NOTES
Cd3P2(c) determined by the torsion effusion method employed in this work is in good agreement with Shchukarev's value2 which was obtained by calorimetry in a KBr-Br2 solution. The results for Cd3P2,like those of SVL5 for Zn3P2, suggest that a is temperature dependent and that the deviation from unity may be caused, in part at least, by an enthalpy barrier to activation for vaporization which exceeds the equilibrium enthalpy change. The foregoing implies that the torsion effusion value for aHoTfor the vaporization process should be smaller than the mass spectrometrically determined values since the latter were determined by measurement of ion intensities in a steady-state system in which the vapor pressure was below saturation. The dependence of pressure on sample surface area which was observed in this work is consistent with a predictionlo which was based on a theoretical model for vaporization from porous solids of substances with low vaporization coefficients. llotzfeld" and Rosenblatt'O have discussed the derivation and limitations of equations of type (1); and, in particular, they have directed attention to the assumptions which are necessary to obtain a simple relationship between a and M . If eq 1 is more than an empirical relation which is useful for extrapolation purposes and if the limiting values for a are meaningful, comparison of values of a from the present work with those of SVL5 suggests that the gross coefficient for vaporization for cadmium phosphide may be considerably smaller than for zinc phosphide. There is some evidence to suggest that in both cases as well as for the vaporization of red phosphorus12 and arsenicI3 the low vaporization coefficient may be the result of an excess enthalpy of activation associated with rearrangement of bond distances and angles in the formation of Xd units which do not exist in the crystal. Cd3P2 has a tetragonal D5$ structure4 in which each phosphorus atom has 12 near neighbors, 4 a t 4.26 A, 4 a t 4.34 A, and 4 at 4.45 A. The P-P distance in P4(g) is 2.21 A.'* Although Zn3P2 and Cd3P2have similar crystal structures, the nearest neighbor phosphorus distances are considerably larger in the latter, and it is possible that the additional separation of phosphorus atoms is responsible for the lower coefficient for vaporization of the cadmium compound. Unfortunately, a test of this hypothesis cannot be extended to mercuric phosphide since apparently there is no well-defined Hg3P2 phase'5 in the mercury-phosphorus system in t h e temperature range of interest. The abnormally high initial rate of vaporization from fresh Of Cd3P2 found in this work is reminiscent of a result previously observed in the Vaporization Of P,(g) from red phosphorus.12 We can
3357
offer no explanation which is preferable to Brewer and Kane's suggestion that the phenomenon may have been the result of distorted or defective crystals in which the atoms were not so rigidly fixed as in a perfect crystal.
Acknowledgment. R. C . Schoonmaker wishes to thank the National Science Foundation for a fellowship and Professor C. A. Coulson, Mathematical Institute, Oxford University, for providing facilities and generous hospitality during a sabbatical leave when this note was prepared. (10) G.M. Rosenblatt, J. Electrochem. Soc., 110, 563 (1963). (11) K. Motzfeld, J. Phys. Chem., 59, 139 (1955). (12) L. Brewer and J. S. Kane, ibid., 59, 105 (1955). (13) G.M. Rosenblatt, P. K. Lee, and M. B. Dowell, J. Chem. Phya., 45, 3454 (1966). (14) C.R.Maxwell, S. B. Hendricks, and V. M.Mosley, ibid., 3, 699 (1935). (15) B. Aronson, T.Lunstrom, and S. Rundquist, "Borides, Silicides, and Phosphides, a Critical Review of their Preparation, Properties, and Crystal Structure," Methuen & Co. Ltd., London, 1965.
A Posteriori Separation of Faradaic and Double-Layer Charging Processes : Analysis of the Transient Equivalent Network for Electrode Reactions
by W. D . Weir Department of Chemistry, Harvard Univereity, Cambridge, Massachusett8 08138 (Received February 84, 1967)
In a recent critique of the assumptions upon which several treatments of nonstationary-state elect,rochemical processes have been based, Delahay has shown that a priori separation of faradaic and double-layer charging processes is without operational justification.' The coordinate conclusion that a posteriori separation can have only formal and not operational significance2 seems unwarranted, however. This communication suggests an operational justification for a posteriori separation under appropriate conditions through an analysis of the transient equivalent network proposed by Weir and Enke.a (1) p. Delahay, J. Phy8. Chem., 70, 2373 (1966). (2) p. Delahay and G. Susbielles, ibid., 70, 3150 (1966). (3) w. D. Weir and C. G. Enke, ibid., 71, 280 (1967).
Volume 71,Number 10
September 1967
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The central assertion of the Delahay critique is that local charge separation or recombination can occur under appropriate circumstances without external current flow, and this idea was developed for the case of an expanding electrode which is potentiostatically poised a t equilibrium with a solution of fixed concentration of a cation ;\I+zto which the electrode is reversible. As the electrode area is increased, the relative surface excess r+ of lI+' can remain constant only by an increase in the absolute surface quantity of M+zthrough (i) ionic transport from solution to the double layer, enforced by the potentiostat, and (ii) local charge separation, without external current, the result of an increase in the rate of the charge separation process, 11 + ^Ilsurface+' Xemetai-, relative to the charge recombination process, Msurface+Z zemetal-+ M,with which it would be in equilibrium were the area of the electrode t o remain constant. The relative contributions of processes i and ii to the charging response depend upon the relative rates of the potentiostatically enforced mass-transport process and the charge-transfer reaction. Based upon this analysis of ionic transport and charge-separation-recombination processes, three equations of general validity were developed by Delahay. These equations, upon solution for appropriate boundary conditions, describe the nonstationarystate electrode response. The set of general equations and the physical interpretation given by Delahay to the response of the expanding electrode as it charges toward equilibrium both show unequivocally the inappropriateness of a priori separation. Direct extension of the physical argument given for this problem by Delahay to the problem of the response of :I constant area electrode which is displaced from equilibrium by a coulombic perturbation is not without difficulty, however.* In the expanding electrode problem, with charging toward equilibrium, a supplementary contribution to double-layer charging through charge separation is a consequence both of instantaneous deficiency of surface excess r+ and instantaneous anodic overpotential, which the potentiostat acts to reduce; for a constant-area electrode, with charging away from equilibrium, such a supplementary contribution to double-layer charging through spontaneous charge separation (in the case of cathodic charging) or recombination (for anodic charging) would require a displacement of the charge-transfer equilibrium in the sense opposite to that dictated by the prevailing surface excesses and the overpotential. For example, in the case of cathodic charging from equilibrium, I'+ increases as a consequence of ionic transport to the double layer, and this increase is reflected in a transient, cathodic overpotential. A supplementary
+
The Journal of Physical Chemistry
+
contribution to double-layer charging would require further enhancement of I'+ through an increase in the zemetai-, relarate of charge separation, XI + M+' tive to the rate of charge recombination, M+z zemetal- --t M, despite the greater-than-equilibrium surface excess of R9+z and the cathodic overpotential, an unreasonable result. Displacement of the chargetransfer equilibrium during charging, if it occurs a t all, must correspond, for this case, to net recombination of charges and, in general, to a faradaic process in the usual sense of that term. For the case of coulombic perturbation of a constantarea electrode, then, it would appear that formal separation of ionic transport and charge-separation-recombination processes corresponds directly to the intuitive sepwation of double-layer charging and faradaic processes. It is necessary to qualify this conclusion through more detailed consideration of the kinetic discharge process over the time interval of charging, however.
+
+
Figure 1. Transient network equivalent to the metal-electrolyte interface for a charge-transfer process which is time-resolved from coupled mass-transport and other prior processes.
Consider that a charge increment (dq)c is applied through an external circuit to an electrode of constant area which is initially a t equilibrium with a fixed concentration of an ion R l + ' to which it is reversible, and that a displacement of electrode potential d p is ob~ e r v e d . ~If, over the time interval to which this displacement of electrode potential corresponds, a change in electrode charge (dq)d occurs through the faradaic discharge process, then the net charge increment for the electrode over this potential interval is dq = (dq)c - (dq)d. The differential quotient dq/dp has the dimensions of capacitance, and for the case where the charge increment is totally nonfaradaic, Le., (dq)d = 0, the customary differential capacitance (4) Discussion by F. C. Anson and by H. Gerischer of a presentation of the physical ideas of P. Delahay [ J . Electrochen. Soc., 113, 967 (1966)] (see pp 972, 973) reflects this difficulty. ( 5 ) Equilibrium between the diffuse layer and the bulk concentration of M+' is assumed; removal of this restriction has been treated by G. C. Barker, J . Electroanal. Chem., 12, 495 (1966).
NOTES
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is defined dq/dp = (dq),/dp = C1. If, on the other hand, (dq)o = 0 (corresponding to open-circuit relaxation) a discharge capacitance can be defined dq/ dp = -(dq)d/dp = C,. Then, because the two contributions are defined over the same potential interval dp, the total capacitance of the interface is C = dq/dp = (dq),/dp - (dq)d/dp = C1 -I- C,.6 If the discharge process is assumed to be formally decoupled from ionic transport, the discharge must involve changes only of surface concentrations, expressed as relative surface excesses; then (dq)d = -zF.dr+ if (dq)d is the change in electrode charge due to a z-electron faradaic process which changes the surface excess of M+z by dr, over the potential interval dp, and the discharge capacitance takes on the form C, = -zF.(dI'+/dp). Subject to the restriction required to give C, this form, the total capacitance is C = C1 C a = (dq), - zF.(dr+/dp), which is customarily identified with the capacitance of an ideal reversible electrode; C1 is that capacitance associated with an ideal polarizable electrode. To accord with intuition, at t = 0 the observed charging behavior must be that of an ideal polarizable electrode, while for t + 03 , charging as an ideal reversible electrode must prevail. In his analysis, Delahay noted that the transition between these two classes of behavior must depend upon the exchange current of the chargetransfer process, but no quantitative interpretation of a transition time constant was given. A deterministic: model for relaxation processes which embodies these ideas has been p r ~ p o s e d . The ~ transient equivalent network, shown in Figure 1, illustrates clearly the relationship between the double-layer charging path (through CJ and the capacitive discharge path (through C, and the charge-transfer resistance R,). The impedance Z(t) represents the kinetic contributions of mass transport and any other coupled processes (adsorption, partial or total desolvation, orientation, surface or volume chemical reaction, preceding chargetransfer step(s), or any combination of these) which can be resolved in time from the charge-transfer process represented by R,.' Over the time interval in a tran-
+
sient charging measurement where the admittance of C, is very large relative to l/Z(t), the surface discharge occurs while the coupled steps, which with it comprise the series mechanism for the over-all reaction, can be neglected. A transition between the charging of C1 alone and parallel charging of C1 and C, over this interval can be directly observed and related to the C1). The magnetwork time constant RaCaC1/(Ca nitudes of CIand C, C1 can be determined by direct measurement of charging slopes a t constant current.* The transient equivalent thus gives quantitative significance to the transition which has been discussed qualitatively by Delahay, and it provides operational justification for the definition of the two capacitances, CI and C,. The identification in the transient equivalent network of two current paths and their associated capacitances lends credibility to an a posteriori separation of double-layer charging (referring only to the current path through C,) and faradaic (referring to the current path through C, and Ra) processes for nonstationarystate electrochemical reactions. I t must be recognized, however, that the charge-transfer reaction associated with the faradaic process does make a contribution to the total charging response of the metal-electrolyte interface.
+
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Acknowledgment. Support from the Research Corporation and from the William F. Milton Fund of Harvard University is gratefully acknowledged. (6) In this discussion, formal separation of nonfaradaic and discharge contributions t o the total charge increment has been made only t o define CI and C, and it corresponds for this case t o the separation of ionic transport and charge separation-recombination processes, as shown above, not t o an a priori separation of these variables without justification. (7) The assumption of formal separation of mass transport and other coupled processes from charge transfer is operationally justified only when these processes can be experimentally resolved in time by the technique applied to the study of a specific electrochemjcal reaction. (8) Charging measurements at very short times may be required. The transition has been observed for constant-current charging at the mercury-electrolyte interface in solutions of mercurous perchlorate for times up to 400 nsec.3
Volume 7 1 , Number 10 September 1967
