Mathematical Analysis of the van der Waals Equation

Abstract

The parametric cubic van der Waals polynomial $p{V}^3-(RT+bp)V^2+aV-ab$ 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.

15 pages, 5 figures

References in corpus (2)

• The Siebeck-Marden-Northshield Theorem and the Real Roots of the Symbolic Cubic Equation
• Isolation Intervals of the Real Roots of the Parametric Cubic Equation and Improved Complete Root Classification