布里奇曼熱力學方程

熱力學裡的布里奇曼熱力學方程是一組基本的熱力學方程，是用一種產生和許多熱力學量有關變量的方法而產生的。此方程得名自美國物理學家珀西·布里奇曼。

系統的外延性質是此方程的基礎。只考慮熵 S、體積 V 以及四個最重要的熱力位能，以下就是這四個熱力位能：

內能｜U

焓｜H

亥姆霍茲自由能｜A

吉布斯能｜G

內能相對外延自然變數S 和V 的一階導數，是系統的內含參數：壓強 P 和溫度 T。針對粒子數N是常數的簡單系統，熱力位能的二階導數只會由三個材料性質表示：

熱容量（定壓）｜CP

熱膨脹係數｜α

等溫壓縮係數｜βT

布里奇曼的方程是上述這些量彼此之間的關係。

## 簡介

許多熱力學方程是由偏微分表示。例如定壓下的熱容表示如下：

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

是在壓力不變的情形下，焓對溫度的偏導數，可以將此方程寫成：

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

布里奇曼有提過這種重寫偏導數的方法（Lewis和Randall也有），用這種方式來表達許多熱力學方程。

以下的內容以熱力位能、狀態參數S, T, P, V，以及以下三個材料性質（可以輕易透過實驗得到＼）的方式列出許多偏導數項。

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

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

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

## 布里奇曼熱力學方程

需注意Lewis和Randall用F和E表示吉布斯能和內能，和此條目用的G和U不同。

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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