Thermodynamic Stability Criteria for Single-Phase Homogeneous Mixtures

THERMODYNAMIC STABILITY CRITERIA FOR SINGLE-PHASE HOMOGENEOUS MIXTURES 

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. 

29-1 ENERGY REPRESENTATION OE THE FUNDAMENTAL EQUATION AND EXACT DIFFERENTIALS 

Extensive thermodynamic state functions such as the internal energy U depend linearly on mass or mole numbers of each component. This claim is consistent with Euler s theorem for homogeneous functions of the first degree with respect to molar mass. If a mixture contains r components and exists as a single phase, then U exhibits r - - 2 degrees of freedom and depends on the following natural variables, all of which are extensive  

T is temperature, p is pressure, and p i is the chemical potential of component i in the mixtnre. The chemical potential is not a partial molar property of U because S and V are held constant during differentiation instead of T and p, based on the definition of /r in (29-3). The total differential of internal energy 

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