Differential geometry is powerful tool to analyze the vapor-liquid critical point on the surface of the thermodynamic equation of state. The existence of usual condition of the critical point $\left( \partial p/\partial V\right) _{T}=0$ requires the isothermal process, but the universality of the critical point is its independence of whatever process is taken, and so we can assume $\left( \partial p/\partial T\right) _{V}=0$. The distinction between the critical point and other points on the surface leads us to further assume that the critical point is geometrically represented by zero Gaussian curvature. A slight extension of the van der Waals equation of state is to letting two parameters $a$ and $b$ in it vary with temperature, which then satisfies both assumptions and reproduces its usual form when the temperature is approximately the critical one.

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热力学表面上汽液临界点的局部形状和范德华状态方程

微分几何是分析热力学状态方程表面汽液临界点的有力工具。临界点$\left( \partial p/\partial V\right) _{T}=0$ 的通常条件的存在需要等温过程，但临界点的普遍性是它独立于采取的任何过程，因此我们可以假设 $\left( \partial p/\partial T\right) _{V}=0$。临界点与表面上其他点的区别使我们进一步假设临界点在几何上由零高斯曲率表示。范德华状态方程的一个轻微扩展是让其中的两个参数 $a$ 和 $b$ 随温度变化，