Legendre transforms 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. 

The complete Legendre transform for the system we are discussing yields the Gibbs-Duhem equation for the system ... 

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... 

This shows that the intensive variables for this system (T, P, pj ) are not independent at equilibrium all but one are independent. The derivation of the Gibbs-Duhem equation by use of a complete Legendre transform is shown in the Appendix. 

There are two ways to derive the Gibbs-Duhem equation for a system (1) Subtract the fundamental equation from the total differential of the thermodynamic potential. (2) Use a complete Legendre transform. As an example of the second method, eonsider fundamental equation 3.3-10 for a single reactant ... 

If all components are included in a Legendre transform, the Gibbs-Duhem equation for a system is obtained. This is useful because it provides a relation between the intensive properties of the system, but to make other calculations at least one component must remain. 

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. 

There are basically two types of Legendre transformations, one type based on the energy and the other type based on the entropy. There are various other types of Massieu functions [15]. The Gibbs - Duhem equation is a very special type of Legendre transformation. It is a full transformation and is a relation of the intensive... 

We recall that the Gibbs - Duhem equation is the complete Legendre transformation of the energy equation, U S, V, n). A relation between the variables T, p, pL emerges. Therefore, the set of the independent variables consists of any two variables in the set T, p, pL. For example, we could rearrange Eq. (7.15) into... 

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