The ideal gas does have a mathematical definition of entropy, Boltzmann used it in the statistical derivation of the second law:https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)

Here is an account of Boltzmann work and the first objections to his conclusions:https://plato.stanford.edu/entries/statphys-Boltzmann/

The ideal gas does have a mathematical definition of entropy, Boltzmann used it in the statistical derivation of the second law:

https://en.wikipedia.org/wiki/Entropy_(statistical_thermodynamics)

Here is an account of Boltzmann work and the first objections to his conclusions:

https://plato.stanford.edu/entries/statphys-Boltzmann/