\[\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial p}{\partial S}\right)_V,\ \left(\frac{\partial T}{\partial p}\right)_S = \left(\frac{\partial V}{\partial S}\right)_p,\ \left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V,\ \left(\frac{\partial S}{\partial p}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_p\]
热力学状态方程（内压）thermodynamic equation of state (internal pressure)
\[\left(\frac{\partial U}{\partial V}\right)_T = T\left(\frac{\partial p}{\partial T}\right)_V - p\]含义Meaning
由基本方程的全微分条件（二阶混合偏导相等）得到的四个关系，把不可直接测量的熵变化转换为可测的 p、V、T 关系。
Four relations from the equality of mixed second derivatives of the fundamental equations, turning unmeasurable entropy derivatives into measurable p–V–T data.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \((\partial V/\partial T)_p\) | 体膨胀（= αV）thermal expansion (= αV) | m³·K⁻¹ |
推导Derivation
以 dG = −S dT + V dp 为例：G 是状态函数，\(\partial^2G/\partial T\partial p = \partial^2G/\partial p\partial T\)。
For dG: mixed second derivatives are equal.
即 \(-(\partial S/\partial p)_T = (\partial V/\partial T)_p\)。其余三个分别来自 dU、dH、dA。
Hence the fourth relation; the others come from dU, dH, dA.
应用：由 \((\partial S/\partial V)_T = (\partial p/\partial T)_V\) 得内压公式；理想气体内压为零，范德华气体为 \(a/V_m^2\)。
Application: internal pressure is 0 for ideal and a/V_m² for van der Waals gases.
适用条件与局限Assumptions & limitations
适用于只做体积功的均相封闭体系。
Closed, homogeneous systems with only expansion work.