Gibbs energy derivatives

One result of the simplification inherent in the CSA treatment is that the same expression is obtained for the entropy of both f.c.c. and b.c.c. lattices which clearly distinguishes it from the differences noted in Fig. 7.3(b). However, Fig. 7.10 shows that the overall variation of Gibbs energy derived from the CSA method agrees well with CVM, falling between the pair approximation, which overestimates the number of AB bonds, and the point approximation, where these are underestimated. As might be expected, if larger clusters are admitted to the CSA approximation the results become closer to the CVM result. However, this is counterproductive if the object is to increase the speed of calculation for multi-component systems. 

The most important themiodynamic property of a substance is the standard Gibbs energy of fomiation as a fimetion of temperature as this infomiation allows equilibrium constants for chemical reactions to be calculated. The standard Gibbs energy of fomiation A G° at 298.15 K can be derived from the enthalpy of fomiation AfT° at 298.15 K and the standard entropy AS° at 298.15 K from... 

The enthalpy of fomiation is obtained from enthalpies of combustion, usually made at 298.15 K while the standard entropy at 298.15 K is derived by integration of the heat capacity as a function of temperature from T = 0 K to 298.15 K according to equation (B 1.27.16). The Gibbs-FIehiiholtz relation gives the variation of the Gibbs energy with temperature... 

When M = the requited derivatives are given by equations 62 and 63. Moreover, the derivative on the left side of equation 64 defines the partial molar Gibbs energy, G. Therefore,... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

Gamma/Phi Approach For many XT E systems of interest the pressure is low enough that a relatively simple equation of state, such as the two-term virial equation, is satisfactoiy for the vapor phase. Liquid-phase behavior, on the other hand, may be conveniently described by an equation for the excess Gibbs energy, from which activity coefficients are derived. The fugacity of species i in the liquid phase is then given by Eq. (4-102), written... 

This equation is derived by considering die transfer of material from a flat surface to a droplet. For the U ansfer of a small mass Sm from the flat surface of vapour pressure p° to the droplet of vapour pressure p, the Gibbs energy of transfer is... 

If we combine the Gibbs energy of formation equations above to derive the equation... 

TABLE 32.2 Standard Gibbs Energy of Transfer and Standard Ion Transfer Potentials for Ion Transfer Between Water and Nitrobenzene Derived from Partition Measurements... 

The relationships of the type (3.1.54) and (3.1.57) imply that the standard electrode potentials can be derived directly from the thermodynamic data (and vice versa). The values of the standard chemical potentials are identified with the values of the standard Gibbs energies of formation, tabulated, for example, by the US National Bureau of Standards. On the other hand, the experimental approach to the determination of standard electrode potentials is based on the cells of the type (3.1.41) whose EMFs are extrapolated to zero ionic strength. 

The derivation for equilibrium between a solution and a gaseous phase is based on the Henry law constant kiy defined on page 5. The standard Gibbs energy of the transfer of HC1 from a solution into water is... 

These considerations can also be used to derive the Dupre equation, where AG(s) is the Gibbs energy of adsorption of the solvent per unit area of the metal surface ... 

Fig. 12.2. Free energy data for electron transfer between the protein cytochrome c and the small acceptor microperoxidase-8 (MP8), from recent simulations [47]. Top Gibbs free energy derivative versus the coupling parameter A. The data correspond to solvated cytochrome c the MP8 contribution is not shown (adapted from [47]) Bottom the Marcus diabatic free energy curves. The simulation data correspond to cyt c and MP8, infinitely separated in aqueous solution. The curves intersect at 77 = 0, as they should. The reaction free energy is decomposed into a static and relaxation component, using the two steps shown by arrows a static, vertical step, then relaxation into the product state. All free energies in kcalmol-1. Adapted with permission from reference [88]...

These arguments are similar to those employed in the derivation of the Butler-Volmer equation for electron-transfer reactions in Chapter 5. However, here the reaction coordinate corresponds to the motion of the ion, while for electron transfer it describes the reorganization of the solvent. For ion transfer the Gibbs energy curves are less symmetric, and the transfer coefficient need not be close to 1/2 it may also vary somewhat with temperature since the structure of the solution changes. 

The pressure derivative of the Gibbs energy (eq. 1.68) at constant temperature is V ... 

In the absence of nonp E-work, an extensive property such as the Gibbs energy of a system can be shown to be a function of the partial derivatives ... 

Figure 2.3 The temperature variation of the Gibbs energy [5], unit-cell volume [4] enthalpy and heat capacity [5] at the second-order a- to /3-quartz transition of SiC>2-Second-order derivatives of the Gibbs energy like the heat capacity have discontinuities at the transition temperature.

The formalism shown above is in general easily extended to multi-component systems. All thermodynamic mixing properties may be derived from the integral Gibbs energy of mixing, which in general is expressed as... 

In order to derive the phase boundaries in Figure 4.22 we need the Gibbs energy of formation of the oxides. This type of data is conveniently given in an Ellingham... 

In general, the first derivative of the Gibbs energy is sufficient to determine the conditions of equilibrium. To examine the stability of a chemical equilibrium, such as the one described above, higher order derivatives of G are needed. We will see in the following that the Gibbs energy versus the potential variable must be upwards convex for a stable equilibrium. Unstable equilibria, on the other hand, are... 

Let us assume the existence of a Taylor series for the Gibbs energy at the equilibrium point. This implies that the Gibbs energy and all its derivatives vary continuously at this point. The Taylor series is given as... 

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