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Available under a Creative Commons Attribution Non-Commercial Share Alike 4.0 International Licence


Applied mathematics, Atomic, Molecular and Chemical Physics


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: the temperature range for applicability of the van der Waals equation, $T > a/(4Rb)$, and the isolation intervals, at any given temperature between $a/(4Rb)$ and the critical temperature $8a/(27Rb)$, of the three volumes on the isobar--isotherm: $3b/2 < V_A \le 3b$, $ 2b < V_B < 4b/(3 - \sqrt{5})$, and $3b < V_C < b + RT/p$. The unstable states of the van der Waals model have also been generically localized: they lie in an interval within the isolation interval of $V_B$. In the case of unique intersection point of an isotherm with an isobar, the isolation interval of this unique volume is also determined. A discussion on finding the volumes $V_{A, B, C}$, on the premise of Maxwell's hypothesis, is also presented.


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