I. R. MILLER
2740
The activation energy for reaction 1 has been reHowever, '~ ported to be less thanabout 1k ~ a l . / m o l e . ~ ~ in view of the chain mechanism postulated above, it is
possible that it could be greater since reaction 1 could be masked by the much faster atomic hydrogen reaction.
Effect of Adsorbed Polyelectrolytes on the Polarographic Currents. I
by I. R. Miller Polymer Department, W e i m n n Institute of Science, Rehovoth, Israel
(Received March 10, 1966)
The diffusion current of an ionic depolarizer through an adsorbed polyelectrolyte layer of equal charge has been calculated. It was assumed that the diffusion coefficient of the depolarizer is considerably smaller in the surface layer than in the solution and that its distribution coefficient between the layer and the aqueous phase is given by the Donnan equilibrium. The experimentally established square root time dependence up to saturation level of the polyelectrolyte surface coverage was applied. Two extreme cases were considered, lateral interaction and no lateral interaction at partly covered surfaces. In the first case, the instantaneous current was assumed to be proportional to the surface coverage. In the second case, the distribution coefficient of the depolarizer and its diffusion coefficient in the surface layer were considered to be linearly dependent on the surface occupancy of the polyelectrolyte. The theory was checked with respect to experimental data obtained on positively charged polyvinylpyridine with Cd2+ as depolarizer.
Introduction The effect of surface-active agents on reversible diffusion currents has not been treated quantitatively although it has been studied experimentally by several a ~ t h o r s . l - ~There were, however, attempts to elucidate quantitatively the influence of surface-active substances on the electrode reactions and thus on the irreversible polarographic currents. Weber, Koutecky, and Koryta6 assumed the over-all rate constant keff to be linearly dependent on the coverage; Kuta and Webera further took into account the variation in potential by the adsorbed surface-active substance, which they assumed to be proportional to the coverage. They obtained for the instantaneous current i, at any mercury drop age t the expression
i,
- e) +
= nFqt*'aCo[ok,(l
x
exp(--x/m - l)(anF/RT)A@ ( 1 ) The Journal of Physical Chemistry
In this case, ok, and 112, are the rate constants for the free and covered surface, respectively, n is the number of electrons involved in the reaction, F is Faradays, p is a constant depending on the flow velocity of mercury, z is the charge of the depolarizer, a is the transfer coefficient, and A$ is the potential difference between the completely covered and the uncovered surfaces. The purposes of the present work is to elucidate the effect of adsorbed charged polyelectrolytes which partly or fully cover the surface on the diffusion current. (1) P. Delahay and I. Trachtenberg, J. Am. Chem. SOC.,80, 2094 (1958). (2) C.Tanford, ibid., 74, 211 (1952). (3) J. Kuta and I. Smoler, 2.Elektrochem, 64, 285 (1960). (4) C. N. Reilly and W. Stumm, Progr. Polaro., 1 , 81 (1962). (5) J. Weber, J. Koutecky, and J. Koryta, 2. Elektrochem., 6 3 , 583 (1959). (6) J. Kuta and J. Weber, Electrochim. Acta, 9, 541 (1964).
EFFECT OF ADSORBED POLYELECTROLYTES ON POLAROGRAPHIC CURRENTS
It is assumed that the current is determined by diffusion of the depolarizer through the aqueous phase and an adsorbed polyelectrolyte layer of finite thickness.
Theoretical a. The Model. We wish to calculate the instantaneous diffusion current when the depolarizer diffuses to the surface from a concentration CO in the aqueous solution (phase I), through a surface layer (phase 11) of thickness f , its concentration at the surface being maintained at zero. K is the distribution coefficient of the depolarizer between phase I and phase 11. The diffusion coefficient in phase I1 (011) differs from that in phase I (01).K and DII may depend on the occupancy 6 of surface phase 11, which is assumed to be equal' to (t/zY)'/', where 19 is the time required for full surface coverage to be reached. This assumption has been shown to be correct for polyelectrolytes of any composition.8-lO In our treatment we shall consider two alternatives, the first consisting of the case in which there is no lateral interaction between the adsorbed polyelectrolyte molecules, which move freely in the surface. I n this case DII and K for any depolarizer entering the surface layer will depend on its chances to hit upon a bare spot or upon an adsorbed polymer molecule. Hence, DII and K are assumed to be linearly proportional to the surface occupancy by polyelectrolyte molecules 8, that is, to (t/6)'/'.
+ = 1 +
= DI
&I K
@sat (KBat
- D~)(t/fi)'/' - 1)(t/6)1/2
(2)
i, =
io,,
- (io,, - is,t)(t/19)'/2
(4)
I n this expression io,,and ia,,are the currents a t time t when the surface is uncovered and when it is fully covered, respectively. b. Solution of the Diflusion Equation. The differential equations in the two phases
and
bC1I --
at
- DII
b2CII BX2
(5)
~
give, for the initial conditions, at X = 0, C = 0 for t 2 0, and at X > 0, C = Co for t = 0, the following solutions CI
co- B I
=
[I - e r f ( ~ / 2 6 t1)for 4
