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

Abstract

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.

Quantum thermochemical engines

“…Classical gases, especially with repulsive interactions, can also exhibit both negative and positive chemical potentials [134]. Nevertheless, the effects of quantum statistics can be neglected when (see Appendix A)…”

Thermodynamics of the relativistic Fermi gas inDdimensions

“…At the end of the 20th century, the experimental realization of quantum degeneration of a trapped Fermi gas of 40 K atoms [4] raised the interest on the theoretical study of the thermodynamical and dynamical properties of the Fermi gas in the ideal approximation [5][6][7][8][9][10][11][12][13][14][15].…”

“…Though, particular attention is paid to the temperature dependence of the chemical potential, which has motivated several discussion of its importance on different levels and contexts [32][33][34][35][36][37][38][39][40], our main results focus on the thermodynamic susceptibilities or response functions, namely the specific heat at constant volume C V and the isothermal compressibility κ T , for which there is a great interest at conditions of extreme densities and/or temperatures. Interestingly, our calculations reveal the appearance of a transition in the temperature dependence of C V and κ T due to pair production, that occurs at a few tenths of the Fermi temperature.…”

Thermodynamics of Low-Dimensional Trapped Fermi Gases

“…Truly, the significance of µ has motivated the discussion of its meaning and/or importance at different levels and contexts [41][42][43][44][45][46][47][48][49][50][51][52]. For the widely discussedtextbook-case, namely the three-dimensional IFG confined by a impenetrable box potential, the chemical potential results to be a monotonic decreasing function of the temperature, diminishing from the Fermi energy, E F , at zero temperature, to the values of the ideal classical gas for temperatures much larger than k −1 B ( 2 /mλ 2 T ), where k B is the Boltzmann's constant, is the Planck's constant divided by 2π, m the mass of the particle and λ T = 2π 2 /mk B T is the thermal wavelength of de Broglie, where T denotes the system's absolute temperature.…”