论文标题
范德华方程的数学分析
Mathematical Analysis of the van der Waals Equation
论文作者
论文摘要
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 $，$ 3B \ le v_c <rt/p + b = v_0 + b $（$ v_0 $是Clapeyron是Clapeyron的理想气量）。对于较低的温度值，root $ v_a $从下面限制为$ b $，而$ v_b $具有本地化间隔$ b <v_b <2a/（r \，τ）$，其中$τ> 0 $是模型的新最低温度。 Van der Waals模型的不稳定状态也已被普遍进行本地化：它们位于$ v_b $的本地化间隔内。还介绍了有关在麦克斯韦假设的前提下查找卷$ v_ {a，b，c} $的讨论。
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 $b < V_B < 2a/(R \, τ)$, 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.
