Complete Legendre transform

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 ... 

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 ... 

On the other hand, we wrote for the energy AU S, V,n) = TAS — pAV + txAn. Therefore, the complete Legendre transformation must be zero ... 

We can use this fact to check whether the variable set and the functional dependence of the variable set obey the demand that the complete Legendre transformation is zero. Only then, we have a function of the energy that obeys thermodynamic similarity. 

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... 

The inequalities of the previous paragraph are extremely important, but they are of little direct use to experimenters because there is no convenient way to hold U and S constant except in isolated systems and adiabatic processes. In both of these inequalities, the independent variables (the properties that are held constant) are all extensive variables. There is just one way to define thermodynamic properties that provide criteria of spontaneous change and equilibrium when intensive variables are held constant, and that is by the use of Legendre transforms. That can be illustrated here with equation 2.2-1, but a more complete discussion of Legendre transforms is given in Section 2.5. Since laboratory experiments are usually carried out at constant pressure, rather than constant volume, a new thermodynamic potential, the enthalpy H, can be defined by... 

Let the system under consideration be composed of k1 components. Let the mixture moreover be kept under constant temperature T. We thus replace (by making an appropriate Legendre transformation) the state variable e by the temperature T and then, since T = const., we shall, in order to simplify the notation, completely omit it. Consequently, we have in this illustration ... 

More complete descriptions of Legendre transforms are provided by Callen (6) and Alberty (15). There is an lUPAC Technical Report on Legendre transforms (18). It is interesting to note that a Legendre transform is used in defining the Hamiltonian for a mechanical system on the basis of the Lagrangian (19). 

In conclusion, let us summarize the main principles of the equilibrium statistical mechanics based on the generalized statistical entropy. The basic idea is that in the thermodynamic equilibrium, there exists a universal function called thermodynamic potential that completely describes the properties and states of the thermodynamic system. The fundamental thermodynamic potential, its arguments (variables of state), and its first partial derivatives with respect to the variables of state determine the complete set of physical quantities characterizing the properties of the thermodynamic system. The physical system can be prepared in many ways given by the different sets of the variables of state and their appropriate thermodynamic potentials. The first thermodynamic potential is obtained from the fundamental thermodynamic potential by the Legendre transform. The second thermodynamic potential is obtained by the substitution of one variable of state with the fundamental thermodynamic potential. Then the complete set of physical quantities and the appropriate thermodynamic potential determine the physical properties of the given system and their dependences. In the equilibrium thermodynamics, the thermodynamic potential of the physical system is given a priori, and it is a multivariate function of several variables of state. However, in the equilibrium... 

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 energy of a system may be completely characterized by its extensive variables, such as S, V, n. If we would choose to replace some extensive variables by the corresponding intensive variables, then information is lost. On the other hand, certain functions of the energy, i.e., the Legendre transformations utilize intensive variables, but there is no loss of information, because the functions can be... 

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