WILLIAM G. STEVENS AND IRVINGSHAIN
2276
served value. Remembering that the triangular distribution, rather than the Schulz distribution is more appropriate for this type of polymer, we may conclude that the polymer S l l l displays Zimm-like, instead of Rouse-like, behavior in Aroclor.
Acknowledgment. Discussions with Professor T. Kotaka of this university have been indispensable during the course of this study. We are also indebted to Professors J. D. Ferry and J. E. Frederick for their kind supply of the solvent and the living polystyrene.
Electrolysis with Constant Potential. Diffusion Currents of Metal Species Dissolved in Spherical Mercury Electrodes
by William G. Stevens1 and Irving Shain Department of Chemistry, University of Wisconsin, Madison, Wisconsin (Received January 26,19fi6)
~
~~
The hanging mercury drop electrode was used in a potentiostatic method for the determination of the diffusion coefficient of metal species in mercury. The amalgam was prepared in a constant-potential preelectrolysis step, during which the quantity of electricity involved in the electrodeposition was measured to determine the amalgam concentration. Then, current-time curves were obtained for the diff usion-controlled dissolution of the amalgam. Diffusion coefficients calculated from the time-dependent current decay were compared with those calculated from the “spherical correction” term in a critical evaluation of the method. As a demonstration system, the diffusion coefficient of cadmium in cm2/sec, relative standard deviation mercury was measured. The value was 1.61 X 3.0%.
The use of stationary spherical mercury electrodes in potentiostatic experiments has been suggested prev i ~ u s l for y ~the ~ ~unambiguous determination of the diffusion coefficients of electroactive species. By evaluating the diffusion coefficients from both the slope and the intercept of potentiostatic current-time curves, it was shown that the uncertainties involving other experimental parameters, particularly the electrode area, could be minimized. This method has been applied previously only for the determination of reducible ions in the solution. It is obvious, however, that the same approach could be used to determine diffusion coefficients of metal atoms in a hanging mercury drop electrode, by analyzing the current-time curves obtained from potentiostatic experiments involving the oxidation (dissolution) of the amalThe Journal of Phyaical Chemistry
gams. Developing the method appeared to be of considerable importance as a result of the possibility of studying intermetallic compound reactions in the merc ~ r y . For ~ any such studies, accurate values of the diffusion coefficients are required. The use of the hanging mercury drop electrode in stripping analysis had indicated that it would be possible to prepare amalgams of known concentration by electrodeposition in a carefully controlled preelectrolysis step.5 This work was carried out to determine exactly (1) National Science Foundation Predoctoral Fellow, 1961-1965. (2) I. S h a h and K. J. Martin, J . Phys. Chem., 65, 254 (1961). (3) I. Shain and D. S. Polcyn, ibid., 65, 1649 (1961). (4) W. Kemula, Z. Galus, and Z. Kublik, Bull. Acad. Polon. Sei. Ser. Sei. Chim., ffeol. Geograph., 6 , 661 (1958); W.Kemula and Z. Galus, ibid., 7 , 553, 607, 729 (1959).
ELECTROLYSIS WITH CONSTANT POTENTIAL
2277
how precisely such hanging amalgam drops could be prepared, to present the detailed theory for the diffusion-controlled dissolution of the amalgam, and to make a careful comparison between the theory and experimental results for the potentiostatic dissolution of a cadmium amalgam.
For the oxidation of a metal dissolved in a stationary spherical mercury electrode R+O+ne
e - (nzroz/DRt)
(6)
n=-1
and since when n = 0, the exponential is unity, eq 5 can be written
i
=
nFA&CR*
[l/ro - ( l / ~ ) & ? - ' " ' r o ' ' D R t ) 1
+ (2/r) (dcR/dr)]
bCR/dt = D R[d2CR/dr2
t t
=
0,O 5 r
5 ro; CR = CR*
(1)
(3)
(2/l/?rDRt)
n=l
e-
-
or, since the integral is symmetrical around zero lim i = nFADRCR* l-
m
The definite integral in eq 9 can be evaluated from standard tables of integrals'
Thus, eq 9 reduces to lim i = ~ F A D R C R * [ ~ /-T O l/ro] t+
(11)
m
and the current goes to zero as the electroactive material in the finite volume of the electrode is depleted.
+ l / ~ om
t - +m
(4)
Here, CR is the concentration of the metal in the amalgam, CR* is its initial uniform concentration in the spherical mercury electrode, DR is its diffusion coefficient, t is the time, r is the distance from the center of the electrode, and ro is the electrode radius. The solution to this boundary value problem can be obtained by a straightforward application of the Laplace transform, or from the solution of the analogous problem in heat, transfer.6
[
lim i = nFADRCR*[l/ro
(2)
> 0, r -+ 0 ; CR remains bounded t > 0,r = ro; CR = O
~FADRCR*-l/(+)
I’
This method was thought to be of interest since it is not necessary to know explicitly the bulk concentration, number of electrons, or electrode area (although the radius must be known, which is equivalent). These results are also included in Table I, but this method of treating the data does not appear to be as precise as the others, possibly because it is more sensitive to small errors in determining the electrode radius. The results summarized in Table I are a realistic evaluation of the method, using relatively conventional instrumentation. Although literature values6Ja-15 of
The Journal of Physical Chemistry
D c ~ / Hranging # from 1.45 to 2.45 X lo4 cm2/sec have been reported, this method would seem to yield a less ambiguous result than the other methods. Any significant improvement in the precision probably would require extensive changes in the experimental approach, including better elimination of vibration and probably a digital read-out system. However, the technique is rapid and convenient, and the precision is adequate for most studies on amalgam systems, where it should find wide application.
Acknowledgment. This work was supported in part by the U.S. Atomic Energy Commission under Contract No. AT(l1-1)-1083. Some of the preliminary experiments were carried out by John Lewinson. (13) N. H. Furman and W. C. Cooper, J. Am. Chem. SOC.,7 2 , 5667 (1950). (14) M. S.Xakharov and A. G. Stromberg, Zh. Fiz. Khim., 38, 130 (1964). (15) A. G. Stromberg and E. A. Zakharova, Elektrokhimiya, 1, 1036 (1965).
