布里奇曼热力学方程

热力学里的布里奇曼热力学方程是一组基本的热力学方程，是用一种产生和许多热力学量有关变量的方法而产生的。此方程得名自美国物理学家珀西·布里奇曼。

系统的外延性质是此方程的基础。只考虑熵 S、体积 V 以及四个最重要的热力位能，以下就是这四个热力位能：

内能｜U

焓｜H

亥姆霍兹自由能｜A

吉布斯能｜G

内能相对外延自然变数S 和V 的一阶导数，是系统的内含参数：压强 P 和温度 T。针对粒子数N是常数的简单系统，热力位能的二阶导数只会由三个材料性质表示：

热容量（定压）｜CP

热膨胀系数｜α

等温压缩系数｜βT

布里奇曼的方程是上述这些量彼此之间的关系。

## 简介

许多热力学方程是由偏微分表示。例如定压下的热容表示如下：

$$C_P=\left(\frac{\partial H}{\partial T}\right)_P$$

是在压力不变的情形下，焓对温度的偏导数，可以将此方程写成：

$$C_P=\frac{(\partial H)_P}{(\partial T)_P}$$

布里奇曼有提过这种重写偏导数的方法（Lewis和Randall也有），用这种方式来表达许多热力学方程。

以下的内容以热力位能、状态参数S, T, P, V，以及以下三个材料性质（可以轻易透过实验得到＼）的方式列出许多偏导数项。

$$\left(\frac{\partial V}{\partial T}\right)_P = \alpha V$$

$$\left(\frac{\partial V}{\partial P}\right)_T = -\beta_T V$$

$$\left(\frac{\partial H}{\partial T}\right)_P = C_P = c_P N$$

## 布里奇曼热力学方程

需注意Lewis和Randall用F和E表示吉布斯能和内能，和此条目用的G和U不同。

$$(\partial T)_P=-(\partial P)_T=1$$

$$(\partial V)_P=-(\partial P)_V=\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial S)_P=-(\partial P)_S=\frac{C_p}{T}$$

$$(\partial U)_P=-(\partial P)_U=C_P-P\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial H)_P=-(\partial P)_H=C_P$$

$$(\partial G)_P=-(\partial P)_G=-S$$

$$(\partial A)_P=-(\partial P)_A=-S-P\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial V)_T=-(\partial T)_V=-\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial S)_T=-(\partial T)_S=\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial U)_T=-(\partial T)_U=T\left(\frac{\partial V}{\partial T}\right)_P+P\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial H)_T=-(\partial T)_H=-V+T\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial G)_T=-(\partial T)_G=-V$$

$$(\partial A)_T=-(\partial T)_A=P\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial S)_V=-(\partial V)_S=\frac{C_P}{T}\left(\frac{\partial V}{\partial P}\right)_T+\left(\frac{\partial V}{\partial T}\right)_P^2$$

$$(\partial U)_V=-(\partial V)_U=C_P\left(\frac{\partial V}{\partial P}\right)_T+T\left(\frac{\partial V}{\partial T}\right)_P^2$$

$$(\partial H)_V=-(\partial V)_H=C_P\left(\frac{\partial V}{\partial P}\right)_T+T\left(\frac{\partial V}{\partial T}\right)_P^2-V\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial G)_V=-(\partial V)_G=-V\left(\frac{\partial V}{\partial T}\right)_P-S\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial A)_V=-(\partial V)_A=-S\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial U)_S=-(\partial S)_U=\frac{PC_P}{T}\left(\frac{\partial V}{\partial P}\right)_T+P\left(\frac{\partial V}{\partial T}\right)_P^2$$

$$(\partial H)_S=-(\partial S)_H=-\frac{VC_P}{T}$$

$$(\partial G)_S=-(\partial S)_G=-\frac{VC_P}{T}+S\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial A)_S=-(\partial S)_A=\frac{PC_P}{T}\left(\frac{\partial V}{\partial P}\right)_T+P\left(\frac{\partial V}{\partial T}\right)_P^2+S\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial H)_U=-(\partial U)_H=-VC_P+PV\left(\frac{\partial V}{\partial T}\right)_P-PC_P\left(\frac{\partial V}{\partial P}\right)_T-PT\left(\frac{\partial V}{\partial T}\right)_P^2$$

$$(\partial G)_U=-(\partial U)_G=-VC_P+PV\left(\frac{\partial V}{\partial T}\right)_P+ST\left(\frac{\partial V}{\partial T}\right)_P+SP\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial A)_U=-(\partial U)_A=P(C_P+S)\left(\frac{\partial V}{\partial P}\right)_T+PT\left(\frac{\partial V}{\partial T}\right)_P^2+ST\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial G)_H=-(\partial H)_G=-V(C_P+S)+TS\left(\frac{\partial V}{\partial T}\right)_P$$

$$(\partial A)_H=-(\partial H)_A=-\left[S+P\left(\frac{\partial V}{\partial T}\right)_P\right]\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]+PC_P\left(\frac{\partial V}{\partial P}\right)_T$$

$$(\partial A)_G=-(\partial G)_A=-S\left[V+P\left(\frac{\partial V}{\partial P}\right)_T\right]-PV\left(\frac{\partial V}{\partial T}\right)_P$$