Capacity: differential Helmholtz

Electrode area Richardson constant Activity coefficient Differential Helmholtz capacity Differential space charge capacity Concentration of species j in solution... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Figure 2.9, it can be seen that the interfacial capacitance does show a dependence on concentration, particularly at low concentrations. In addition, whilst there is some evidence of the expected step function away from the pzc, the capacitance is not independent of V. Finally, and most destructive, the Helmholtz model most certainly cannot explain the pronounced minimum in the plot at the pzc at low concentration. The first consequence of Figure 2.9 is that it is no longer correct to consider that differentiating the y vs. V plot twice with respect to V gives the absolute double layer capacitance CH where CH is independent of concentration and potential, and only depends on the radius of the solvated and/or unsolvated ion. This implies that the dy/dK (i.e. straight lines joined at the pzc. Thus, in practice, the experimentally obtained capacitance is (ddifferential capacitance. (The value quoted above of 0.05-0,5 Fm 2 for the double-layer was in terms of differential capacitance.) A particular value of (di M/d V) is obtained, and is valid, only at a particular electrolyte concentration and potential. This admits the experimentally observed dependence of the double layer capacity on V and concentration. All subsequent calculations thus use differential capacitances specific to a particular concentration and potential. 

Now, the cosh function gives inverted parabolas [Fig. 6.65(b)]. Hence, according to the simple diffuse-charge theory, the differential capacity of an electrified interface should not be a constant. Rather, it should show an inverted-parabola dependence on the potential across the interface. This, of course, is a welcome result because the major weakness of the Helmholtz-Perrin model is that it does not predict any variation in capacity with potential, although such a variation is found experimentally [Fig. 6.65(b)],... 

Fig. 6.66. The Stem model, (a) A layer of ions stuck to the electrode and the remainder scattered in cloud fashion, (b) The potential variation according to this model, (c) The corresponding total differential capacity C is given by the Helmholtz and Gouy capacities in series.

This result is formally identical to the expression for the total capacity displayed by two capacitors in series [Fig. 6.66(c)]. The conclusion therefore is that an electrified interface has a total differential capacity that is given by the Helmholtz and Gouy capacities in series. Let s examine two extreme cases. 

After all this analysis, can we say that the Stem model is consistent with experimental results In other words, is the Stem model able to reproduce the differential capacity curves Under certain conditions, it is. So, to some extent, the Stem model was successful. However, what are the restrictions the model imposes Recall that in the Helmholtz-Perrin model the ions lay close to the electrode on the OHP. The condition for the Stem model to succeed is that ions not be in close proximity to the electrode they are not to be adsorbed. Thus the model proved to be valid only for electrolytes such as NaF (Graliame, 1947).45 Both of these ions, Na+ and F, are known to have a hydration layer strongly attached to them in such a way that even in the proximity of the electrode they are almost not interacting with the electrode surface. The Stem model works well representing noninteracting ions. 

When we revised the different models of the interface, namely, the Helmholtz-Perrin, Gouy-Chapman, and Stem models, we left the corresponding section (Section 6.6.6) with the idea that these models were not able to reproduce the differential capacity curves [Fig. 6.65(b)]. We said that when ions specifically adsorb on the electrode, the models fail to explain the experimental facts. 

Starting with the above equations (principally the four fundamental equations of Gibbs), the variables U, S, H, A, and G can be related to p, T, V, and the heat capacity at constant volume (Cy) and at constant pressure (Cp) by the differential relationships summarized in Table 11.1. We note that in some instances, such as the temperature derivative of the Gibbs free energy, S is also an independent variable. An alternate equation that expresses G as a function of H (instead of S) is known as the Gibbs-Helmholtz equation. It is given by equation (11.14)... 

This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems. 

To obtedn a general relation between and the enthalpy H (which bears a more direct relation to experimentally measured molar heat capacity), we differentiate the Gibbs-Helmholtz equation (5.2.13) with respect to Nk and use (5.3.1) to obtain... 

The simplest model is that of a plate capacitor developed very early by Helmholtz. The idea is that the ions of the electrolyte, which form the excess charge there, can approach the metal surface only up to the distance of the radius which includes the irmer solvation sphere in liquid solutions. Measurements of the differential capacity of smooth electrodes yielded values for the Helmholtz double-layer capacity, Ch, on tlie order of 20 to 30 pF cm . The model of a plate capacitor gives for the differential capacity... 

Since the capacity of the diffuse double layer is in series with the Helmholtz double layer, the best chance to see it in the measurements requires Cd < Ch. Equations (2.50) and (2.54) indicate that Cd increases rapidly with Q. In accordance with this conclusion, a minimum of the differential capacity was found in measurements on metals in very dilute electrolyte solutions (<10 M), which is an indication for the absence of excess charge on the electrode. 

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