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C O M M U N I C A T I O N S TO THE E D I T O R
Comments on Weir’s Note on “ A Posteriori Separation of Faradaic and Double-Layer Charging Processes : Analysis of the Transient Equivalent Network for Electrode Reactions”
determined in part by another process, namely mass transfer. Charge separation or recombination is not introduced in this reasoning, but the conclusion previously reached2 about the major defect of the classical approach is confirmed. The idea of charge separation or recombination arose in the chronological develop ment, Le., in the analysis of doublelayer charging of an electrode of varying area. (2) Weir states:’ “Based upon this analysis of ionic transport and charge separation-recombination, three equations of general validity were developed by Delahay.” It should be emphasized that these three equations do not involve the concept of charge separation or recombination as they are based on the balance of transport through the electrodksolution interface and the rates of storage at this interface.2 The first equation equates the faradaic current, as given by any suitable expression, to the sum of the reactant flux and the rate of reactant storage (multiplied by the charge zF involved per mole of reactant). The second equation balances the fluxes of reactant and product and the storage rates of these species. These two equations are the boundary conditions for which the mass transfer problem must be solved. The third general equation equates the current being measured to the sum of the flux of reactant and the storage rates of all other ions (flux and rates being multiplied by z F ) . These equations involve a model, which hopefully has been correctly described,* but they do not require the introduction of charge separation or recombination. (Cf. also a recent note.4) The introduction of this concept showed the need for some general equations, but the concept is not contained in these equations. (3) Weir writes’ in his initial paragraph, “The COordinate conclusion that a posteriori separation can have only formal and not operational significance seems unwarranted, however” (see also his further discussion). We examine the general forms of if and 6ql and we suppose that the mass-transfer problem is solved as indicated above. The changes 6E, 6a. . . , 6u.. . are then functions of time for a nonsteady-state perturbation and of frequency for a periodic perturbation (steady-state solution in the latter case). The measured current i is some function i = h(6E, 6 a . . ., I
Sir: These comments on Weir’s note‘ pertain to the following points: (1) coupling between faradaic and double-layer charging processes in general and more particularly for electrodes of constant area; (2) the three previously proposed2 general equations in relation to charge separation or recombination; (3) a posteriori separation of faradaic and charging processes. (1) The difficulty in extending intuitive analysis, based on charge separation or recombination, to electrodes of constant area can be avoided by considering the general forms of the equations for faradaic and charging proce~ses.~The faradaic current if for an electrode of constant area, in general, depends on the departure from the equilibrium values of the potential E,, the concentrations a,, b e . . . of reactants and products, and the concentrations u,, v,. . . of other ions present in. solution (supporting electrolyte). If a, b . . . and u,v. . . are the nonequilibrium concentrations just outside the double layer and E the potential corresponding to if,one has if = f(6E, 6a. . . , 6u. . .) where 6E = E - E,, 6a = a - a,. . ., 6u = u - U , and f represents some general function whose explicit form is not needed. (In many instances 6u. . . can be neglected for a small perturbation.) Likewise, the perturbation 6q of the charge density q on the electrode is some general function 6q = g(6E, 6a. . . , 6u. . .). The classical treatment of nonsteady-state or periodic electrode processes involves the sole consideration of i t in the analysis of mass transfer, the charging current depending only on a constant differential capacity of the double layer. It follows from the general forms of i f and 6q that this a priori separation of faradaic and charging processes is not justified, in all rigor, since both it and 6q depend on the same variables 6E, 6a. . . , 6u. . . . It is only when the changes of concentration 6a, 6b. . . do not practically affect 6q that the classical approach is satisfactory for all practical purposes. When this simplification is not justified, the mass transfer problem must be solved by considering simultaneouslzi Analvsis can be ” its effect on ir and 6a. extended to an electrode of varying area. by including the change of area 6A in if and 6q. Thus, coupling between faradaic and charging processes from the dependence of if and 6q on variables which are The Journal of Physical Chemistry
‘l7 3357 (1967). (l) w. Weir, J * Phus. (2) P. Delahay, ibid., 70, 2067, 2373 (1966). (3) P. Delahay and K. Holub, submitted for publication. (4) p. K. Holub, G. Susbielles, and G. Tessari, J . Phgs. Chem., 71, 779 (1967). Ds
COMMUNICATIONS TO THE EDITOR
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6u. . .). In principle, it does not seem possible to devise a measurement at any finite time, different from zero, which would separate the current i into two components attributed to the faradaic and charging processes, respectively. This seems impossible because both processes are coupled as indicated before. It is perhaps possible to isolate the charging process by measurement at, time zero if one neglects the finite rate of double layer relaxation. This was already discussed beforea2g5It should be noted that this operation seems possible provided one assumes that charging is instantaneous. whereas the faradaic process proceeds a t a finite rate for a finite exchange current. Thus one isolates one process, but one does not separate one process from the other when both processes occur simultaneously . Finally, one should remember that the description of the electrode response involves a large number of parameters. For instance, for an amalgam electrode one must consider three variables ( E , c M + ~ , and car) and four functions (if, q and the surface excesses of M + L and 31). Description of a perturbation of small amplitude thus requires 12 partial derivatives in the most general case (ten partial derivatives if one introduces the exchange current instead of the derivatives of if with respect to the three variables). These coefficients must somehow appear in the equivalent circuit, and unambiguous analysis of experimental data appears very difficult except when some of the partial derivatives can be dropped.6 (5) P.Delahay and G. Susbielles, J. Phys. Chem., 70, 3150 (1966). (6) K. Holub, G. Tessari, and P. Delahay, ibid., 70, 2612 (1967).
DEP.4RTMENT O F CHEMISTRY NEWYORKUNIVERSITY NEW YORK, NEW YORK 10003
proton of the amine, by the assumption that only the nonassociated form of diethylamine exists in cyclohexane over this range of concentrations. We have also observed the chemical shift of the n" proton of this amine as a function of concentration in cyclohexane. Samples were distilled from CaO through a Hempel column and stored in dry nitrogen or a desiccator. Sample tubes were oven dried and similarly stored until use. The data, obtained at 31", are shown in Table I alongside those obtained at 40" by Springer and Meek. The Varian A-60 nmr spectrometer was used in both studies. At values of X of 0.11 and less, practically the same chemical shifts of the 9 H proton were obtained, despite the different temperatures. An alternative explanation for the constant chemical shift obtained at low amine concentrations follows from consideration of the 60-;\lc/sec spectrum of cyclohexane, which includes not only the characteristic, sharp signal 87 cps to the low-field side of the tetramethylsilane signal, but also two signals arising from 13C-proton coupling, i . e . , 13C satellites, approximately 9 cps broad at half-height, occurring about 62 cps to the high- and low-field sides of the main solvent signal. I n a 0.117X diethylamine solution in cyclohexane the number of amine protons bonded to nitrogen atoms
Table I : Chemical Shift of Diethylamine S H Proton as a Function of Concentration in Cyclohexane
x
of amine
"
"
chemical shift,
of amine
chemical shift, cpsb
1.000 0.6710 0.4815
44 47 52
0.1109
63
0.09204
63
0.05986 0,02606
64 63
cpsa
PAUL DELAHAY
RECEIVED b l a ~22, 1967
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Carbon 13 Satellite Interference with Chemical Shift Measurements in Cyclohexane-Diethylamine Solutions
Sir: A recent publication of a study of diethylamine hydrogen bonding by nmr by Springer and Meek2 reports a constant chemical shift for the n" proton a t COnCentratiOnS of less than 0.1109 mole fraction amine ( X ) in cyclohexane. They have taken this constant chemical shift to be that of the nonhydrogen-bonded
0.1864 0.1618 0.1286 0.1142
59 60 61 62
0.1082 0.1008
63 63
0 . OSt%
64
x
* Data of Springer and a Relative to cyclohexane signal. Meek,' converted from units to cps from cyclohexane signal. (1) This work was supported in part by the National Science Foundation and the Office of Saline Water, U. S. Department of the Interior. (2) C. S. Springer, Jr., and D. W.Meek, J. Phys. Chem., 70, 481 (1966).
Volume 71, il'umber I O
September 1967
