, they involve the charge transfer reaction at the phase boundary and the transport of the reacting species from the bulk solution to the interface. For the theoretical analysis of these phenomena it is necessary to deduce a basic law for the rate of the charge transfer reaction with particular attention being paid to its dependence on the Galvani potential difference A~. A simple approach has been put forward [3,6] based on analogy with electron transfer reactions at metallic electrodes. We shall discuss this point in more detail starting from the present theory of chemical reactions in polar media [8--13]. GENERAL REMARKS The transfer reactions in question are in fact the charge redistributions between two polar solvents. According to the qualitative considerations [8--13], the strong electrostatic interaction of the charged reactant with the dipole solvent molecules will result in changes in the solvent state, namely in the change of the mean dipole m o m e n t of unit volume (i.e. polarization of medium). Conversely, because the electrostatic interaction is a reciprocal one, the change in the polarization of the solvent can induce a change in the source of the electrostatic field, especially the electron energy levels of the reactants will be displaced with respect to their positions in vacuum. Both effects depend mainly on the charge of the reactants and on dielectric properties of the solvent. In the course of the charge transfer of the type (1) or (2) the electron state of the reacting particles as well as the polarization state of both solvents in contact and, in the case of the ion transfer (1) the space configuration of the ion X ~, can change considerably. In the more general case it is not also possible to neglect the change in the molecular structure of the reactants including the deformation of the valence bonds, their rupture or formation. The role of the different parts of the total system (electrons, solvent, intramolecular vibrations, reactant as a whole) in the transfer reactions (1) or (2) can be analysed with the help of the theory of the radiationless electron and atom transfer reactions in polar media [8--13]. In particular, the thermal fluctuations in the classical subsystem (mainly the orientation vibrations of solvent molecules and the classical intramolecular vibrations of reactants) must bring the system to an activated state favourable for the adiabatic or non-adiabatic transition of the quantum subsystem (elec-

199

trons and the quantum intramolecular degrees of freedom of the reactants). In the quantum theory of the charge transfer reactions in polar media [10-13,1__7] the reaction rate is expressed by means of the mean transition probability W of the elementary reaction path. In the case of the transfer reactions (1) and (2) the reciprocal value of W is the mean time which is necessary for one ion or one n-electron transfer to occur through the interface. The net specific rate v(1 ~ 2), i.e. the number of the ion or n-electron transfers through the unit area of the interface in unit time from the phase 1 (here the initial state of the system) to the phase 2 (the final state), can be written as V(1 "~ 2)

= (1/A)(Wif

--

Wfi )

(7)

where A is the interfacial area and Wif or Wfi are the probabilities of the transition from the initial (i) to the final (f) state or vice versa, respectively. If the Gibbs distribution may be used for the energy levels of the non-equilibrium system, a simple relation follows from the principle of detailed balancing [15] W~i = Wif e x p ( A F / k T )

(8)

where k is the Boltzmann constant and the quasi-equilibrium quantity AF = F ( 2 ) - F(1) is the difference between the electrochemical Helmholtz energies * (related to one particle) of the reactants in the phases 2 and 1. For the ion transfer reaction (1) AF can be expressed as A F = z e ( A G -- AGo) + k T ln[c(2)/c(1)]

(9)

where e is the elementary electronic charge. Analogously, in the case of the electron transfer reaction (2) , ~ = - - n e ( A G - AG°) + k T In[cox1(1) Cred:(2)/Credl(1) Cox2(2)]

(10)

Under equilibrium conditions AF and v(1 -~ 2) tend to zero and from the equations (9) and (10) we get the equations (3) and (5), respectively. The mathematical form of the expression for the mean transition probability Wif depends on the physical model of the system. For the present discussion we shall suppose that a non-reacting supporting electrolyte is present in each phase at concentration much larger than that of species participating in the transfer reactions (1) or (2). As proposed by Gavach et al. [16] (cf. also ref. 19) at the plane of the contact between two immiscible electrolyte solutions a compact layer of the oriented molecules of both solvents is formed which separates two space-charge layers (diffuse double layer). This situation is schematically illustrated in Fig. 1. It may be expected that the short-range repulsive forces between a particle and the solvent molecules in the compact layer impede the penetration of the particle into the compact layer. While in the solution including the space-charge regions the behaviour of a particle as a whole is usually classical, the change to quantum behaviour may take place close to the phase boundary if this repulsion is such that the potential energy of the particle increases very sharply starting at some distance from the plane of contact. Then it is possi* As m e n t i o n e d in ref. 12 the change in volume during the reaction in solution is n o t usually large, so that the change in Helmholtz energy may be practically replaced by the change in Gibbs energy.

200

bulk solution 2

dilluse layer

x2(2} compact layer

plane o! conlsct

x2(1) diffuse layer

bulk solution 1

Fig. 1. Schematic diagram of the electrical double layer at the interface of two immiscible electrolyte solutions 1 and 2.

ble to postulate the existence of two outer planes of the closest (classical) approach (the planes x2(1) and x2(2) in Fig. 1), which separate the space regions of the classical and quantum behaviour of a particle as a whole *. For the sake of simplicity we shall assume that the movement of the supporting electrolyte ions as well as of the reacting particles outside the compact layer is n o t only classical b u t also quasistatic. Actually we shall neglect the dynamic effect of the ionic atmosphere [ 17] on the charge transfer kinetics. For the ion transfer reaction (1) a more detailed model of the c o m p a c t layer including the dynamic behaviour of the solvent molecules would be probably necessary. Here we shall only suppose that the ion participating in the reaction (1) must penetrate through a static repulsion barrier of a certain limited hight. Lastly, the intramolecular degrees of freedom of reactants as well as those of the solvent molecules in the solvation shell of reactants will be included into the quantum subsystem. Their possible contributions to the transition probability may be taken into account in the second approximation. Thus, the classical subsystem contains only the classical degrees of freedom of the solvent around the solvated reactants. Based on the non-adiabatic treatment of the charge transfer reactions in polar media [ 10,17], there exists an optimum configuration {R*} of the reactants in the transfer reactions (1) or (2) which results from the competition of two factors. The first factor is the requirement of the maximum overlap of the wave * If, along with the repulsion some attraction is possible, the particle may be located inside the compact layer at an inner plane of closest approach [16 ] or, in other words, it is adsorbed at the interface. This case will not be considered here.

201

functions of the quantum subsystem in the initial and final states and the second factor is the repulsion between the reacting particles and the solvent molecules in the compact layer at the short distances. Obviously, the points {R*~ are located in the planes of the closest approach x~(1) and x~(2). If the quasistatic approximation is used for t h e m o v e m e n t of heavy particles, the formula for the mean transition probability Wit can be written in this approximative form Wif = W i f ( { R * ) ,

AJ*) ~(i)({R*}) 5{R*}

(11)

where Wit( (R* }, AJ*) is the transition probability at the fixed optimum configuration (R*) of the reactants, ~P(i)(,(R*} ) is the density of probability of the reactants having the configuration (R } in the initial state (the particle distribution function) and 5 (R*} is the product of the elements of the configuration space of the reactants which give the maximum contribution to the mean transition probability Wi~"In general, the transition probability at fixed configuration of reactants Wi~((R*), AJ*) can be expressed as [10,17] Wit( {R *),/~J*) = B* exp(--Ea( AJ*) /k T)

(12)

where the pre-exponential factor B* = B( (R* t) involves the parameters of the quantum subsystem, namely the overlap integral of the wave functions of the quantum subsystem in the initial and final states, and E a is the activation energy of the classical subsystem which is a function of the difference between the ground state energies of the initial and final states of the whole system •J* = AJ(~R*) ). In the case of atom transfer reactions, the theory of which may be applied to the ion transfer of the type (1), the penetration of the reactants into the classically forbidden region may be necessary and B* will involve the tunneling factor depending on the potential energy of the reactants in this region [10]. If the change of classical to quantum behaviour of a particle as a whole, close to the phase boundary, is rather gradual, the theoretical analysis complicates considerably (cf. the theory of atom transfer reaction [ 10]). In this case also the concept of the plane of the closest approach becomes obscure. The dependence of the activation energy E a o n AJ* is of particular importance for the establishment of the relationship between the charge transfer rate and the Galvani potential difference A~. In the simplest case, to which we shall confine here, the classical subsystem contains only the classical degrees of freedom of both solvents. Using the dielectric formalism [ 10,15 ], according to which the solvent is represented by a set of harmonic oscillators, the following equation can be deduced for activation energy Ea [10,13--15] Ea = (~ + AJ*)2/4~ = ~/4 + ~ J *

(13)

where ~ is the reorganization energy of both solvents in contact and the function a = 1/2 + ,5J*/4~

(14)

is introduced to simplify further treatment. The quantity/~J* practically coincides with the change in the electrochemical internal energy of the system (see e.g. ref. 12) during one ion, or one n-electron transfer from the plane of the closest approach in the phase 1 to that in the phase 2. If the classical subsystem contains only the classical degrees of freedom of both solvents, the entropy changes may be neglected [12,15] and the change in the electrochemical

202

Helmholtz energy may be substituted for AJ*. Thus, in the case of the ion transfer reaction (1) AJ* = ze ~0(x2(2)) -- ze ~(x2(1)) -- ze A ~ ° = z e ( A ~ - - A~o° +

~2(2) -- ~2(1))

(15)

where ~2 is the difference of the electrical potential at the plane of the closest approach and of the potential in the bulk of the solution ~2(1) = ~(x2(1)) -- ~(1)

(16)

~2(2) = ~(x2(2)) -- ¢(2)

(17)

Analogously, for the electron transfer reaction (2) we have AJ* = --ne ~(x2(2)) + ne ~(x2(1)) + ne A~O =

- - n e ( A ~ -- A~ ° + ¢2(2) -- ¢2(1))

(18)

RATE OF ION TRANSFER REACTION

In the case of the monomolecular reaction (1) the element of the configuration space 6 {R*} = 6R~ in the phase 1 giving the maximum contribution to the mean transition probability Wi~ may be replaced by the volume of the surface monolayer {R* } = A d

(19)

where d is the linear dimension of the ion, and the classical expression may be used for the single-particle distribution function ~P(i)({R* }) [17,18] ~(i)( {R* }) ----( N ( 1 ) / V , ) exp(--ze ~2(1)/hT)

= c(1) exp(--ze ~ 2 ( 1 ) / k T )

(20)

where N(1) is the number of the ions X z in the phase 1 and V, is the volume of the phase 1. Then, after defining the true rate constant k t k t =

B*d exp(--k/4 k T )

(21)

and the apparent rate constant ha°pp k°av, = kt exp{--ze[(1 -- a)~2 (1) + a~2(2)]/hT}

(22)

we can formulate the basic law of the ion transfer kinetics by a combination of eqns. (7)--(9), (11)--(13), (15), (19) and (20) Vion(1 ~ 2) = k' ~2 c(1) -- k 2 -,1 c(2)

(23)

where the rate constants h' -+ 2 and h 2 -+ ' depend on the Galvani potential difference A~ h ' -* 2 = k oa~ e x p [--aze( A~o -- A~°) /h T]

(24)

h2 -* 1 = k0app exp[(1 -- a ) z e ( A ~ -- A ~ ° ) / k T ]

(25)

The formal similarity between the equations (22) to (25) and the equations describing the rate of the electron transfer reaction at a metallic electrode is obo

203 vious. Here, of course, the apparent rate constant (22) involves the contributions from both sides of the electrical double layer. The parameter a, which may be called the formal charge transfer coefficient, also depends on A~ a = 1/2 + ( z e / 4 ~ ) ( A ~ - - A ~ ° + ~2(2) -- ~2(1)) It is to be stressed, however, that a has no physical significance. The quantity which is of real physical significance is the true charge transfer coefficient at characterizing the change of the activation energy E a with d~J* [10,13] a t = a E a / a A J * = 1/2 + A J * / 2 ~

= 1/2 + ( z e / 2 ~ ) ( A ~ - - A ~ ° + ~ ( 2 ) -- ~ ( 1 ) )

(27)

According to the basic theory [10,12], at may vary only between 0 and 1, so that the expressions (13), (14), (26) and (27) are applicable only for IAJ*I ~< (so called normal region of transition). In the case when AJ* > X, at = const = 1 (barrierless region) and in the equations (22), (24) and (25) the parameter a must be replaced by a = 1. If, on the other hand, AJ* < -- X, a t = const -- 0 (activationless region), and a must be replaced by zero. It may be expected that, if the Galvani potential difference A~ does not differ much from A~0, both charge transfer coefficients ~ and at are close to one half. RATE OF ELECTRON TRANSFER REACTION The electron transfer across the interface of two immiscible electrolyte solutions is the result of a bimolecular reaction (2). Therefore, the two-particle distribution function must be substituted for ~¢i)((R* }) in eqn. (11) and ~ (R*~ is the product of the space elements 8R~ and 5R~ of the phases 1 and 2, respectively. Apparently the points R* may be located anywhere in the planes of the closest approach, but only those two-particle configurations, with as small as possible an interparticle distance, will be preferred. Thus, while the most favoured configuration of one of the reactants may vary throughout the surface layer of the dimension ~ R * = A d , the configuration of the second reactant will be restricted to the small volume of molecular dimensions ~R* = Vm adjacent to the phase boundary and 5(R*} = 5R*~ ~R*2 = A V m d

(28)

We shall assume that the contribution from the mutual interaction of the reactants to the potential energy of the reactants as a whole may be neglected in comparison with the contributions from the interactions of the reacting particles with the electrical field of the supporting electrolyte ions. This assumption does not influence the bilinear form of two-particle distribution function with respect to the concentrations of both reactants, but only simplifies the calculation of the configuration integral in ~¢i)( (R* }), so that ~¢i~((R* }) may be expressed as the product of two independent single-particle distribution functions of the type (20) ~(i)((R* }) --- Credl(1) Cox2(2) exp{--[Zrea, e~2(1) + Zox 2 e ~ 2 ( 2 ) ] / k T }

(29)

Then, by a combination of eqns. (7), (8), (10)--(13), (18), (28) and (29) we get

204

for the rate of the electron transfer reaction vel(1 ~ 2) = k I -*2 Credl(1) Cox2(2 ) _ k 2-~ , Coxl(1 ) Cred2(2)

(30)

where the rate constants k I -* 2 and k 2 -* 1 d e p e n d on the Galvani potential difference A~ according to the relationships kl -* 2 = ka0pv exp[ane(A~0 -k2-~ 1 = kapp0

A~°)/kT]

exp[--(1 -- a ) n e ( A ~ --

(31)

A~°)/kT]

(32)

The apparent and true rate constants k°app and kt, respectively, are defined b y k°app = kt exp{--e[(Zred, + an)~2(1) + (Zox2 --

cm)~P2(2)]/kT}

(33)

and

kt = B*Vmd e x p ( - - k / 4 kT)

(34)

The formal charge transfer coefficient a in eqns. (31) to (33) can be expressed as

a = 1/2 --

ne(A~o -- A~ ° + ~o2(2) -- ~02(1))/4X

(35)

The restrictions imposed u p o n the applicability of the eqns. (31)--(33) and (35) are the same as m e n t i o n e d at the end of the previous section. The solvent reorganization energy X for the electron transfer reaction (2) was calculated by Kharkats [14] w h o used the m o d e l in which the reactants are treated as spheres with h o m o g e n e o u s surface charge distribution [9,11]. The expression for X has the f o r m X = (l/87r)(1/eopl -- 1/esl) It + (l/87r)(l/eop2 -- lies2) I2

(36)

where the indexes 1 or 2 refer to the phases 1 or 2, respectively, Cop is the square of the refractive index, es is the static dielectric c o n s t a n t and I is the f u n c t i o n of the geometric and dielectric parameters of the system which is p r o p o r t i o n a l to

(ne) 2. In conclusion we n o t e t h a t the oversimplifications involved in the physical model of the system used in the present paper m a y be r e m o v e d whenever necessary, b u t this seems to be of little use until the a m o u n t of d a t a on charge transfer kinetics is gathered and c o m p a r e d with the simple t r e a t m e n t outlined above. REFERENCES

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