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 $1 / (4 R b)$ ], the localization intervals of the three volumes on the isobar-isotherm are: $3 b / 2 < V_{A} \leq 3 b$ , $2 b < V_{B} < (3 + \sqrt{5}) b$ , and $3 b \leq V_{C} < R T / p + b = V_{0} + b$ (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 $b < V_{B} < 2 a / (R τ)$ , where $τ > 0$ 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.

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 $1 / (4 R b)$ ], the localization intervals of the three volumes on the isobar-isotherm are: $3 b / 2 < V_{A} \leq 3 b$ , $2 b < V_{B} < (3 + \sqrt{5}) b$ , and $3 b \leq V_{C} < R T / p + b = V_{0} + b$ (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 $b < V_{B} < 2 a / (R τ)$ , where $τ > 0$ 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.

文献摘要