Helmholtz differential

There are several differential equations that are related to Laplace s equation, e.g. the Poisson equation for the distribution of electric potential in the presence of electric charges, the wave equation for the propagation of a disturbance or the Helmholtz differential equation for the time-invariant distribution of harmonic fields. The latter is of particular relevance for scattering phenomena it has the form ... 

Spherical harmonics Y 9,(l>) are the angular contribution to the solution of Laplace s equation (or Helmholtz differential equation) in spherical coordinates (i.e. Eqs. (C.9) and (C.IO)). They are hence the product of the associated Legendre polynomial of cos0 and the general sine of the azimuth (/> ... 

The spherical vector wave functions (SVWF) are the general solution of the vectorial Helmholtz differential equation in spherical coordinates (Xu 1995) ... 

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. 

According to the literature [21], all reported electrochemical oscillations can be classified into four classes depending on the roles of the true electrode potential (or Helmholtz-layer potential, E). Electrochemical oscillations in which E plays no essential role and remains essentially constant are known as strictly potentiostatic (Class I) oscillations, which can be regarded as chemical oscillations containing electrochemical reactions. Electrochemical oscillations in which E is involved as an essential variable but not as the autocatalytic variable are known as S-NDR (Class II) oscillations, which arise from an S-shaped negative differential resistance (S-NDR) in the current density (/) versus E curve. Oscillations in which E is the autocatalytic variable are knovm as N-NDR (Class III) oscillations, which have an N-shaped NDR. Oscillations in which the N-NDR is obscured by a current increase from another process are knovm as hidden N-NDR (HN-NDR Class IV) oscillations. It is known that N-NDR oscillations are purely current oscillations, whereas HN-NDR oscillations occur in both current and potential. The HN-NDR oscillations can be further divided into three or four subcategories, depending on how the NDR is hidden. 

Thus, we now have a reasonable model of the interface in terms of the classical Helmholtz model that can explain the parabolic dependence of y on the applied potential. The various plots predicted by equation (2.18) are shown in Figures 2.5(a) to (c). The variation in the surface tension of the mercury electrode with the applied potential should obey equation (2.18). Obtaining the slope of this curve at each potential V (i.e. differentiating equation (2.18)), gives the charge on the electrode, [Pg.49]

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. 

Another important and fundamental consequence of Figure 2.9 is that the Helmholtz theory cannot cope with a concentration dependence of the differential capacitance nor can it be modified in such a way as to incorporate a minimum in (derM/d V) at low concentrations of electrolyte. Instead, a second theory must be invoked, having a dependence on electrolyte concentration but in such a way that it is only important at low concentration. A first step... 

We can progress from here provided that we can find expressions for the partial derivatives of equation (2.99). Provided that the concentration of supporting electrolyte is sufficiently high that all the potential difference across the interface is accommodated within the Helmholtz layer, then transport of O and R near the electrode will only take place via diffusion (i.e. we can neglect migration). The equation of motion for either O or R is given by the differential form of Fick s equation, as discussed in chapter I ... 

Table 8 gives the results of this thermodynamic analysis for the spreading of film types I and II from the bulk, and the direct transition from film types I and II. It is obvious that the Helmholtz free energies, entropies, and enthalpies are differentiated stereochemically. 

As the procedure is the same for both the Gibbs function and the Helmholtz function, we shall consider in detail only the derivation for the Gibbs function G. After Equation (7.14), we obtained the differential of the function, which was later to be defined as G, as... 

By an analogous procedure, it can be shown that the total differential of the Helmholtz function is given by the expression... 

Equation (7.85) frequently is called the Gibbs-Helmholtz equation. From it, the temperature coefficient of the free energy change (0AGm /. 7-/0T)p can be obtained if AGiji and AH are known. By differentiating Equation (7.83), we obtain... 

The Gibbs-Helmholtz equation (Eq. (3.25) below) can be conveniently used to calculate the enthalpy if the rate of change of Gibbs energy with temperature is known. AS is obtained from Eq. (3.24a) by differentiating it with respect to temperature, so dAG/dT = AS. Substituting back into Eq. (3.24a) gives the relationship... 

Extensive theoretical analyses of the compensatory enthalpy-entropy relationship were first carried out by Leffler and later by Leffler and Grunwald, Exner, and Li. The empirical linear relationship between the thermodynamic or activation parameters AH and AS) directly leads to Eq. 11, where the proportional coefficient p, or the slope of the straight line in Figure 9, has a dimension of temperature. Merging Eq. 11 into the differential form of the Gibbs-Helmholtz Eq. 12 gives Eq. 13 ... 

Combining Eq. 14 with the differential form of the Gibbs-Helmholtz equation (Eq. 12), we obtain Eq. 16 ... 

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. 

For example, from the right (G) edge, the arrow from natural variable T ends up at — S (taken minus because it lies at the tail of the arrow), whereas that from variable P ends up at +V (taken plus because it lies at the head of the arrow), giving the differential form dG = (—S)dT + (+V)dP. Similarly, from the top (A) edge, we can read the differential form for the Helmholtz energy as dA = ( S) dT + (— P) dV, because both coefficients fall at arrow tails. 

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