Simple Gibbs Energy Diagrams

To make up Gibbs energy diagrams we need the derivatives of G with respect to T and P. These are shown in 

But we know that for any uniform mass of matter, from the property equation (Chapter 2) 

This result was shown as derived from a Bridgman table in Section 2.10. Here we see a somewhat more intuitive route to it. From Eq. 4.31 the two necessary derivatives follow by inspection  

We can obviously divide each of these by the number of mols or the mass to obtain derivatives on a per-mol, per-pound, or per-kilogram basis. 

The S and s in Eqs. 4.33 and 4.34 are absolute entropies, not entropies relative to some arbitrary datum like steam table entropies (see Appendix E). 

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Let us consider a simple isomerization reaction A B in the solvents I and II, whose abilities to solvate A and B are different. This corresponds to the Gibbs energy diagram shown in Fig. 4-1. 

Many models for the liquid phase have been used in the analysis of phase diagrams. Some of these are discussed briefly here to provide a basis for the choice of model we made and to cite earlier work. Most of the relevant points can be made by considering models for two component liquids. A number of simple models are special cases of one that is defined by giving the excess Gibbs energy of mixing as... 

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... 

Thermochemical data on the separate phases in equUibrium are needed to constmct accurate phase diagrams. The Gibbs energy of formation for a pure substance as a function of temperature must be calculated from experimentally determined temperature-dependent thermodynamic properties such as enthalpy, entropy, heat capacity, and equihbrium constants. By a pure substance, one generally means a stoichiometric compound in which the atomic constituents ate present in an exact, simple reproducible ratio. 

The proposed approach leads directly to practical results such as the prediction—based upon the chemical potential—of whether or not a reaction runs spontaneously. Moreover, the chemical potential is key in dealing with physicochemical problems. Based upon this central concept, it is possible to explore many other fields. The dependence of the chemical potential upon temperature, pressure, and concentration is the gateway to the deduction of the mass action law, the calculation of equilibrium constants, solubilities, and many other data, the construction of phase diagrams, and so on. It is simple to expand the concept to colligative phenomena, diffusion processes, surface effects, electrochemical processes, etc. Furthermore, the same tools allow us to solve problems even at the atomic and molecular level, which are usually treated by quantum statistical methods. This approach allows us to eliminate many thermodynamic quantities that are traditionally used such as enthalpy H, Gibbs energy G, activity a, etc. The usage of these quantities is not excluded but superfluous in most cases. An optimized calculus results in short calculations, which are intuitively predictable and can be easily verified. 

The ideal solution assumes equal strength of self- and cross-interactions between components. When this is not the case, the solution deviates from ideal behavior. Deviations are simple to detect upon mixing, nonideal solutions exhibit volume changes (expansion or contraction) and exhibit heat effects that can be measured. Such deviations are quantified via the excess properties. An important new property that we encounter in this chapter is the activity coefficient. It is related to the excess Gibbs free energy and is central to the calculation of the phase diagram. 

Sodium hydroxide has been the most commonly used base in experimental nitroalkane proton transfer reaction studies.However, the computational studies of these reactions have generaUy been with hydroxide ion without the sodium counter ion. Recently a computational study of the proton transfer reactions of the three simple nitroalkanes in the presence of NaOH in water has been carried out and it was found that the presence of Na had an enormous effect on the energetics of the reactions. Double potential energy well diagrams, much like those found for the Sn2 reactions, were recorded for the proton transfer reactions of NM, NE and 2-NP with hydroxide ion in water. The computations included two explicit water molecules in the water cavity. The Gibbs free energies and enthalpies observed for the reactant complex (CPI), the TS and the product complex (CP2) both in the presence and absence of sodium ion and two explicit water molecules are summarized in Table 1.24. 

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