Particle Entropies and Entropy Quanta. III. The van der Waals Gas

The statistical thermodynamics of the van der Waals gas is treated by means of the particle entropies σ_s=ka_s. The non-dimensional number a_s measures the particle entropy σ_s in multiples of the Boltzmann constant k, which is the atomic entropy unit. a_s can be calculated by an extended Schrödinger equation containing the temperature T. The attraction between the particles is introduced by perturbation theory using the mean field approximation, and the particle volume is considered by suitable boundary conditions in solving the eigenvalue problem. Eigenvalues and eigenfunctions are determined by the translational quantum numbers. – The occupation numbers f_s of the entropy levels s are given by the Boltzmann distribution, which is rewritten in terms of the non-dimensional particle entropy a_s and the α-function. The latter has the physical meaning of a dimensionless chemical potential, α=μ/kT. In the case of a closed system α can be determined from the entropy quanta a_s and the condition of the conservation of the particle number N. The α-function is the connecting link between the entropy quanta a_s and the Helmholtz free energy F of the system, which can be calculated by integrating the α-function. From the thermodynamic potential F we deduce all the other relevant thermodynamic quantities as derivatives of F, e.g. the equation of state of the van der Waals gas, its entropy S and internal energy E. – The critical properties of the van der Waals gas are also deduced from the α-function. For that reason we expand α at the critical point c in terms of the small quantities τ=(T-T_c)/T_c and Δρ=ρ-ρ_c; T_c, ρ_c critical temperature and particle density respectively. The discussion can be simplified by introducing the order parameter ξ=(ρ-ρ_c)/ρ_c and the conjugate thermodynamic field Δα=α(ξ,τ)-α(0,τ). The spontaneous symmetry breaking at the critical point c, the jump ΔC_V of the heat capacity C_V, the susceptibility χ and its relation to the isothermal compressibility κ_T, the critical exponents α, β, γ, δ and other topics are discussed in some detail.