Thermodynamics of a classical ideal gas at arbitrary temperatures

We propose a fundamental relation for a classical ideal gas that is valid at all temperatures with remarkable accuracy. All thermodynamical properties of classical ideal gases can be deduced from this relation at arbitrary temperature. The famous equation of state for an ideal classical gas is PV = NkBT . (1) Interestingly, a classical gas obeys this relation at all temperatures as long as it is ideal, i.e., the Hamiltonian of the system does not depend on the co-ordinates of the particles at all. The proof is simple, and appears in most textbooks [1]. We will also provide the proof later. An equation of state, however, does not specify a system completely [2]. For example, from the equation of state in Eq. (1), we cannot find the entropy of the system, and many other properties for that matter. Of course if we have a fundamental relation for the system, it contains all thermodynamic information about the system including the equations of state themselves [2]. These are, for example, relations of the type S = S(U, V, N) or U = U(S, V, N), which express the entropy or the internal energy as a function of other extensive parameters of the system. Legendre transforms of these equations work just as well, like the Helmholtz free energy A as a function of T , V and N . However, it is often difficult to obtain such relations in closed forms which would be valid for any temperature. The Sackur-Tetrode relation, for example, is a fundamental relation of the form S = S(U, V, N), but unlike the relation in Eq. (1), it is valid only if the gas is nonrelativistic, i.e., if the temperature is small in the sense that βmc 1, where m is the mass of the gas particles. Our aim in this article is to suggest a fundamental relation for the classical ideal gas that can be used at any temperature. Since the gas is assumed to be ideal, the energy of any particle in the gas depends only on its momentum. At a momentum p, let us denote the energy of a particle by ε(p). The