Chemical Potentials in Gas Mixtures

The analysis in Sect. 7.2.2 of the entropy change associated with mixing of ideal gases at fixed T and p was based on the absence of an entropy change if the pure gases are at the partial pressures that they will have in the mixture. Since the gases are ideal, neither is there an enthalpy change in the mixing process. With both the enthalpy and entropy of each species unaltered, the Gibbs free energy must also remain constant during this mode of mixing. Since the partial molar Gibbs free energy of a species in a solution or a gas mixture is the same as its chemical potential, the above argument can be summarized by the equation:

3 Proofs of Eqs (7.40) - (7.42) are left as an exercise (problem 7.4).

Contrary to condensed phases, the Gibbs free energy of a pure gas is pressure-dependent. In order to provide a common pressure reference for all pure gases (arbitrarily chosen at 1 atm), gi in the above equation is expressed in terms of g°, which is the molar Gibbs free energy of species i at temperature T and 1 atm pressure. The effect of the difference in pressure between pi and 1 atm is obtained by combining dgi/dpi = vi with the ideal gas law:

Integrating this equation from pi to 1 atm gives the molar free energy of pure i at pi, which, when substituted into Eq (7.3) yields:

The reference condition denoted by the superscript zero is called a standard state of the pure gas. In the above analysis, it has been set at 1 atm, which is convenient when dealing with permanent gases such as air. With this standard state, pi must be expressed numerically in units of atmospheres.

Reference

1. D. R. Gaskell, Introduction to Metallurgical Thermodynamics, 2nd Ed., McGraw-Hill (1981)

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