Contribution to stationary curve of current vs potential of electron-transfer between two immiscible electrolyte solutions

CONTRIBUTION TO STATIONARY CURVE OF CURRENT VS POTENTIAL OF ELECTRON-TRANSFER BETWEEN TWO IMMISCIBLE ELECTROLYTE SOLUTIONS EMANUEL MAKRL~K Nuclear Research Institute, 25068 tie!& Czechoslovakia (Received

Abstract-Theoretical

electron-transfer

24 March

1983; in

revisedfirm 11 May 1983)

stationary curves DCcurrent us potential that are mathematical expressions of an

reaction,

01 (w) + R2 (non) +Rl(w)+02(non), proceeding at the interface of two immiscible phase-sbetweenthe redox couple Ol/Rl in the aqueous phase (w) and the redox couple 02/R2 in the non-aqueous phase(non) has been derived. Procedures for the determination of the diffusion coefficients of the reactants in the corresponding phase and for the evaluation ofthermodynamic and kinetic parameters of the electron-transfer reaction from the curve of mean current vs potential of this electron-transfer at the electrolyte dropping electrode have been suggested.

INTRODUCTION The ion-transfer and the electron-transfer raresent two alternative ways of the charge-transfer across the interface of two immiscible electrolyte solutions[l, 23. Theoretical fundamentals oftheir kinetics based on the present theory of the charge-transfer processes in polar media[3] were worked out and the first- or the secondorder kinetic law was established for the ion or the electron-transfer reaction, respe?-*7ly. iU”F BP.*. The equation of the current-Prential curves was derived and analysed using the stationary Nernstdiffusion layer treatment with the migration and adsorption effects neglected[4]. The method was suggested for the analysis of the curve of the mean current vs potential of the electron-transfer reaction at the electrolyte dropping electrode[4]. The electron-transfer reaction bet ween hexacyanoferrate redox couple in water and ferrocene in &trobenzene was det&ted by cyclic voltammetry with a four-electrode systern[5, 6-J. The phenomenological relations found were discussed in view of the theory of the stationary current-potential curve providini a qualitative understanding of the observed phenomena[6]. Application of the convolution potential sweep voltammetry to the quantitative analysis of the linear potential sweep or cyclic voltammetry of ion- or electron-transfer across the interface between two immiscible electrolyte solutions was suggested and discussed[7]. This method was used for a study of the above-mentioned electron-transfer reaction between hexacyanoferrate redox couple in water and ferrocene in nitrobenzene[8]. The aim of our communication is to present the possibilities of the determination of thermodynamic and kinetic parameters of the electron-transfer rcaction from the stationary curve of current us potential or the curve of mean current us potential of the electrontransfer reaction at the electrolyte dropping electrode

in somewhat more general form than in the previous

paperC41.

RESULTS

AND DISCUSSION

Let us consider the electron-transfer 0 1 (w) + R2 (non) *Rl

reaction,

(w) + 02(non),

(I)

which takes place at the boundary of the aqueous(w) and the non-aqueous(non) electrolyte solutions, between the redox couple Oi/R1 in the aqueous phase and the redox couple 02/R2 in the non-aqueous phase. The electrical current I connected with the transfer of the positive charge from the aqueous phase to the non-aqueous phase has been defined as positive[l]. If the adsorption of reactants at the interface can be neglected, the boundary condition for the electrontransfer reaction is formulated by equation (2)[7], - I/nFA

= k~+nonca,(w)cOz(non)

--kp”‘“e,,(w)c,(non),

(2)

where ci’s are the concentrations of reactants at the interface, n is the number of electrons transferred in reaction (l), A is the interfacial area, and k;l_ nonor k$‘“-w are the potential-dependent rate constants of the electron-transfer from the aqueous to the nonaqueous phase or the reverse, respectively. The latter two quantities are inter-related with each other according to the general equation (3)[7],

kT -

w= k;‘-‘exp

[nF(E-E’)/Rq,

(3)

where E is the potential, which is the Galvani potential difference ArO:,,lp= q(w) --(non) between the phases in contact related to the constant potential differences involved in the reference electrodes[9]. The formal potential of the electron-transfer reaction (1). E“, is connected with the formal potentials EoIIRI(w) and E ozBal(non) of the redox couple 0 l/R 1 in the aqueous phase and the redox couple 02/R2 in the non-aqueous

12

EWANUELMAKRLIK

phase, respectively (both related to the same reference electrode) by relation (4)[7], E0 = Eoz,iRz(non) - Eo,/lU(w).

(4)

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. With the use of the stationary Nernst-diffusion layer treatment the following relation can be written[4]: i = x01(w) Ccb,(w) - co,(w)3 = - KR1(w)C&w) - CR1WI

= -

~0dnOn) Cc%noQ - c0dnon)l

of the stationary current-potential curve at E + + co (or E -S - Q)). Consequently, this limiting current is proportional to the bulk concentration of one of the reactants on the left (or right)-hand side of (1) and independent of the bulk concentration of the other reactant. If c”,, (w) = 0 or c&(non) = 0, then Ii, = 0 [see (S), (9) and (13)], and therefore the current I belonging to the electron-transfer reaction (1) takes evidently positive values only. In the case that c&(w) = 0 or ci,(non) = 0, then from relations (7), (10) and (12) it follows that f&, = 0, so that, on the contrary, the mentioned current takes only negative values (I < 0) for any exoerimentallv accessible ootential values E. . If & (w), ci, (w), &,(non), c”,,(non), # 0, so that I&, current-... IL‘:- i 0, and therefore the stationary potential curve may be obviously character&d by the point of intersection of this curve with the axis 1 = 0. By the substitution I = 0 into relation (11) and by the simple solution of the equation resulted in this way we have the equilibrium potential E,[4], I

=

~R2(non)CC~2(non)-cR2(non)l,

(5)

where co is the bulk concentration coefficient ICYis given by K~

=

of a reactant

nFAD,/S,,

and the (6)

where Di is the diffusion coefficient and 6, is the thickness of the Nernst-diffusion layer. When the concentration of a reactant at the interface turns to zero, its material flux approaches the limiting value to which the limiting diffusion current IdVi corresponds[4]: ILO, = K01 (w) c&1 (w),

(7)

I&Rl = %,(w)c”,,(w),

(8)

z d. 02

=

(9)

I,,,,

= K&on)cL(non).

%2

bon)

&bonh

(IO1

By the substitution of concentrations of reactants at the interface from (5) into (2) and by the combination of (2) and (3) we obtain the general equation of the stationary current-potential curve in the following form:

~OIWKdnon)

E, = E” + (RThF) x In LcoR,bv)~O,~ Wn)/c&

@on)1.

=

(I +1d,R,)(l+1d,02)

KR1(W)K02(non)

x exp[-nF(E--E”)/R7-jI.

xexp[-nnF(E-E’)/RT]

(14)

Two limits may be distinguished as regards the effects of the electron-transfer kinetics on the stationary current-potential curve. If the equilibrium in the electron-transfer reaction (1) is preserved at the interface even under current flow, then the electron-transfer process is controlled only by the diffusion of reactants (reversible electron-transfer). This situation arises with the potential values E for which the second term on the right-hand side of (11) is negligible compared with the first term of this equation, ie (15) results from (11) for the case of the reversible electron-transfer,

(Id,,, - l)(fd,,,-J)

1

(w) CL

1

(15)

On the contrary, if the first term on the right-band side of (11) can be neglected with regard to the second one of the same side, then from [ll) we obtain the following relation: This equation (1l)corresponds to the situation that the electron-transfer process is controlled both by the diffusion and by the electron-transfer reaction at the interface (quasi-reversible electron-transfer). As E t + co, the right-hand side of (11) diminishes to zero and I approaches its limiting value either f li:, = idSo (when Id,0, d Id,R2) or Gm = Id,R2 (when Id,,2 d

Id.,,),

ie I&

=

min

{ld,Ol~

'd+Rd-

WI

From the simple analysis of (11) it follows that for E -+ - 03 the current I approaches the limiting value either I,;,,,= --I,,,,, when 14R, g IdSo or l,;, = -I,,,,, when Idlo < I,,,,, so that the limiting current 1&, is given by &, = -min

{ld.R1,

fd.02).

(13)

It is apparent that the lower one of the limiting diffusion currents I,,, of two reactants on the left (or right)-band side of (1) determines the limiting current

(Id,o, - I )(I,,,,-1)/I

= ,

Ko1(W)KR2@on), (16) nFAk;P” -‘*

which describes the irreversible electron-transfer. In thiscase the backward rqtion of the electron-transfer (1) from right to left is ptactically neglected. The method of the electrolysis with the electrolyte dropping electrode suggested in previous works[ 1, 10-j is based on the same principles as the polarography with the dropping mercury elecvode[l I]. As the polarographic conditions can be handled with sufficient accuracy by the stationary diffusion layer description[ 111. the relationships derived in the present communication or in Samec’s paper[4] may be used in the analysis of the mean current vs potential curves of the electron-transfer reaction (1) at the electrolyte dropping electrode. In such a case the coefficient K, coincides with the coefficient of the IlkoviE equation[4, 111, ?ci = 3.573 nFD!j2 r~?‘~ c;‘~,

(17)

Electron

transfer between two immiscibleelectrolyte

where m is the electrolyte flow rate (in cm3 s ‘) and t, is the electrolyte drop time (in s). The mean electrolyte drop area (A),
(18)

is substituted for the interfacial area A. The analysis of the experimental curves of mean current us potential should proceed as follows. (a) From the dependences of the limiting mean current on the bulk concentrations of reactants-see (12), 13 and (7)-( lOFthe coefficients K* are found, from which the diffusion coefficients of reactants may be calculated according to (17). (b) Analysis of the data corresponding to the reversible electron-transfer on the basis of (15) yields the value of the formal potential E” and the number n of electrons transferred in reaction (l), ie the value of the thermodynamic parameters characterizing this electron-transfer reaction. Under these conditions, the plot of the logarithm of the left-hand side of (15) vs potential E gives rise to the straight line having the reciprocal slope -2.303 RT/nF. This straight line intersects the potential axis at B = E” + (RT/nF) XIII [KcJI(W)

K~~(non,/Kn~(w)K,,(non)l.

In the case that c& (w), c$, (w), c& (non), c&(non) # 0, then the value of the formal potential E0 may also be inferred from the equilibrium potential E, using (14).

solutions

13

(c) Finally, the kinetic law on the electron-transfer reaction (I), ie the dependence of the rate constant ky-‘” on potential E, can be obtained on the basis of (11).

REFERENCES 1. J. Koryta, P. Vangsek and M. Bfezina, J. electroanaL them. 75, 211 (1977). 2. Z. Samec, J. electroanal. Chem. 99, 197 (1979). Spec. Period. Repl., 7’he 3. P. P. Schmidt, Electrochemistry Chemical Society 5, 21 (1975). 4. Z. Samee, J. elecrrounal. Chem. 103, 1 (1979). 5. Z. Samec, V. Mare&k and J. Weber, J. electroanul. Chem. 96, 245 (1979). 6. Z. Samec, V. MareEek and J. Weber, J. electroanal. Chem. 103, 11 (1979). 7. Z. Sarnec, 1. electroanal. Chem. 111, 211 (1980). J. Weber and D. Homolka, J. 8. Z. Samec, V. Mar&k, electroanal. Chem. 1245,105 (1981). 9. Z. Samec, V. Mare&k and J. Weber, J. electroanal. Chem. 100, 841 (1979). and M. Bfaina, J. elerrroonal. 10. J. Koryta, P. Van$ek Chem. 67, 263 (1976). 11. J. Heyrovsky and J. K&a, Principles of Polurography, Publishing House of the Czechoslovak Academy of Sciences, Prague (1965).