Thermodynamic identity

By making use of classical thermodynamic identities, this is found to be equivalent to... 

What is the difference between CP and CV1 We first express this difference in terms of a standard thermodynamic identity for 1 mole of a simple gas ... 

Joule-Thomson inversion typically occurs at temperatures far above the critical temperature (Ti > Tc). We shall later prove (Sidebar 5.5) the general thermodynamic identity... 

Maxwell relations are a powerful tool for deriving thermodynamic relationships. Their use should be considered whenever it is desirable to replace thermodynamic derivatives involving S with equivalent derivatives involving variables P, V, T only. Sidebars 5.4-5.6 illustrate this derivation techniques for a number of standard thermodynamic identities. 

Problem Prove that the heat capacity difference CP — Cv satisfies the thermodynamic identity (3.56) CP — Cv= TVap//3T. 

Note that many alternative methods of solution are possible using various thermodynamic identities.)... 

As mentioned in the Preface, our goal in Part III is not merely to re-generate the material of Parts I and II (as summarized in Section 8.9) in new mathematical dress. We re-derive (rather trivially) many earlier thermodynamic identities and stability conditions to illustrate the geometrical techniques, but our primary emphasis is on thermodynamic extensions (particularly, to saturation properties, critical phenomena, multicomponent Gibbs-Konowalow-type relationships, higher-derivative properties, and general reversible changes... 

Taken together, (11.22) and (11.23) lead to various thermodynamic identities between measured response functions, as will be illustrated below. Equation (11.23) shows that the inverse metric matrix M-1 plays a role for conjugate vectors R/) that is highly analogous to the role played by M itself for the intensive vectors R,). In view of this far-reaching relationship, we can define the conjugate metric M,... 

Let us return to (11.23) to obtain a group of thermodynamic identities simultaneously. From Table 11.1, the metric matrix M and its inverse are... 

The thermodynamic identities (11.73a, b) can be derived by more conventional means [see, e.g., J. T. Rowlinson. Liquids and Liquid Mixtures (Academic Press, New York, 1959), Chap. 2], but their derivation here illustrates rather general and systematic matrix-algebraic procedures that remain effective when traditional methods are unduly cumbersome. 

The practical applications of the theory just outlined divide themselves into two broad classes (1) Those which are based on the existence and properties of the functions U and S and some others related to them—all "thermodynamic identities being merely the integrability condition for the total differentials of these functions and (2) those which aie based on die Principle of Increase of Entropy the entropy of the actual state of an adiabatically enclosed system being greater than that of any neighboring virtual state. 

From these basic equations a whole spectrum of useful thermodynamic identities may be derived by partial differentiation... 

Form other thermodynamic identities one obtains the following relation between the coefficients k, B and X ... 

Both equations (1.6) and (1.7) are thermodynamically identical, but equation (1.6) correlates better with the classical kinetics and, therefore, is more appropriate for writing kinetic equations in terms of thermody namics (see following). 

Coupled with the thermodynamic identity that relates the temperature to the energy dependence of the entropy, namely,... 

As a corollary to this section, we derive a relation between the density derivative of the chemical potential and an integral involving g(R). Recall the thermodynamic identity... 

We notice that this equation is similar to that obtained for the ideal solubility of solids (Section 6.4). Indeed the two situations are thermodynamically identical, but the enthalpy of fusion is that of the solute in the ideal solubility equation. In the case now being examined the solid state is the pure solvent (Fig. 6.8). We may integrate as before from Xj = 1, T — Tfus to xx = xt> T —T and obtain... 

Since we can experimentally determine/and obtain fs from the thermodynamic identity [Maxwell relationship resulting from equation (6-17)],... 

Equation 14 follows from Equation 13 because of the thermodynamic identity (3S/3V) p = (3P/6T)v. Because (3P/3T)v can be determined directly, the internal pressure can be measured. Note that the term T(3P/3T)v, and hence (Pj +P), also arises in Equation 9 in accounting for the effect of volume changes that occur on mixing. 

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