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Chemical potential for the interacting classical gas and the ideal quantum gas obeying a generalized exclusion principle

In this work, we address the concept of the chemical potential μ in classical and quantum gases towards the calculation of the equation of state μ = μ(n, T) where n is the particle density and T the absolute temperature using the methods of equilibrium statistical mechanics. Two cases seldom discussed in elementary textbooks are presented with detailed calculations. The first one refers to the explicit calculation of μ for the interacting classical gas exemplified by van der Waals gas. For this purpose, we used the method described by van Kampen (1961 Physica 27 783). The second one refers to the calculation of μ for ideal quantum gases that obey a generalized Pauli's exclusion principle that leads to statistics that go beyond the Bose–Einstein and Fermi–Dirac cases. The audience targeted in this work corresponds mainly to advanced undergraduates and graduate students in the physical–chemical sciences but it is not restricted to them. In regard of this, we have put a special emphasis on showing some additional details of calculations that usually do not appear explicitly in textbooks.