On the Sackur-Tetrode equation in an expanding universe

in this work we investigate the thermodynamic properties satisfied by an expanding universe filled with a monoatomic ideal gas. we show that the equations for the energy density, entropy density and chemical potential remain the same of an ideal gas confined to a constant volume v. in particular the sackur-tetrode equation for the entropy of the ideal gas is also valid in the case of an expanding universe, provided that the constant value that represents the current entropy of the universe is appropriately chosen.