GaIvanostatic Method for Measuring Rates of Fast Electrode Reactions II.
Modified Method with Precharging of the Double Layer
RONALD L. BIRKE and DAVID K. ROE Department o f Chemistry and Laboratory for Nuclear Science, Massachusetts Institute o f Technology, Cambridge, Mass. 02 139 The mathematical and experimental interpretation of galvanostatic overpotential-square root of time curves is shown to b e simplified b y precharging of the electrode double layer capacity. This precharging is accomplished by differentiation of the leading edge of the current step by a parallel RC circuit. A complete mathematical treatment is given, and it is shown that good linearity between overpotential and square root of time can b e expected at times as short as 5 microseconds after the start of electrolysis. Experimental confirmation is offered for the case of the Hg2+2/Hg electrode reaction. Certain experimental aspects are discussed in relation to the use of this modified galvanostatic technique.
T
HE EXPERIMENTAL SIMPLICITYof the single-pulse galvanostatic method for kinetic studies of electrode processes is offset to a large extent by complications arising from the presence of a charging current (3). The faradaic current is therefore not equal to the applied current, and it is not constant with time. While the mathematical treatments of the dual charging faradaic process have been exact, a time restriction is usually imposed to simplify the equation t o linearity in W. The first portion of the experimental curve is thus unusable and some uncertainty results in the intercept at zero time because of the extrapolation. Further refinements of the equations are possible, but the advantage is lost in the introduction of additional time dependent terms and subsequent tedium in manipulations. The galvanostatic method was modified so that the simplicity of a straight line extrapolation would be maintained over an extended time region of the overpotential-time curve. The principle behind the modification was to rapidly charge the electrode double layer to the extent that the faradaic current would become a nearly constant fraction of the applied constant current a t times sufficiently short so that a linear graph could be obtained. The
overpotential at zero time becomes obtainable from a very short extrapolation. The need for extending the kinetically useful part of the measurement to very short times (> C, and R > R1 results in a cell current with the following time dependence:
I n most cases, R >> R1and the constant current is then V / R . The rise time of the current step is simply RlCl and the contribution of C, can decrease the rise time to zero. As C, is increased, a spike appears superimposed on the current step. The duration of this current spike can be made very short, and it is assumed in the following that the charge transfer process does not occur to a significant extent during this charging sec. process of ca. 2 X Equation 1, written in the form I = 1 2 (1 q exp-mt), is used in the boundary condition for the solution of Fick's
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R, lOOn
1
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Figure 1.
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Circuit for generation of charging spike and constant current VOL. 37, NO. 4, APRIL 1965
455
to their Equation 22 (see Part I, Equation 8) which then becomes valid a t t >
equation for linear diffusion as given in the Appendix. The analysis closely parallels the derivation first given by Bereins and Delahay (5), hut with the somewhat simpler factorization of the transformed equation shown in Equsr tion 12 of the analysis of the double pulse method by Matsuda, O h , and Delahay (8). Clearing Equation 7 of the Appendix of terms which are negligible after one microsecond results in
ca. 5 psec. by this simple and noncritical experimental adjustment. KINETIC PARAMETERS OF Hgr+*/Hg ELECTRODE
Figure 2. Oscilloscope display of overpotentiol VI. time for Hg?f2 (4 X
lo4' m~le/cm.~)/Hgin IM HClOa Anodic current h = 60.9 mo., with q = 0, 2, 3, and 4 from bottom lo lop. Voltoge axis,
1 mr/cm.;
The overpotential is defined as q = E E., and 1%is positive for a cathodic proess. Comparison of this equation with the results of the derivation of Berzins and Delahay (5) (see Part I, Equation 7) shows that the current spike modifies the coefficientsof the exp-erfc terms. A quantitative appraisal of the added coefficients can be obtained by substitution of a series expansion for exp (p)erfc (y), as wns done to obtain Equation 10 in Part I. The result, in terms of the constants a and b, is
2b)
+ (3)
where the square bracketed terms are the series approximation to the combined experfc terms. The first term within the brackets represents the effect of the double layer capacity on the overpotential, and the time dependence is the expected t-"*. At times less than 50 @see., and especially in the region of 5 to 10 psec., this term shows the slow rise of q with time due to double layer charging. The term in q and m is due to the current spike and can always decrease the coefficient of 1-l)' because 2b is larger than az and m is larger than 2b. A numerical example will clarify the effect of the current spike. For an exchange current of 0.1 amp./crn.l a t a concentration of 10-6 mole/cm.' and diffusion coefficient of 9 X 10- cm.*/ sec. for both species, a double layer capacity of 2.0 X 10-6fd./cm.*and a twoelectron change, the constants a and b a r e 3 X 10*and4 X 106, respectively. A 456
ANALYTICAL CHEMISTRY
time oxis: 4.06 ~yec./cm_
suitable and easily attained value for m is lo' set.-' In the absence of forced double layer charginci.e., q = 0the ratio of the term in t - U 2 to the term in tu1 is -0.22 a t 10-$ sec.; the latter term is negative. A current spike corresponding to q = 35,40, or45decreases the ratio to -0.045, -0.020, or +0.005, respectively. Thus the effect of double layer charging on the observed overpotential is clearly decreased to insignificance at very short times and the observed 7 - t'J2 curve is made linear for more accurate extrapolation to zero time. Perhaps the most important conclusion from this example is that the adjustment of C. is not critical; good linearity a t less than sec. is obtained over a 25% variation of C.. This is a very attractive and convenient feature of the method, particularly in comparison with the double pulse method. Additional terms in the series expansion of experfc can be included in the above approximation. It can he easily shown that the coefficient of each additional term can be decreased hy m and q. For most caxa the approximation given above is adequate. In fact, to estimate probable values of q, the terms in Equation 3 can he simplified since 2b > at and m' > b'. The ratio of charging to faradaic terms then becomes (q/m - 2/b)/2t; for a chosen time, m, and estimated b, the appropriate range of q is evident. When the exchange current density is relatively small, about 0.01 amp./cm.l, the required value of q becomes a b u t 100 if the charging term is to be
