Planck's experiment on gas mixing, Gibbs' paradox and the definition of entropy in statistical mechanics

Abstract

The interpretation of entropy provided by statistical thermodynamics is not adequate to represent the thermodynamic entropy of the gas of noninteracting particles considered in this theory. Planck's thought experiment on reversible mixing and Gibbs' paradox provide perhaps the best-known evidence of this. The assumption that the internal energy of an ideal gas depends only on its temperature is introduced both in the kinetic theory of gases and in the classical thermodynamics. Such an assumption is no doubt adequate to deal with real gases at appropriately low pressures and high temperatures. However, the present paper shows that the same assumption implies that the entropy of an ideal gas, like its internal energy, must also depend only on temperature. The paper calculates the expression of the entropy function that is consistent with the internal energy function of the gas. From this expression, the thermodynamic entropy of the ideal gas – as distinct from its statistical entropy – is finally expressed in terms of statistical mechanics variables.