Mathematical Analysis of the van der Waals Equation

The parametric cubic van der Waals polynomial $p V^3 - (R T + b p) V^2 + a V - a b$ is analysed mathematically and some new generic features (theoretically, for any substance) are revealed - if the pressure is not allowed to take negative values [temperatures not lower than $1/(4Rb)$], the localization intervals of the three volumes on the isobar-isotherm are: $3b/2 < V_A \le 3b$, $\,\, 2b < V_B < (3 + \sqrt{5})b$, and $3b \le V_C < RT/p + b = V_0 + b$ (with $V_0$ being Clapeyron's ideal gas volume). For lower values of the temperature, the root $V_A$ is bounded from below by $b$, while $V_B$ has the localization interval $b < V_B < 2a/(R \, \tau)$, where $\tau > 0$ is the new minimum temperature of the model. The unstable states of the van der Waals model have also been generically localized: they lie in an interval within the localization interval of $V_B$. A discussion on finding the volumes $V_{A, B, C}$, on the premise of Maxwell's hypothesis, is also presented.