H. RUBINAND F. C. COLLINS
958
Vol. 58
CONCENTRATION OVERPOTENTIAL A T REVERSIBLE ELECTRODES1i2 BY H. RUBINAND F. C. COLLINS Department of Chemistry, Polytechnic Institute of Brooklyn, Brooklyn, N . Y . Received December SO, 1065
A study has been made of rate- and diffusion-controlled currents at a planar reversible electrode under applied potentials such that the reverse reaction makes an appreciable contribution to the over-all process. The ferrocyanide, ferricyanide and molybdocyanide systems were studied. The data obtained confirm an earlier prediction that the current under a potential substantially less than that corresponding to “limiting” current will converge at sufficiently long times t o the value of the “limiting” current at the corresponding time. Values of the specific rate constants were calculated from the data and enable an approximate determination of the fraction of the ions which react upon impinging on the electrode. The relation between the specific rate constants and the applied potential agrees with the Eyring theory of absolute reaction rates and reasonable values of the activation free energies are obtained.
Introduction Concentration overpotential effects at a planar electrode under potentials corresponding to limiting current flow have been investigated previously by Laitinen and Kolthoff.* Here it may be assumed that every ion diffusing up to the electrode surface reacts. The present investigation extends these studies to electrolysis under lower applied potentials where the rate limiting process at the electrode and the reverse reaction make appreciable contributions to the over-all reaction rate. The particular systems studied appeared t o have but a single ratedetermining step at the electrode, probably electron transfer, and t o be free from complicated side effects other than diffusion. The primary purpose of the investigation was the experimental verification of the previously proposed boundary condition4s6
tial period has expired. This circumstance is of Borne interest as it tends to mask the effect of reaction probabilities less than unity. I n the present studies, the diffusion and reaction processes in both the forward and reverse reactions were taken into account by solving Fick’s law for the interdependent diffusion processes as described in the next section of this paper. The specific rate constants for the ‘forward and reverse reactions then were selected t o provide the best fit between the experimental data and the theoretical equation for the current as a function of time. This led to values of the rate constants within a precision of about one per cent. Theoretical Discussion.-Where both forward and reverse reactions must be taken into account, the diffusion processes involved in these reactions interact only at the electrode and the equations to be solved are6
for solving Fick’s second law (3)
with the initial conditions for diffusion-controlled reactions. In these equations, c(z, t ) denotes the concentration of the reactive species a t the distance, x,from the electrode a t the time, t, and D and k are the diffusion and specific rate constants, respectively. It is to be noted that the specific rate constant, IC, as here used has the dimensions of length times reciprocal time. As previously pointed out,@ the solutions obtained from eq. 2 using the boundary condition (1) lead to the interesting conclusion that, in the one-dimensional case, the reaction rate should eventually become independent of the fraction of the molecules arriving a t the electrode which undergo an electron transfer. Thus the effect of a reaction probability less than unity is confined to an initial period and the observed rate converges to the rate corresponding to a reaction probability of unity after this ini(1) Dissertation of H. Rubin submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy a t the Polytechnic Institute of Brooklyn. Present address: Department of Chemistry, Syracuse University, Syracuse, N. Y. (2) Presented a t the Meeting-in-Miniature of the Metropolitan Long Island Subsection of the American Chemical Society’s New York Section, February, 1953. (3) H. A. Laitinen and I. M. Kolthoff, J . Am. Chem. Soc., 61, 3344 (1939). (4) (a) F. 0. Collins and G. E. Kiniball, J. Colloid Sei., 4 , 425 (1049); (b) F. C. Collins, ibid., 6, 499 (1950). ( 5 ) P. Delahay, J. Am. Chem. Soc., 7 8 , 4944 (1951).
c ~ ( z0, ) = c 1 O = constant c 2(q 0) = c2” = constant
and the boundary conditions D1( $ ) z = ,
=
klcl(O, t )
- k d 0 , t)
c l ( m , t ) = cl0 cz(m, t ) = c2O
I n eq. 3 t o 5 the subscripts 1 and 2 refer to reactant and product, respectively. The problem to be solved is equivalent to the conduction of heat in two semi-infinite slabs of different thermal conductivities with contact through a film at z = 0. The solution of this problem is well-known’ and is
(6) The assumption that D is independent of the concentration is probably a good approximation because the data upon which the rate constants below are calculated begin a t two minutes, by which time the major changes in concentration at. the electrodes have occurred. (7) H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” P. 71, Oxford University Press, New York, N. Y., 1947, p. 71.
CONCENTRATION OVERPOTENTIAL AT REVERSIBLE ELECTRODES
Nov., 1954 where kl hi = D1
and erfc y =
2
7r'/2J
r" II
exp
(-22)
dz
The concentration of the product, c2(x, t ) , is given by a similar expression with the subscripts 1 and 2 permuted. The current flow is given by the number flux converted to amperes I = nFAD1 =
(2)
x=o
nFA(klclo
- kmo)[exp (h12Dlt)erfc ($.I 5
0
a
0
DI
io8
D2
Initial ooncn., nolee/ cm.8 X 108 ClO
EX area, om. A
Current const. x amp.sec.'/z
I f i
Ferrocyanide oxidation
8.04 6.78 3.000
0.0633
28.2
Ferricyanide reduction
6.78 8.04
3.000
0.0611
28.30
Molybdocyanide oxidation 7.38 (7.38) 3.000
0.0620
27.51
-
-
-
0.05 I
I
I
I
I
I
I
I
I
I
I
I
I
CONCENTRATION OVERPOTENTIAL AT REVERSIBLE ELECTRODES
Nov., 1954
961
TABLE I1 SUMMARY OF THE FINAL RESULTS GIVINQSPECIFIC RATECONSTANTS, kl AND ka, THE MEANSTANDARD DEVIATION OF kl, u,,,,AND THE ELECTRODE REACTION PROBABILITIES, a1AND a2 A pp1ied potential,
Q) BV.,
kn
cm./sec.
(kdav.
ai X 10'
2.78 1.25 0.31 0.20 0.14
0.90 2.55 4.22 7.41 12.4
3.5 1.6 0.39 0.25 0.18
Ferricyanide reduction 0.004 ,002 .005 .008 .003
3.72 2.14 1.36 0.73 0.57
1.12 1.62 2.53 3.61 6.45
5.49 3.15 2.00 1.08 0.84
urn
V.
0.0500 .loo0 ,1500 .2000 .2500
0.121 110 .121 .200 .328
Ferrocyanide oxidation 0.002 0.608 .003 1.73 .004 2.86 .005 5.03 ,006 8.41
0.1250 .loo0 .0750 .0500 .0250
0.175 .128 .124 .130 .205
0.90 1.30 2.04 2.90 5.19
.
0.4000 0.350 .4250 .350 .320 .4500 .400 .4750 .435 .5000 a These values were computed assuming the literature.
aa
X 10'
Molybdocyanide oxidation 1.86 0.005 7.665 2.52 10.39 3.52 .012 5.98" 4.77 8 10 7.38 4.42 .011 3. 26a 5.44 11.91 2.84 .009 2. 0ga 8.78 14.67 1.36 .014 1.005 10.82 D1to be equal to D2. A value of the diffusion constant, Dz,was not available in
The calculated values of the rate constants kl and kz for the oxidation of ferrocyanide ion a t various applied potentials are given in Table I1 and illus-
k2t
=kT exp ( - A F 2 r / R T ) exp [-(1
h
*
*
- P)nS(EEo)IRTl (15)
trated in Fig. 5, as a logarithmic plot vs. the applied where AFl and AFz are the respective standard potential. The mean standard deviations of kl activation free energies, E is the applied single eleccorresponding to the finally selected values of y trode potential, E" is the standard electrode potenare also given in Table 11. The deviations lie within the expected precision of the experimental measureOXIDATION OF FERROCYANIDE 0 0ments and indicate the validity of eq. 7 as an expression for the current and thus the applicability REDUCTION OF FERRICYANIDE of the boundary condition (1) for the present electrode process. Now kl and k2 for the reduction of ferricyanide should be equal to kz and IC,, respectively, for the oxidation of ferrocyanide a t the same applied potential. These rate constants, together with the corresponding values of a, the fraction of the ions reacting a t the electrode, are given in Table I1 and Fig. 5. From Fig. 5, it appears that the agreement between these two sets of calculated rate constants is roughly of the same order as the internal agreement within each set. In spite of the careful preparation of the reagents, the absolute values of the rate constants may be systematically in error due to very small amounts of impurities in the reagents and by contamination by the deKhotinsky cement. The calculated rate constants and reaction probabilities, a, for the oxidation of molybdocyanide are given in Table I1 and Fig. 3 and have the same dependence upon the applied potential as in the ferrocyanide-ferricyanide case. It will be noted -5 that the values of a corresponding to the tabulated 0 0.05 0.10 0.15 0.20 0.25 values of kl are extraordinarily small in all cases. Applied potential in volts. The relation between the rate constant, kl and 5.-The logarithmic values of the rate constants, kz, for an oxidation electrode process and the over- kl Fig. and kz, a t various values of the applied potential for the potential is given by the absolute reaction rate the- oxidation-reduction of the ferrocyanide-ferricyanide couple. ory and may be expressed as The points, o 0 , were obtained with the microelectrode as an anode and the points, eo-,with the microelectrode as a
*
t
k ~ '=
kT
cxp ( - AF1* / R T ) exp [BnS(E'
- E")/RT]
(1-1)
cathode. The second half-cell was a 0.1 N calomel electrode.
962
WILLIAMP. BANKS,BERNARD 0. HESTONAND FORREST F. BLANKENSHIP
tial for the given half reaction and p is the fraction of the potential promoting the forward reaction. Here E o is the potential of the given half-cell with all substances in their standard states as customarily defined. The rate constants kl ' and k2' are the conventional first-order chemical rate constants and have the dimensions of reciprocal time. They may be readily calculated from the virtual rate constants kl and kz which are defined by eq. 1. At any instant, the number of moles of ions available for electrolysis a t the electrode is Apc(0, t ) where p is the order of the mean diffusional flight length. From eq. 7, it is clear that kl and k2 must be divided by p in order t o become equal t o the conventional first-order rate constants. For this purpose, p has been assumed t o be equal to 10-8 em. It should be noted that the potential difference, E - Eo,is the potential of the half-cell referred t o the standard potential as zero, rather than the overpotential where the zero of potential is taken as the reversible potential at the given concentrations of reactant and product. The calculated activation free energies AFI* and AFz+ will depend on the choice of the standard states of the substances involved as this will determine the reference zero of potential EO. I n addition, calculated activation free energies will depend on the magnitude chosen for p. The uncertainty in p is of the order of 3 which leads t o an uncertainty of the order of 2 kcal./ mole in the calculated activation energies. Under the present convention, with both reactant and product present at unit activity and with the potential applied t o the half-cell equal to the standard electrode potential, then E Eo = 0, no current flows, kl = kz, and hence the two calculated activation free energies should be equal. As lcl and kz are separately determined from the experimental data, Fig. 4 enables independent calculation of AFl* afid AFz* as well as bobs and (1 - p)obs. The values so obtained are presented in Table 111. The calculated activation free energies are of a reasonable order of magnitude and agree within 2 kcal. Similarly the values of bobs and (1 - p).b,are of the order of 0.5 as might be expected from a priori calculations. Thus the data
-
Vol. 58
obtained appear to be consistent with eq. 14 and 15. TABLE111 ACTIVATION ENERGIESOF THE FORWARD AND REVERSE REACTIONS AFl* AND AFg*, AND THE FRACTION Dobs OF THB
ELECTRICAL POTENTIAL PROMOTING THE FORWARD
-
REACTION.THE FRACTION(1 D)obs PROMOTINQ THE REVERSE REACTION, AS CALCULATED, ALSO IS GIVEN AP,*
AF~*,
kcal.)
kcal./ mole
Bobs
Ferrocyanide oxidn. 12.8 Molybdocyanide oxidn.
11.0
0.41 0.46
System
mole
.. .
...
(1
-
B)oba
0.42
...
Conclusion.-The agreement between theory and experiment shown by the small magnitude of the standard deviations of the specific rate constants a t the various applied potentials shows that eq. 7 is a correct expression for the current as a function of time. Hence eq. 1 constitutes an appropriate boundary condition for electrode processes which are partially or completely diffusion-controlled. Where the applied potential is so small that only a fraction, a, of the ions diffusing up to the electrode react, the observed current is smaller than the current corresponding to saturation potential only during an initial induction period and becomes equal to the saturation current for longer times. The fracor less tion, a,is required to be of the order of in order for this initial period to be sufficiently long for measurements t o show a reduction of the current. While the data for a and the calculated specific rate constants show excellent internal precision, the absolute values of these quantities may be somewhat in doubt because of the possible presence of trace amounts of capillary-active substances at the electrode. The calculated specific rate constants depend on the applied potential in the manner predicted by the Eyring absolute reaction rate theory. The activation free energies and the constants /3 and (1 p) describing the fractions of the applied potential which promote the forward and reverse reactions respectively are found to be of the correct order of magnitude and to be internally consistent.
FORMULA AND PRESSURE-TEMPERATURE RELATIONSHIPS OF THE HYDRATE OF DICHLOROFLUOROMETHANE1 BY WILLIAMP. BANKS,BERNARD 0. HESTONAND FORREST F. BLANKENSHIP Depart.ment of Chemistry, University of Oklahoma, Norman, Oklahoma Received January 16, 1964
The pressure-temperature relationships of various Freon 21 hydrate systems were determined. hydrate was found to be CHClzF.17 f 0.5M20by the method of de Forcrand.
The formula and phase relationships of Freon 21 hydrate were determined as part of a series of studies intended to clarify further the nature of the hydrocarbon-type or lattice hydrates. (1) From a thesis b y William P. Banks, submitted in partial fulfillment of the requirements for the Ph.D. degree, 1953.
The formula of Freon 21
Experimental Materials.-Freon 21 of 99.0% purity obtained from The Matheeon Company was used. Apparatus.-The apparatus for obtaining the pressuretemperature relationships of Freon 21 hydrate consisted principdly of a reaction vessel located in a Constant tmemperature bath and connected to a mercury manometer.
.
