Chemical potential, Gibbs-Duhem equation and quantum gases

Thermodynamic relations like the Gibbs-Duhem are valid from the lowest to the highest temperatures. But they cannot by themselves provide any specific temperature behavior of thermodynamic functions like the chemical potential. In this work, we show that if some general conditions are attached to the Gibbs-Duhem equation, it is possible to obtain the low temperature form of the chemical potential for the ideal Fermi and Bose gases very directly.