The thermodynamics of liquid-vapor coexistence for a van der Waals fluid. Analytical solution of the Clausius-Clapeyron equation

This work presents a pedagogical derivation of the thermodynamics of a van der Waals fluid by explicitly incorporating pairwise molecular interactions and the finite size of particles into the statistical-mechanical description. Starting from the Lennard-Jones potential, we evaluate the second virial coefficient to infer the virial expansion of the equation of state and recover the van der Waals equation using only its leading correction. The corresponding partition function allows us to obtain all thermodynamic potentials for both monoatomic and diatomic fluids in a transparent and instructive manner. Building on this framework, we formulate and solve analytically the Clausius-Clapeyron equation in the vicinity of the critical point, obtaining the liquid-vapor coexistence curve in closed form. This approach not only clarifies the microscopic origin of van der Waals thermodynamics but also complements-and in several aspects improves upon-traditional treatments that rely heavily on numerical methods or heuristic arguments. In addition, because the van der Waals equation naturally predicts the liquid-vapor equilibrium, the existence of critical points, and the functional form of the saturation curve of the pressure as a function of temperature, it provides an analytically tractable framework for studying a 150-year-old problem that has historically been addressed using graphical constructions or numerical solutions. As such, the formulation developed here offers a coherent, accessible, and conceptually unified route for students and instructors to understand phase coexistence in simple fluids from first principles.