Gibbs energy integral

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... 

Evaluation of the integrals requires an empirical expression for the temperature dependence of the ideal gas heat capacity, (3p (8). The residual Gibbs energy is related to and by equation 138 ... 

In this context G itself is often referred to as the integral Gibbs energy. 

The value of this standard molar Gibbs energy, p°(T), found in data compilations, is obtained by integration from 0 K of the heat capacity determined by the translational, rotational, vibrational and electronic energy levels of the gas. These are determined experimentally by spectroscopic methods [14], However, contrary to what we shall see for condensed phases, the effect of pressure often exceeds the effect of temperature. Hence for gases most attention is given to the equations of state. 

Since AmixGA = fiA - pA, the integral molar Gibbs energy of mixing can alternatively be expressed in terms of the chemical potentials as... 

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... 

Figure 3.12 The integral molar Gibbs energy of liquid Ge-Si at 1500 K with pure liquid Ge and solid Si as standard states. Data are taken from reference [4].

This type of defect equilibrium treatment has been used extensively to model the defect chemistry and non-stoichiometry of inorganic substances and has the great advantage that it easily takes several simultaneous defect equilibria into account [22], On the other hand, the way the mass action laws are normally used they are focused on partial thermodynamic properties and not on the integral Gibbs energy. The latter is often preferred in other types of thermodynamic analyses. In such cases the following solid solution approach is an alternative. 

For thermochemical uses, an expression for the integral Gibbs energy of formation of the compound ABC can be derived by integration of eq. (9.104), but in order to show clearly some of the main implications of the model a more detailed analysis starting from the partition function is preferred [23]. 

Considerable use has been made of the thermodynamic perturbation and thermodynamic integration methods in biochemical modelling, calculating the relative Gibbs energies of binding of inhibitors of biological macromolecules (e.g. proteins) with the aid of suitable thermodynamic cycles. Some applications to materials are described by Alfe et al. [11]. 

The difference in the Gibbs energies of formation corresponds to the integral of the cell voltage as a function of the composition between the limits of the starting and the final compositions. The value for ASf° is determined from the variation of the cell voltage as a function of the temperature... 

Figure 3.7, Gibbs energy vs composition diagrams of an A-B system showing relationship between partial and integral Gibbs energies.

One method would be to use Eq. (3.62) and utilise a Newton-Raphson technique to perform a Gibbs energy minimisation with respect to the composition of either A or B. This has an advantage in that only the integral function need be calculated and it is therefore mathematically simpler. The other is to minimise the difference in potential of A and B in the two phases using the relationships... 

The electromotive force (EMF) generated by electrochemical cells can be used to measure partial Gibbs energies which, like vapour pressure measurements, distinguishes these methods from other techniques that measure integral thermodynamic quantities. Following Moser (1979), a typical cell used to obtain results on Zn-ln-Pb is represented in the following way ... 

The integral Gibbs energy, of a pure species or stoichiometric compound is... 

The integral Gibbs energy for this model is then given by... 

However, AmicH° and TA, determined from the temperature dependence of the Gibbs energy, are less sensitive to the association number than is AmicG° itself (Yang ef al. 1995). Assuming that Am cH° is approximately constant within a certain temperature range, eqn 3.7 can be integrated to yield... 

Liquid-Solution Models. The simple-solution model has been used most extensively to describe the dependence of the excess integral molar Gibbs energy, Gxs, on temperature and composition in binary (142-144, 149-155), quasi binary (156-160), ternary (156, 160-174), and quaternary (175-181) compound-semiconductor phase diagram calculations. For a simple multicomponent system, the excess integral molar Gibbs energy of solution is expressed by... 

The Gibbs energy of the gas mixture is again obtained by integrating Equation (4.25) under conditions of constant temperature and constant mole... 

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