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

TL;DR

This paper addresses deriving the thermodynamics and phase behavior of a van der Waals fluid from microscopic principles. It begins with the Lennard–Jones potential to compute the second virial coefficient $B(T)$, enabling a virial expansion that recovers the van der Waals equation $P=RT/(v-b)-a/v^2$ from the partition function, and it derives all thermodynamic potentials for monoatomic and diatomic fluids. The authors then analytically solve the Clausius–Clapeyron equation near the critical point to obtain a closed-form liquid–vapor coexistence curve, presenting zero-, first-, and second-approximation expressions for the saturation pressure $p_s(t_s)$; the best approximation achieves sub-percent accuracy compared with Maxwell’s construction. By extending the framework to diatomic rotors and maintaining the same functional form of the EOS, the work provides a coherent, pedagogical, and analytically tractable route from microscopic interactions to phase coexistence in simple fluids.

Abstract

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.

Figures (5)

• Figure 1: Isotherms according to the van der Waals equation of state. In the upper graphs, one isotherm corresponds to $t_r > 1$, another to $t_r = 1$, and two others to $t_r < 1$. In addition, the spinodal pressure is plotted. Note that $p_{\mathrm{esp}}$ passes through the minima and maxima of the isotherms for $t_r \leq 1$. Plotting the spinodal pressure is useful because it delineates the unstable thermodynamic states, since below this curve $\kappa_T$ becomes negative. The procedure to determine the vapor pressure, according to Maxwell’s rule, is shown in the lower panel. It consists of replacing the section of the isotherm between $v_{r1}$ and $v_{r3}$ with a horizontal line such that the area between this line and the curve from $v_{r1}$ to $v_{r2}$, $A_1$, is equal to the area between the curve from $v_{r2}$ to $v_{r3}$ and the same line, $A_2$.
• Figure 2: The Lennard–Jones intermolecular potential and the corresponding Mayer function are shown. The blue curve illustrates the Lennard–Jones potential, while the red curve represents the Mayer function evaluated with $\phi\left(q_{2\,1}\right)$. Spheres representing monatomic molecules are depicted below; if these spheres are taken to be hard, two regions appear: a repulsive region for $q\leq q_0$ and a short-range attractive region beyond this point. Within the repulsive region, the Mayer function exhibits an inflection point at $d$, which is interpreted as the diameter of the hard spheres.
• Figure 3: Phase diagrams in the $P-v$ and $P-T$ planes, showing the three states of aggregation in which a substance composed of fermions, such as the isotope He$^3$ or NO, can exist. The red curve, which begins at the triple point and ends at the critical point, represents the liquid–vapor coexistence. The diagrams are not drawn to scale.
• Figure 4: Phase diagrams in the $P-v$ and $P-T$ planes, showing the four states of aggregation in which a substance composed of bosons, such as the isotope He$^4$ or N$_2$, can exist. The superfluid phase is a macroscopic manifestation of Bose–Einstein condensation. The diagrams are not drawn to scale.
• Figure 5: Liquid-vapor coexistence curves for the zero and first approximations, where the saturation pressure is plotted as a function of temperature in both cases.