Differential surface excess quantities

By adopting the usual conventions of chemical thermodynamics, we are able to derive from the surface excess chemical potential pa a number of useful surface excess quantities. Our purpose here is to draw attention to the difference between the molar and the differential surface excess quantities. 

We now return to the definition of the surface excess chemical potential fta given by Equation (2.19) where the partial differentiation of the surface excess Helmholtz energy, Fa, with respect to the surface excess amount, rf, is carried out so that the variables T and A remain constant. This partial derivative is generally referred to as a differential quantity (Hill, 1949 Everett, 1950). Also, for any surface excess thermodynamic quantity Xa, there is a corresponding differential surface excess quantity xa. (According to the mathematical convention, the upper point is used to indicate that we are taking the derivative.) So we may write ... 

The differential quantities of adsorption , are the differences between the differential surface excess quantities and the same molar quantity, as in Equation (2.46) or... 

The quantity dyl3 In a2 at the potential of the electrocapillary maximum is of basic importance. As the surface charge of the electrode is here equal to zero, the electrostatic effect of the electrode on the ions ceases. Thus, if no specific ion adsorption occurs, this differential quotient is equal to zero and no surface excess of ions is formed at the electrode. This is especially true for ions of the alkali metals and alkaline earths and, of the anions, fluoride at low concentrations and hydroxide. Sulphate, nitrate and perchlorate ions are very weakly surface active. The remaining ions decrease the surface tension at the maximum on the electrocapillary curve to a greater or lesser degree. 

Electroneutral substances that are less polar than the solvent and also those that exhibit a tendency to interact chemically with the electrode surface, e.g. substances containing sulphur (thiourea, etc.), are adsorbed on the electrode. During adsorption, solvent molecules in the compact layer are replaced by molecules of the adsorbed substance, called surface-active substance (surfactant).t The effect of adsorption on the individual electrocapillary terms can best be expressed in terms of the difference of these quantities for the original (base) electrolyte and for the same electrolyte in the presence of surfactants. Figure 4.7 schematically depicts this dependence for the interfacial tension, surface electrode charge and differential capacity and also the dependence of the surface excess on the potential. It can be seen that, at sufficiently positive or negative potentials, the surfactant is completely desorbed from the electrode. The strong electric field leads to replacement of the less polar particles of the surface-active substance by polar solvent molecules. The desorption potentials are characterized by sharp peaks on the differential capacity curves. 

The surface tension depends on the potential (the excess charge on the surface) and the composition (chemical potentials of the species) of the contacting phases. For the relation between y and the potential see - Lipp-mann equation. For the composition dependence see -> Gibbs adsorption equation. Since in these equations y is considered being independent of A, they can be used only for fluids, e.g., liquid liquid such as liquid mercury electrolyte, interfaces. By measuring the surface tension of a mercury drop in contact with an electrolyte solution as a function of potential important quantities, such as surface charge density, surface excess of ions, differential capacitance (subentry of... 

Integral and differential thermodynamic relationships between the different excess quantities defined by Eq, (2) and the set of experimental variables (P, r,y,) or (T,nf) can be derived analogously to those for conventional bulk-phase thermodynamic properties [9], However, an additional intensive property called the surface potential (cf>, ca /g) is necessary to completely define the Gibbsian adsorbed phase. The surface potential can be calculated by using the relationship [9] ... 

For a multicomponent gas system (/ = 1, 2,..., A), a total of N experiments must be carried out around the base saturation conditions of T, P, y (corresponding surface excess of component i — rtf ), by separately introducing differential quantities (AA/, mol/g) of each pure gas i into the sample side. This will generate N independent equations (/= 1, 2,..., A) analogous to Eq. (46) that relate N different unknowns (, ). Consequently, the isosoteric heat of adsorption of component i can be estimated as functions of and nf. 

The derivation leading to Equation 1.26 involves a particular choice of the referenee surface S. We may naturally ask about the sensitivity of quantities such as interfacial tension y, the surface excess entropy per unit area F5, and the surface excess eoneentrations F, to the location of the reference surface. For a plane interfaee, the value of y is independent of the position of S, as may be shown for instanee by differentiating Equation 1.4.v of Problem 1.4 with respect to the radius a of the intraface and taking the limit as a becomes very large and the pressure difference (p - Pe) small. Evrai for an interface that is slightly curved, y should not vary greatly with small shifts in the position of S. 

The linear differential equation given by Eq. (8) uniquely characterizes the interfaces using the surface tension and the various excess quantities. Applying the Young-Schwartz theorem, the Gibbs adsorption equation for a solid/fluid interface s = S fl can be deduced directly ... 

The information one can derive from measurements of the surface tension as a function of potential is specified in Eqs. (9.9)-(9.14). It includes the dependence of charge and the double-layer capacitance on potential and the relative surface excess of all the species in solution, except the solvent), as a function of potential and of the composition of the solution. It should be noted, however, that all these quantities must be obtained by numerical differentiation of the experimental results. This requires very high accuracy in measurement, since differentiation inherently amplifies experimental errors (while integration tends to smooth them out). The range of values of y in most cases is (0.250—0.426) N m . The best measurements recorded claim an accuracy of 0.1 mN m which amounts to about (0.04-0.02)%. 

This quantity provides information about the excess of component-adsorbent interactions averaged over all surface domains from which the solvent has been displaced by the adsorbing solute species. In consequence, it is not easy to monitor subtle changes in the adsorption mechanism based on usually small variations of the Adpih values with increasing quantity of adsorption. Compared to the differential molar enthalpy of displacement, the enthalpy Adpih is less sensitive to the energetic heterogeneity of the solid surface. 

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