Particle Entropies and Entropy Quanta. I. The Ideal Gas

Abstract

We consider an ideal gas of monatomic independent particles, which is enclosed in a cubic box. At temperature T the particles are in thermal equilibrium. All relevant properties of the gas can be deduced from the particle statistics on the assumption that each particle of the ensemble has the particle entropy σ = ε/T = ka. ε is the translational energy of the particle. The non-dimensional number a measures the particle entropy σ in multiples of the Boltzmann constant k, which acts as an atomic entropy unit. a obeies an eigenvalue equation and satisfies boundary conditions. Eigenvalues and eigenfunctions are determined by the translational quantum numbers. Using particle entropies it is easy to calculate the Bose-Einstein and the Boltzmann distribution; and in combination with the density function we immediately get the internal energy E and the Helmholtz free energy F, the total entropy S, the chemical potential μ, the equation of state of the ideal gas at ordinary temperatures and at low temperatures near absolute zero, inclusively Bose-Einstein condensation. Entropy quanta are used to introduce the temperature into the equations of statistical thermodynamics and to calculate the thermal and the actual de Broglie wavelength at temperature T.