Differential relationships from Gibbs equation

Chapter 3 starts with the laws, derives the Gibbs equations, and from them, develops the fundamental differential thermodynamic relationships. In some ways, this chapter can be thought of as the core of the book, since the extensions and applications in all the chapters that follow begin with these relationships. Examples are included in this chapter to demonstrate the usefulness and nature of these relationships. 

Esin-Markov coefficient — Various cross-differential relationships can be obtained from the - Gibbs-Lippmann equation because it is a complete differential. For instance,... 

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... 

The macroscopic description of the adsorption on electrodes is characterised by the development of models based on classical thermodynamics and the electrostatic theory. Within the frames of these theories we can distinguish two approaches. The first approach, originated from Frumkin s work on the parallel condensers (PC) model,attempts to determine the dependence of upon the applied potential E based on the Gibbs adsorption equation. From the relationship = g( ), the surface tension y and the differential capacity C can be obtained as a function of E by simple mathematical transformations and they can be further compared with experimental data. The second approach denoted as STE (simple thermodynamic-electrostatic approach) has been developed in our laboratory, and it is based on the determination of analytical expressions for the chemical potentials of the constituents of the adsorbed layer. If these expressions are known, the equilibrium properties of the adsorbed layer are derived from the equilibrium equations among the chemical potentials. Note that the relationship = g( ), between and , is also needed for this approach to express the equilibrium properties in terms of either or E. Flere, this relationship is determined by means of the Gauss theorem of electrostatics. 

The linear AH° vs. AS° relationship observed experimentally leads to Eq. (3), where the proportional coefficient fi has the dimension of temperature. From a combination of Eq. (3) and the differential form of the Gibbs-Hehnholtz equation (Eq. (4)), we obtain Eq. (5). 

Taking the total differential of this equation and subtracting it from the previous equation produces the following Gibbs-Duhem relationship [73Hir]... 

The measurement of formal potentials allows the determination of the Gibbs free energy of amalgamation (cf Eq. 1.2.27), acidity constants (pATa values) (cf. Eq. 1.2.32), stability constants of complexes (cf. Eq. 1.2.34), solubility constants, and all other equilibrium constants, provided that there is a definite relationship between the activity of the reactants and the activity of the electrochemical active species, and provided that the electrochemical system is reversible. Today, the most frequently applied technique is cyclic voltammetry. The equations derived for the half-wave potentials in dc polarography can also be used when the mid-peak potentials derived from cyclic voltammograms are used instead. Provided that the mechanism of the electrode system is clear and the same as used for the derivation of the equations in dc polarography, and provided that the electfode kinetics is not fully different in differential pulse or square-wave voltammetry, the latter methods can also be used to measure the formal potentials. However, extreme care is advisable to first establish these prerequisites, as otherwise erroneous results will be obtained. 

By means of the substitution y = jc/ /7 , these equations can be converted into ordinary differential equations. An analogous equation could also be written for the change in concentration of iron. However, because of eq. (7-17) and because of thermodynamic relationships in the ternary system (Gibbs-Duhem equation), this equation would not be independent. Because of the higher mobility of carbon, it is observed that the carbon reaches equilibrium after a relatively short time (as compared to the diffusion of silicon). From eq. (7-15) we may write ... 

---