Natural variables Gibbs-Duhem equation

These four Legendre transforms introduce the chemical potential as a natural variable. The last thermodynamic potential U T, P, /<] defined in equation 2.6-6 is equal to zero because it is the complete Legendre transform for the system, and this Legendre transform leads to the Gibbs-Duhem equation for the system. 

In discussing one-phase systems in terms of species, the number D of natural variables was found to be Ns + 2 (where the intensive variables are T and P) and the number F of independent intensive variables was found to be Ns + 1 (Section 3.4). When the pH is specified and the acid dissociations are at equilibrium, a system is described in terms of AT reactants (sums of species), and the number D of natural variables is N + 3 (where the intensive variables are T, P, and pH), as indicated by equation 4.1-18. The number N of reactants may be significantly less than the number Ns of species, so that fewer variables are required to describe the state of the system. When the pH is used as an independent variable, the Gibbs-Duhem equation for the system is... 

The number C of components is equal to the number of terms in the summations in equation 8.1-12 minus the number N% of independent equilibria between phases, that is, C = 2NS — Ns = Ns. Equation 8.1-13 shows that there are D = C + 2 = jVs + 2 natural variables. The Gibbs-Duhem equations for the two phases are... 

Equation 8.5-3 indicates that the number of natural variables for the system is 6, D = 6. Thus the number D of natural variables is the same for G and G, as expected, since the Legendre transform interchanges conjugate variables. The criterion for equilibrium is dG 0 at constant T,P,ncAoi, ncA(3, /icC, and The Gibbs-Duhem equations are the same as equations 8.4-8 and 8.4-9, and so the number of independent intensive variables is not changed. Equation 8.5-3 yields the same membrane equations (8.4-13 and 8.4-14) derived in the preceding section. 

Natural Variables Legendre Transforms Isomer Group Thermodynamics Gibbs-Duhem Equation References... 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

The Gibbs-Duhem equation is derived from the Euler and Gibbs equations (e.g., Prausnitz et al. [108], app D Slattery [132], p. 443). The mass based form of the Gibbs-Duhem equation is outlined next. The total differential of the Gibbs free energy G in terms of the natural variables (i.e., T, p, ms) is ... 

In fluid mechanics it might be natural to employ mass based thermodynamic properties whereas the classical thermodynamics convention is to use mole based variables. It follows that the extensive thermodynamic functions (e.g., internal energy, Gibbs free energy, Helmholtz energy, enthalpy, entropy, and specific volume) can be expressed in both ways, either in terms of mass or mole. The two forms of the Gibbs-Duhem equation are ... 

---