Gibbs free energy differential relationships

For nF = 0, we immediately get the relationship between nB and q. We now want to change both uB and nF in a direction that decreases G. More precisely, we will exchange one free for one base variable at a time as long as the Gibbs free energy can be decreased. The last equation can be differentiated as... 

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)... 

One consequence of practical significance is that eq. (7.4) provides a relationship between saturation pressure and temperature. To extract this relationship, we return to eq. (fj.ij). which gives the differential of the Gibbs free energy. We write this equation for the saturated vapor and liquid ... 

This establishes the relationship between Gibbs free energy and fugacity, and shows that if we know one of these properties, the other one can be easily obtained. Taking the differential of the above equation at constant temperature, and using eq. fc .i7l for dG, the differential of fugacity is... 

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. 

FIGU RE 1.152 The temperature dependence of the concentration of the unfrozen water (C ,) and relationship between differential Gibbs free energy and for wetted APMS/A-300 powder in (a) air and (b) CDClj medium. 

We will show in Chapter 8 that the state of equilibrium in many laboratory experiments is defined by the total differential dG = 0, where the Gibbs free energy G(T, p) is a function of temperature T and pressure p. However, suppose that you want to know the relationship between the measurable quantities T and p at equilibrium when dG = 0. This is the basis for phase equilibria. How is the relationship of T and p defined at constant G We U return to the ph> sics later. For now let s look at the mathematics. 

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