Stationary curve of current versus potential of facilitated transfer of cation across the interface between two immiscible electrolyte solutions

Stationary Curve of Current versus Potential of Facilitated Transfer of Cation across the Interface between Two Immiscible Electrolyte Solutions The facilitated transfer of a cation across the interface between two immiscible electrolyte solutions was first quantitatively characterized by Koryta (1). Analogous transfers of cations facilitated by some electroneutral ionoph0res (2-4) were observed during a study of the water/ nitrobenzene interface by cyclic voltammetry. The aim of this communication is to present the possibilities for the determination of transport, thermodynamic, and kinetic parameters of the facilitated transfer of a cation across the interface of two immiscible electrolyte solutions from the stationary curve of current vs potential or the curve of mean current vs potential of the abovementioned facilitated transfer at the electrolyte dropping electrode (5, 6). The simplest mechanism according to which the facilitated transfer o f a metal cation M "+ from the aqueous (w) to the nonaqueons (n) phase may proceed is represented by the following reaction [1] (4): rM=+(w) + sEX-(n) ~ M~L,('z-sx)(n)

[11

where the stoichiometry of the complex is assumed to be r:s (cation to hydrophobic ligand) and r, x, or (rz - sx) are the charge numbers of the metal cation, ligand, or complex, respectively. In such a case, the formation of the complex MrL~ occurs in one step, i.e., the formation of the activation complex requires the intimate presence of r particles M z+ and s particles Lx-. The electrical current I connected with the transfer of the positive charge from the aqueous phase to the nonaqueous phase has been defined as positive (6). The boundary condition for the facilitated transfer of the cation M ~+from the aqueous to the nonaqueous phase is formulated by Eq. [2] (4): I / r z F A = k ~-~" {[cld(w)]r[CL(n)] s -- polCML(n)}

[2]

where I is the faradaic current flowing through the interface, A is the interfacial area, gs are the concentrations of reactants M ~+, L x-, and M~L]=-'~) (shortly ME) at the interface, and k ' ~ " (or k n-~) is the potential-dependent rate constant of the charge transfer from the aqueous to the nonaqueous phase (or the reverse, respectively). The latter two quantities are interrelated with each other according to general Eq. [3] (4): Po = k ~ " / k . . . .

exp[rzF(E - E * ) / R T ]

[3]

where E is the potential, which is the Galvani potential difference A,~O = ~O(w) -- ~O(n)between the phases in contact related to the constant potential differences involved

in the reference electrodes (7). Based on a thermodynamic cycle, the formal potential E ~ for the charge-transfer reaction [1] can be expressed by Eq. [4] (4): E ~ = E ~ - ( R T / r z F ) In K

[4]

where E ~ is the formal potential for the ion transfer reaction M~+(w) ~ MZ+(n) [5] and Kis the stability constant of the complex MrLs in the nonaqueous phase, i.e., K is the apparent equilibrium constant of the chemical reaction [6]: rMZ÷(n) + sEX-(n) ~ MrL,(~-~)(n).

[6]

Let us suppose that an indifferent base electrolyte is present in excess in each phase so that the migration contributions to the material fluxes may be neglected. If the adsorption of reactants at the interface can be also neglected, then with regard to the results of a paper (4) and with the use of the stationary Nernst-diffusion layer treatment the following relation [7] may be written: I = KM(W)[C° -- CM(W)] = rKML(n)CML(n) = (r/S)rL(n)[COL -- CL(n)]

[7]

where c o (i = M, L) is the bulk concentration of a reactant and the coefficients K~(1 = M, ML, L) are given by Eqs.

[8]-[10]: r ~ w ) = zFADM(W)/SM(W),

[8]

KML(n) = zFADML(n)/~ML(n),

[9]

~L(n) = zFADL(n)/bL(n)

[10]

where Dj is the diffusion coefficient and ~j is the thickness of the Nernst diffusion layer. When the concentration of a reactant turns to zero, its material flux approaches the limiting value to which the limiting diffusion current la.~ corresponds: Id,M = KM(W)CO ,

[1 1]

Id,L

[12]

=

(r/s)rL(n)c °.

Finally, the substitution of concentrations of reactants from Eq. [7] into the boundary condition [2] yields the general equation of the stationary current-potential curve. Two limiting cases can be distinguished. I. If c ° ~> c ° , the concentration of the ligand in the nonaqueous phase is practically constant, i.e, cL(n)

595 0021-9797/84 $3.00 Journal of Colloid and Interface Science, Vol. 97, No, 2, February 1984

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596

NOTES

-~ c °, and the equation of the stationary current-potential curve is given by (/d,M -- I) r I

[KM(W)] r rzFA(C°L)Sk ~

q

POI[KM(w)]r rrML(n)(COL) s"

[13]

Equation [ 13] corresponds to the situation that the charge transfer process is controlled both by the diffusion and by the charge transfer reaction at the interface (so called quasi-reversible charge transfer). At the sufficiently negative values of E the second term on the right-hand side of Eq. [13] should dominate (we suppose that the rate constant k ~-~" increases with E less sharply than the function Po) and (Ia,M -- I)r/I = PffI[KM(W)y/rKML(n)(cO)S.

[14]

In this case the equilibrium in the charge transfer reaction is preserved at the interface even under current flow, i.e., the charge transfer process is controlled solely by the diffusion of reactants (reversible charge transfer). When E ~ + ~ , the right-hand side of Eq. [13] decreases to zero and I approaches its limiting value I~m = Id,M.

[15]

2. On the other hand, if c ° ~ c ° , the concentration of the metal cation in the aqueous phase can be considered as practically constant, so that CM(W)= C° , and the stationary current-potential curve of the quasi-reversible charge transfer has the form (Id, L -- I ) ~

[rL(n)]Sr s-l

I

zFA(c°)rsSk '~"

eol[rL(n)]Sr s-I

+

(c°)rs ~

[161

Analogously, at the sufficiently negative values of E the second term on the right-hand side of Eq. [16] will dominate, so that Eq. [17] results from Eq. [16] for the case of the reversible charge transfer (k "~" ~ +oo): (Id,e - I)S/I = Pol[rL(n)]Sr~-l/(c°)'s.

[17]

From the simple analysis of Eq. [16] it follows that for E ---o + oo the current I approaches the limiting value I[im = Id,L-

[18]

The method of the electrolysis with the electrolyte dropping electrode suggested in previous works (5, 6) is based on the same principles as polarography with the dropping mercury electrode (8). As the polarographic conditions can be handled with sufficient accuracy by the stationary diffusion layer description (8), the relationships derived in the present communication may be used in the analysis of the mean current vs potential curves of the charge transfer reaction [ 1] at the electrolyte dropping electrode. In such a case the coefficient Kj is identical with the coefficient of the Ilkovi6 equation (8, 9) rj = 3.573zFDyZm2/3t~/6

[19]

Journal of Colloid and Interface Science, Vol. 97, No. 2, February 1984

where m is the electrolyte flow rate (in cm 3 sec-~) and h is the electrolyte drop time (in sec). The mean electrolyte drop area (A) (A~ = 2.90mZ/3tl/6

[20]

is substituted for the interfacial area A. The analysis of the experimental curves of mean current vs potential should proceed as follows. (a) From the dependences of the fimiting mean current on the bulk concentrations of reactants--see Eqs. [11], [12], [15], and [18]----~e coefficients rM(w) and (r/S)KL(n) are found. From these coefficients and from the value of (r/s) obtained in step (b), the diffusion coefficients DM(W) and DL(n) can be calculated using Eq. [19]. (b) Logarithmic analyses of the data corresponding to the reversible charge transfer on the basis of Eqs. [ 14] and [17] yield the value of the formal potential E ~ and the values of r and s under the assumption that the diffusion coefficients of the ligand and the complex are the same, i.e., DL(n) = DML(n). From Eq. [4] the stability constant K of the complex M,Ls in the nonaqueous phase can be found. (c) Finally, the kinetic law for the charge transfer reaction [1], i.e., the dependence of the rate constant k"~" on potential E, can be obtained on the basis of Eq. [13] or [16], respectively. REFERENCES 1. Koryta, J., Electrochim. Acta 24, 293 (1979). 2. Homolka, D., Hung, L. Q., Hofmanovfi, A., Khalil, M. W., Koryta, J., Mare~ek, V., Samec, Z., Sen, S. K., Van~sek, P., Weber, J., B~ezina, M., Janda, M., and Stibor, I., Anal. Chem. 52, 1606 (1980). 3. Hofmanov~i, A., Hung, L Q., and Makrllk, E., Proceedings of J. Heyrovsk~ Memorial Congress on Polarography, Prague, August, 25-29, 1980, p. 66. 4. Samec, Z., Homolka, D., and Mare~ek, V., J. Electroanal. Chem. 135, 265 (1982). 5. Koryta, J., Van~sek, P., and B[ezina, M., 3". Electroanal. Chem. 67, 263 (1976). 6. Koryta, J., Van~sek, P., and B[ezina, M., J. Electroanal. Chem. 75, 211 (1977). 7. Samec, Z., Mare~ek, V., and Weber, J., J. Electroanal. Chem. 100, 841 (1979). 8. Heyrovsk~,, J., and Kfita, J., "Principles of Polarography." Publishing House of the Czechoslovak Academy of Sciences, Prague, 1965. 9. Samec, Z., J. Electroanal. Chem. 103, l (1979). EMANUEL MAKRLfK Nuclear Research Institute 250 68 l~e~, Czechoslovakia Received March 29, 1983; accepted June 27, 1983