热力学全部336个偏导数系统性推导与分类结果（完整版）

零、前言

热力学中共有8个描述系统的核心状态量：压强 $p$、体积 $V$、温度 $T$、熵 $S$、内能 $U$、焓 $H$、自由能 $F$（亥姆霍兹自由能，部分教材记作 $A$）和自由焓 $G$（吉布斯自由能）。
从这8个量中任选3个，分别作为被求导量、求导变量和恒定条件，可构成 $\left(\frac{\partial a}{\partial b}\right)_c$ 形式的偏微分，共计 $8 \times 7 \times 6 = 336$ 种情况。
本文系统给出全部336个偏导数的推导方法与完整分类结果，所有偏微分均可由简单状态量、物态方程系数（体胀系数 $\alpha$、压缩系数 $\kappa$、压强系数 $\beta$）和热容系数（$C_p$、$C_v$、$\gamma$）表示。

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一、前置准备与推导方法

1.1 基本定义与系数关系

体胀系数：
$$\alpha = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_p$$

等温压缩系数：
$$\kappa = -\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_T$$

压强系数：
$$\beta = \frac{1}{p}\left(\frac{\partial p}{\partial T}\right)_V$$

三者关系：
$$\alpha = \beta \kappa p$$

热容关系：
$$C_p - C_v = \frac{TV\alpha^2}{\kappa} = T\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_p$$

热容比：
$$\gamma = \frac{C_p}{C_v}$$

1.2 偏微分恒等式

倒易关系：
$$\left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial x}\right)_z = 1$$

循环关系：
$$\left(\frac{\partial x}{\partial y}\right)_z\left(\frac{\partial y}{\partial z}\right)_x\left(\frac{\partial z}{\partial x}\right)_y = -1$$

复合关系1（同条件链式）：
$$\left(\frac{\partial x}{\partial y}\right)_w\left(\frac{\partial y}{\partial z}\right)_w = \left(\frac{\partial x}{\partial z}\right)_w$$

复合关系2（变换约束条件）：
$$\left(\frac{\partial x}{\partial y}\right)_w = \left(\frac{\partial x}{\partial y}\right)_z + \left(\frac{\partial x}{\partial z}\right)_y\left(\frac{\partial z}{\partial y}\right)_w$$

1.3 热力学基本方程与麦克斯韦关系

热力学基本微分方程（粒子数不变）：

$$\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V$$
$$\mathrm{d}H = T\mathrm{d}S + V\mathrm{d}p$$
$$\mathrm{d}F = -S\mathrm{d}T - p\mathrm{d}V$$
$$\mathrm{d}G = -S\mathrm{d}T + V\mathrm{d}p$$

由此推出基本偏导关系：

$$\left(\frac{\partial G}{\partial T}\right)_p = \left(\frac{\partial F}{\partial T}\right)_V = -S$$
$$\left(\frac{\partial G}{\partial p}\right)_T = \left(\frac{\partial H}{\partial p}\right)_S = V$$
$$\left(\frac{\partial H}{\partial S}\right)_p = \left(\frac{\partial U}{\partial S}\right)_V = T$$
$$\left(\frac{\partial U}{\partial V}\right)_S = \left(\frac{\partial F}{\partial V}\right)_T = -p$$

麦克斯韦关系：

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

1.4 雅可比行列式算法（程序化求解）

条件偏导数可写成雅可比行列式之比：

$$\left(\frac{\partial a}{\partial b}\right)_c = \frac{\partial(a,c)}{\partial(b,c)} = \frac{\frac{\partial(a,c)}{\partial(p,T)}}{\frac{\partial(b,c)}{\partial(p,T)}} = \frac{\left(\frac{\partial a}{\partial p}\right)_T\left(\frac{\partial c}{\partial T}\right)_p - \left(\frac{\partial c}{\partial p}\right)_T\left(\frac{\partial a}{\partial T}\right)_p}{\left(\frac{\partial b}{\partial p}\right)_T\left(\frac{\partial c}{\partial T}\right)_p - \left(\frac{\partial c}{\partial p}\right)_T\left(\frac{\partial b}{\partial T}\right)_p}$$

因此只需计算8个热力学量对 $(p,T)$ 的16个基础偏导数，即可推出全部336个偏导数。

1.5 16个基础偏导数（对 $p,T$ 求导）

$$\left(\frac{\partial p}{\partial p}\right)_T = 1,\quad \left(\frac{\partial p}{\partial T}\right)_p = 0$$
$$\left(\frac{\partial V}{\partial p}\right)_T = -\kappa V,\quad \left(\frac{\partial V}{\partial T}\right)_p = \alpha V$$
$$\left(\frac{\partial T}{\partial p}\right)_T = 0,\quad \left(\frac{\partial T}{\partial T}\right)_p = 1$$
$$\left(\frac{\partial S}{\partial p}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_p = -\alpha V,\quad \left(\frac{\partial S}{\partial T}\right)_p = \frac{C_p}{T}$$
$$\left(\frac{\partial U}{\partial p}\right)_T = T\left(\frac{\partial S}{\partial p}\right)_T - p\left(\frac{\partial V}{\partial p}\right)_T = \kappa pV - \alpha TV$$
$$\left(\frac{\partial U}{\partial T}\right)_p = \left(\frac{\partial H}{\partial T}\right)_p - p\left(\frac{\partial V}{\partial T}\right)_p = C_p - \alpha pV$$
$$\left(\frac{\partial H}{\partial p}\right)_T = T\left(\frac{\partial S}{\partial p}\right)_T + V\left(\frac{\partial p}{\partial p}\right)_T = V - \alpha TV$$
$$\left(\frac{\partial H}{\partial T}\right)_p = C_p$$
$$\left(\frac{\partial F}{\partial p}\right)_T = -S\left(\frac{\partial T}{\partial p}\right)_T - p\left(\frac{\partial V}{\partial p}\right)_T = \kappa pV$$
$$\left(\frac{\partial F}{\partial T}\right)_p = -S\left(\frac{\partial T}{\partial T}\right)_p - p\left(\frac{\partial V}{\partial T}\right)_p = -\alpha pV - S$$
$$\left(\frac{\partial G}{\partial p}\right)_T = V,\quad \left(\frac{\partial G}{\partial T}\right)_p = -S$$

1.6 纸笔推导的化简策略（30个基础一阶式）

将 $U,H,S,F,G$ 记为大写 $X,Y$（热力学势），$p,V,T$ 记为小写 $a,b,c$（可观测量）。所有偏导可归约为最简一阶形式 $\left(\frac{\partial X}{\partial y}\right)_z$，其中 $X$ 是热力学势，$y,z \in {p,V,T}$，共 $5\times 6=30$ 个基础式子。

化简规则：

1. $\left(\frac{\partial b}{\partial a}\right)_Z$（约束为热力学势）：
   $$\left(\frac{\partial b}{\partial a}\right)_Z = -\frac{\left(\frac{\partial Z}{\partial a}\right)_b}{\left(\frac{\partial Z}{\partial b}\right)_a}$$

2. $\left(\frac{\partial X}{\partial Y}\right)_a$（求导量、约束都是势）：
   $$\left(\frac{\partial X}{\partial Y}\right)_a = \frac{\left(\frac{\partial X}{\partial b}\right)_a}{\left(\frac{\partial Y}{\partial b}\right)_a}$$

3. $\left(\frac{\partial X}{\partial a}\right)_Y$（约束为势）：用复合关系2展开降阶。

4. $\left(\frac{\partial X}{\partial Y}\right)_Z$（三者全是势）：用复合关系2展开降阶。

1.7 30个基础一阶偏导数的完整推导

1.7.1 $C_p$ 与 $C_v$ 的关系

$$C_p - C_v = T\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_p$$

证明：

$$C_p - C_v = \left(\frac{\partial H}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_V = \left[\frac{\partial(U+pV)}{\partial T}\right]_p - \left(\frac{\partial U}{\partial T}\right)_V = \left(\frac{\partial U}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p - \left(\frac{\partial U}{\partial T}\right)_V$$

由复合关系2：

$$\left(\frac{\partial U}{\partial T}\right)_p = \left(\frac{\partial U}{\partial T}\right)_V + \left(\frac{\partial U}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_p$$

所以：

$$C_p - C_v = \left(\frac{\partial U}{\partial V}\right)_T\left(\frac{\partial V}{\partial T}\right)_p + p\left(\frac{\partial V}{\partial T}\right)_p = \left[p + \left(\frac{\partial U}{\partial V}\right)_T\right]\left(\frac{\partial V}{\partial T}\right)_p$$

由 $\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V$ 得：

$$\left(\frac{\partial U}{\partial S}\right)_V = T,\quad \left(\frac{\partial U}{\partial V}\right)_S = -p$$

由复合关系2：

$$\left(\frac{\partial U}{\partial V}\right)_T = \left(\frac{\partial U}{\partial V}\right)_S + \left(\frac{\partial U}{\partial S}\right)_V\left(\frac{\partial S}{\partial V}\right)_T = T\left(\frac{\partial S}{\partial V}\right)_T - p = T\left(\frac{\partial p}{\partial T}\right)_V - p$$

（麦克斯韦关系）

所以：

$$C_p - C_v = \left[p + T\left(\frac{\partial p}{\partial T}\right)_V - p\right]\left(\frac{\partial V}{\partial T}\right)_p = T\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial T}\right)_p$$

证毕。

1.7.2 $U$ 的6个偏微分关系

1. $\left(\frac{\partial U}{\partial T}\right)_V = C_v$

2. $\left(\frac{\partial U}{\partial p}\right)_V = C_v\left(\frac{\partial T}{\partial p}\right)_V$

3. $\left(\frac{\partial U}{\partial T}\right)_p = C_p - p\left(\frac{\partial V}{\partial T}\right)_p$

4. $\left(\frac{\partial U}{\partial V}\right)_p = C_p\left(\frac{\partial T}{\partial V}\right)_p - p$

5. $\left(\frac{\partial U}{\partial V}\right)_T = (C_p - C_v)\left(\frac{\partial T}{\partial V}\right)_p - p$

6. $\left(\frac{\partial U}{\partial p}\right)_T = -(C_p - C_v)\left(\frac{\partial T}{\partial p}\right)_V - p\left(\frac{\partial V}{\partial p}\right)_T$

1.7.3 $H$ 的6个偏微分关系

（$H$ 与 $U$ 基本对偶，将 $C_v$ 换成 $C_p$，$V$ 和 $p$ 调换，注意负号。）

1. $\left(\frac{\partial H}{\partial T}\right)_p = C_p$

2. $\left(\frac{\partial H}{\partial V}\right)_p = C_p\left(\frac{\partial T}{\partial V}\right)_p$

3. $\left(\frac{\partial H}{\partial T}\right)_V = C_v + V\left(\frac{\partial p}{\partial T}\right)_V$

4. $\left(\frac{\partial H}{\partial p}\right)_V = C_v\left(\frac{\partial T}{\partial p}\right)_V + V$

5. $\left(\frac{\partial H}{\partial p}\right)_T = V - (C_p - C_v)\left(\frac{\partial T}{\partial p}\right)_V$

6. $\left(\frac{\partial H}{\partial V}\right)_T = V\left(\frac{\partial p}{\partial V}\right)_T + (C_p - C_v)\left(\frac{\partial T}{\partial V}\right)_p$

1.7.4 $S$ 的6个偏微分关系

（由麦克斯韦关系直接得出前两个。）

1. $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial p}{\partial T}\right)_V$

2. $\left(\frac{\partial S}{\partial p}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_p$

3. $\left(\frac{\partial S}{\partial T}\right)_V = \frac{C_v}{T}$

4. $\left(\frac{\partial S}{\partial p}\right)_V = \frac{C_v}{T}\left(\frac{\partial T}{\partial p}\right)_V$

5. $\left(\frac{\partial S}{\partial T}\right)_p = \frac{C_p}{T}$

6. $\left(\frac{\partial S}{\partial V}\right)_p = \frac{C_p}{T}\left(\frac{\partial T}{\partial V}\right)_p$

1.7.5 $F$（亥姆霍兹自由能）的6个偏微分关系

1. $\left(\frac{\partial F}{\partial T}\right)_V = -S$

2. $\left(\frac{\partial F}{\partial V}\right)_T = -p$

3. $\left(\frac{\partial F}{\partial p}\right)_V = -S\left(\frac{\partial T}{\partial p}\right)_V$

4. $\left(\frac{\partial F}{\partial p}\right)_T = -p\left(\frac{\partial V}{\partial p}\right)_T$

5. $\left(\frac{\partial F}{\partial T}\right)_p = -S - p\left(\frac{\partial V}{\partial T}\right)_p$

6. $\left(\frac{\partial F}{\partial V}\right)_p = -S\left(\frac{\partial T}{\partial V}\right)_p - p$

1.7.6 $G$（吉布斯自由能）的6个偏微分关系

1. $\left(\frac{\partial G}{\partial T}\right)_p = -S$

2. $\left(\frac{\partial G}{\partial p}\right)_T = V$

3. $\left(\frac{\partial G}{\partial V}\right)_p = -S\left(\frac{\partial T}{\partial V}\right)_p$

4. $\left(\frac{\partial G}{\partial V}\right)_T = V\left(\frac{\partial p}{\partial V}\right)_T$

5. $\left(\frac{\partial G}{\partial T}\right)_V = -S + V\left(\frac{\partial p}{\partial T}\right)_V$

6. $\left(\frac{\partial G}{\partial p}\right)_V = -S\left(\frac{\partial T}{\partial p}\right)_V + V$

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二、全部336个偏导数分类结果（按被求导量分类）

2.1 $p$ 开头（42个）

2.1.1 对 $V$ 求偏导

$$\left(\frac{\partial p}{\partial V}\right)_T = -\frac{1}{\kappa V}$$
$$\left(\frac{\partial p}{\partial V}\right)_S = -\frac{\gamma}{\kappa V}$$
$$\left(\frac{\partial p}{\partial V}\right)_U = \frac{p(\gamma - \beta\gamma T - 1)}{\alpha T V}$$
$$\left(\frac{\partial p}{\partial V}\right)_H = \frac{\beta\gamma p T}{V(1-\gamma-\alpha T)}$$
$$\left(\frac{\partial p}{\partial V}\right)_F = -\frac{\alpha p V + S}{\kappa S V}$$
$$\left(\frac{\partial p}{\partial V}\right)_G = \frac{S}{\alpha V^2 - \kappa S V}$$

2.1.2 对 $T$ 求偏导

$$\left(\frac{\partial p}{\partial T}\right)_V = \beta p$$
$$\left(\frac{\partial p}{\partial T}\right)_S = \frac{\beta\gamma p}{\gamma - 1}$$
$$\left(\frac{\partial p}{\partial T}\right)_U = \beta\left(p - \frac{C_v}{\alpha V - \alpha\beta T V}\right)$$
$$\left(\frac{\partial p}{\partial T}\right)_H = \frac{C_p}{V(\alpha T - 1)}$$
$$\left(\frac{\partial p}{\partial T}\right)_F = \beta\left(p + \frac{S}{\alpha V}\right)$$
$$\left(\frac{\partial p}{\partial T}\right)_G = \frac{S}{V}$$

2.1.3 对 $S$ 求偏导

$$\left(\frac{\partial p}{\partial S}\right)_V = \frac{\beta p T}{C_v}$$
$$\left(\frac{\partial p}{\partial S}\right)_T = -\frac{1}{\alpha V}$$
$$\left(\frac{\partial p}{\partial S}\right)_U = \frac{\gamma - \beta\gamma T - 1}{\alpha V}$$
$$\left(\frac{\partial p}{\partial S}\right)_H = -\frac{T}{V}$$
$$\left(\frac{\partial p}{\partial S}\right)_F = -\frac{\alpha p T V + S T}{\alpha S T V - \kappa p V C_v}$$
$$\left(\frac{\partial p}{\partial S}\right)_G = -\frac{S T}{\alpha S T V - V C_p}$$

2.1.4 对 $U$ 求偏导

$$\left(\frac{\partial p}{\partial U}\right)_V = \frac{\beta p}{C_v}$$
$$\left(\frac{\partial p}{\partial U}\right)_T = \frac{1}{\kappa p V - \alpha T V}$$
$$\left(\frac{\partial p}{\partial U}\right)_S = \frac{\beta\gamma}{\alpha V}$$
$$\left(\frac{\partial p}{\partial U}\right)_H = \frac{\beta\gamma T}{V(\gamma + T(\alpha - \beta\gamma) - 1)}$$
$$\left(\frac{\partial p}{\partial U}\right)_F = \frac{\beta(\alpha p V + S)}{\alpha V(C_v + S - \beta S T)}$$
$$\left(\frac{\partial p}{\partial U}\right)_G = -\frac{S}{V(-C_p - \kappa p S + \alpha p V + \alpha S T)}$$

2.1.5 对 $H$ 求偏导

$$\left(\frac{\partial p}{\partial H}\right)_V = \frac{\beta p}{C_v + \beta p V}$$
$$\left(\frac{\partial p}{\partial H}\right)_T = \frac{1}{V - \alpha T V}$$
$$\left(\frac{\partial p}{\partial H}\right)_S = \frac{1}{V}$$
$$\left(\frac{\partial p}{\partial H}\right)_U = \frac{\gamma - \beta\gamma T - 1}{V(\gamma + T(\alpha - \beta\gamma) - 1)}$$
$$\left(\frac{\partial p}{\partial H}\right)_F = \frac{\alpha p V + S}{V(\kappa p C_v + \alpha p V + S - \alpha S T)}$$
$$\left(\frac{\partial p}{\partial H}\right)_G = \frac{S}{V(C_p + S - \alpha S T)}$$

2.1.6 对 $F$ 求偏导

$$\left(\frac{\partial p}{\partial F}\right)_V = -\frac{\beta p}{S}$$
$$\left(\frac{\partial p}{\partial F}\right)_T = \frac{\beta}{\alpha V}$$
$$\left(\frac{\partial p}{\partial F}\right)_S = \frac{\beta\gamma p}{\alpha p V - \gamma S + S}$$
$$\left(\frac{\partial p}{\partial F}\right)_U = \frac{\beta(\gamma p(\beta T - 1) + p)}{\alpha\beta p T V - (\gamma - 1)S(\beta T - 1)}$$
$$\left(\frac{\partial p}{\partial F}\right)_H = \frac{\alpha\beta\gamma p T}{\alpha p V(\gamma + \alpha T - 1) - (\gamma - 1)S(\alpha T - 1)}$$
$$\left(\frac{\partial p}{\partial F}\right)_G = -\frac{S}{V(\kappa(-p)S + \alpha p V + S)}$$

2.1.7 对 $G$ 求偏导

$$\left(\frac{\partial p}{\partial G}\right)_V = \frac{\alpha}{\alpha V - \kappa S}$$
$$\left(\frac{\partial p}{\partial G}\right)_T = \frac{1}{V}$$
$$\left(\frac{\partial p}{\partial G}\right)_S = \frac{\beta\gamma p}{\beta\gamma p V - \gamma S + S}$$
$$\left(\frac{\partial p}{\partial G}\right)_U = \frac{\alpha p V - C_p}{V(-C_p - \kappa p S + \alpha p V + \alpha S T)}$$
$$\left(\frac{\partial p}{\partial G}\right)_H = \frac{C_p}{V(C_p + S - \alpha S T)}$$
$$\left(\frac{\partial p}{\partial G}\right)_F = \frac{\alpha p V + S}{V(\kappa(-p)S + \alpha p V + S)}$$

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2.2 $V$ 开头（42个）

2.2.1 对 $p$ 求偏导

$$\left(\frac{\partial V}{\partial p}\right)_T = -\kappa V$$
$$\left(\frac{\partial V}{\partial p}\right)_S = -\frac{\kappa V}{\gamma}$$
$$\left(\frac{\partial V}{\partial p}\right)_U = -\frac{\alpha T V}{p(\gamma(\beta T - 1) + 1)}$$
$$\left(\frac{\partial V}{\partial p}\right)_H = -\frac{\kappa V(\gamma + \alpha T - 1)}{\alpha\gamma T}$$
$$\left(\frac{\partial V}{\partial p}\right)_F = -\frac{\kappa S V}{\alpha p V + S}$$
$$\left(\frac{\partial V}{\partial p}\right)_G = V\left(\frac{\alpha V}{S} - \kappa\right)$$

2.2.2 对 $T$ 求偏导

$$\left(\frac{\partial V}{\partial T}\right)_p = \alpha V$$
$$\left(\frac{\partial V}{\partial T}\right)_S = \frac{\alpha V}{1 - \gamma}$$
$$\left(\frac{\partial V}{\partial T}\right)_U = \frac{C_v}{p(1 - \beta T)}$$
$$\left(\frac{\partial V}{\partial T}\right)_H = \frac{\kappa C_p}{1 - \alpha T} + \alpha V$$
$$\left(\frac{\partial V}{\partial T}\right)_F = -\frac{S}{p}$$
$$\left(\frac{\partial V}{\partial T}\right)_G = \alpha V - \kappa S$$

2.2.3 对 $S$ 求偏导

$$\left(\frac{\partial V}{\partial S}\right)_p = \frac{\alpha T V}{C_p}$$
$$\left(\frac{\partial V}{\partial S}\right)_T = \frac{\kappa}{\alpha}$$
$$\left(\frac{\partial V}{\partial S}\right)_U = \frac{T}{p}$$
$$\left(\frac{\partial V}{\partial S}\right)_H = \frac{\kappa(\gamma + \alpha T - 1)}{\alpha\gamma}$$
$$\left(\frac{\partial V}{\partial S}\right)_F = \frac{S T}{\beta p S T - p C_v}$$
$$\left(\frac{\partial V}{\partial S}\right)_G = \frac{T(\kappa S - \alpha V)}{\alpha S T - C_p}$$

2.2.4 对 $U$ 求偏导

$$\left(\frac{\partial V}{\partial U}\right)_p = \frac{\gamma - 1}{p(\gamma(\beta T - 1) + 1)}$$
$$\left(\frac{\partial V}{\partial U}\right)_T = -\frac{1}{p(1 - \beta T)}$$
$$\left(\frac{\partial V}{\partial U}\right)_S = -\frac{1}{p}$$
$$\left(\frac{\partial V}{\partial U}\right)_H = -\frac{\gamma + \alpha T - 1}{p(\gamma + T(\alpha - \beta\gamma) - 1)}$$
$$\left(\frac{\partial V}{\partial U}\right)_F = -\frac{S}{p(C_v + S - \beta S T)}$$
$$\left(\frac{\partial V}{\partial U}\right)_G = \frac{\alpha V - \kappa S}{C_p + \kappa p S - \alpha p V - \alpha S T}$$

2.2.5 对 $H$ 求偏导

$$\left(\frac{\partial V}{\partial H}\right)_p = \frac{\alpha V}{C_p}$$
$$\left(\frac{\partial V}{\partial H}\right)_T = \frac{\kappa}{\alpha T - 1}$$
$$\left(\frac{\partial V}{\partial H}\right)_S = -\frac{\kappa}{\gamma}$$
$$\left(\frac{\partial V}{\partial H}\right)_U = \frac{\alpha T}{p(\gamma + T(\alpha - \beta\gamma) - 1)}$$
$$\left(\frac{\partial V}{\partial H}\right)_F = -\frac{\kappa S}{\kappa p C_v + \alpha p V + S - \alpha S T}$$
$$\left(\frac{\partial V}{\partial H}\right)_G = \frac{\alpha V - \kappa S}{C_p + S - \alpha S T}$$

2.2.6 对 $F$ 求偏导

$$\left(\frac{\partial V}{\partial F}\right)_p = -\frac{\alpha V}{\alpha p V + S}$$
$$\left(\frac{\partial V}{\partial F}\right)_T = -\frac{1}{p}$$
$$\left(\frac{\partial V}{\partial F}\right)_S = -\frac{\alpha V}{\alpha p V - \gamma S + S}$$
$$\left(\frac{\partial V}{\partial F}\right)_U = -\frac{C_v}{p(C_v + S - \beta S T)}$$
$$\left(\frac{\partial V}{\partial F}\right)_H = -\frac{\kappa C_v + \alpha V}{\kappa p C_v + \alpha p V + S - \alpha S T}$$
$$\left(\frac{\partial V}{\partial F}\right)_G = \frac{\kappa S - \alpha V}{\kappa(-p)S + \alpha p V + S}$$

2.2.7 对 $G$ 求偏导

$$\left(\frac{\partial V}{\partial G}\right)_p = -\frac{\alpha V}{S}$$
$$\left(\frac{\partial V}{\partial G}\right)_T = -\kappa$$
$$\left(\frac{\partial V}{\partial G}\right)_S = -\frac{\alpha V}{\beta\gamma p V - \gamma S + S}$$
$$\left(\frac{\partial V}{\partial G}\right)_U = -\frac{\kappa C_v}{C_v + \kappa p(\beta T - 1)(\beta p V - S)}$$
$$\left(\frac{\partial V}{\partial G}\right)_H = -\frac{\kappa C_p + \alpha V(1 - \alpha T)}{C_p + S - \alpha S T}$$
$$\left(\frac{\partial V}{\partial G}\right)_F = -\frac{\kappa S}{\kappa(-p)S + \alpha p V + S}$$

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2.3 $T$ 开头（42个）

2.3.1 对 $p$ 求偏导

$$\left(\frac{\partial T}{\partial p}\right)_V = \frac{\kappa}{\alpha}$$
$$\left(\frac{\partial T}{\partial p}\right)_S = \frac{\alpha T V}{C_p}$$
$$\left(\frac{\partial T}{\partial p}\right)_U = \frac{V(\alpha T - \kappa p)}{C_p - \alpha p V}$$
$$\left(\frac{\partial T}{\partial p}\right)_H = \frac{V(\alpha T - 1)}{C_p}$$
$$\left(\frac{\partial T}{\partial p}\right)_F = \frac{\kappa p V}{\alpha p V + S}$$
$$\left(\frac{\partial T}{\partial p}\right)_G = \frac{V}{S}$$

2.3.2 对 $V$ 求偏导

$$\left(\frac{\partial T}{\partial V}\right)_p = \frac{1}{\alpha V}$$
$$\left(\frac{\partial T}{\partial V}\right)_S = -\frac{\beta p T}{C_v}$$
$$\left(\frac{\partial T}{\partial V}\right)_U = \frac{p(1 - \beta T)}{C_v}$$
$$\left(\frac{\partial T}{\partial V}\right)_H = \frac{1 - \alpha T}{\kappa C_v + \alpha V}$$
$$\left(\frac{\partial T}{\partial V}\right)_F = -\frac{p}{S}$$
$$\left(\frac{\partial T}{\partial V}\right)_G = \frac{1}{\alpha V - \kappa S}$$

2.3.3 对 $S$ 求偏导

$$\left(\frac{\partial T}{\partial S}\right)_p = \frac{T}{C_p}$$
$$\left(\frac{\partial T}{\partial S}\right)_V = \frac{T}{C_v}$$
$$\left(\frac{\partial T}{\partial S}\right)_U = \frac{T(1 - \beta T)}{C_v}$$
$$\left(\frac{\partial T}{\partial S}\right)_H = \frac{T(1 - \alpha T)}{C_p}$$
$$\left(\frac{\partial T}{\partial S}\right)_F = -\frac{T}{\beta S T - C_v}$$
$$\left(\frac{\partial T}{\partial S}\right)_G = -\frac{T}{\alpha S T - C_p}$$

2.3.4 对 $U$ 求偏导

$$\left(\frac{\partial T}{\partial U}\right)_p = \frac{1}{C_p - \alpha p V}$$
$$\left(\frac{\partial T}{\partial U}\right)_V = \frac{1}{C_v}$$
$$\left(\frac{\partial T}{\partial U}\right)_S = \frac{\beta T}{C_v}$$
$$\left(\frac{\partial T}{\partial U}\right)_H = \frac{1 - \alpha T}{C_p(1 - \kappa p) + \alpha p V(\alpha T - 1)}$$
$$\left(\frac{\partial T}{\partial U}\right)_F = \frac{1}{C_v + S - \beta S T}$$
$$\left(\frac{\partial T}{\partial U}\right)_G = \frac{1}{C_p + \kappa p S - \alpha p V - \alpha S T}$$

2.3.5 对 $H$ 求偏导

$$\left(\frac{\partial T}{\partial H}\right)_p = \frac{1}{C_p}$$
$$\left(\frac{\partial T}{\partial H}\right)_V = \frac{1}{C_v + \beta p V}$$
$$\left(\frac{\partial T}{\partial H}\right)_S = \frac{\alpha T}{C_p}$$
$$\left(\frac{\partial T}{\partial H}\right)_U = \frac{\alpha(\beta T - 1)}{(\beta - \alpha)C_p + \alpha\beta p V(\alpha T - 1)}$$
$$\left(\frac{\partial T}{\partial H}\right)_F = \frac{\kappa p}{\kappa p C_v + \alpha p V + S - \alpha S T}$$
$$\left(\frac{\partial T}{\partial H}\right)_G = \frac{1}{C_p + S - \alpha S T}$$

2.3.6 对 $F$ 求偏导

$$\left(\frac{\partial T}{\partial F}\right)_p = -\frac{1}{\alpha p V + S}$$
$$\left(\frac{\partial T}{\partial F}\right)_V = -\frac{1}{S}$$
$$\left(\frac{\partial T}{\partial F}\right)_S = \frac{\gamma - 1}{\alpha p V - \gamma S + S}$$
$$\left(\frac{\partial T}{\partial F}\right)_U = -\frac{1 - \beta T}{C_v + S - \beta S T}$$
$$\left(\frac{\partial T}{\partial F}\right)_H = \frac{\alpha T - 1}{\kappa p C_v + \alpha p V + S - \alpha S T}$$
$$\left(\frac{\partial T}{\partial F}\right)_G = \frac{1}{S(\kappa p - 1) - \alpha p V}$$

2.3.7 对 $G$ 求偏导

$$\left(\frac{\partial T}{\partial G}\right)_p = -\frac{1}{S}$$
$$\left(\frac{\partial T}{\partial G}\right)_V = \frac{1}{\beta p V - S}$$
$$\left(\frac{\partial T}{\partial G}\right)_S = -\frac{\alpha T}{\alpha S T - C_p}$$
$$\left(\frac{\partial T}{\partial G}\right)_U = \frac{\alpha T - \kappa p}{C_p + \kappa p S - \alpha p V - \alpha S T}$$
$$\left(\frac{\partial T}{\partial G}\right)_H = -\frac{1 - \alpha T}{C_p + S - \alpha S T}$$
$$\left(\frac{\partial T}{\partial G}\right)_F = \frac{\kappa p}{\kappa(-p)S + \alpha p V + S}$$

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2.4 $S$ 开头（42个）

2.4.1 对 $p$ 求偏导

$$\left(\frac{\partial S}{\partial p}\right)_V = \frac{\alpha V}{\gamma - 1}$$
$$\left(\frac{\partial S}{\partial p}\right)_T = -\alpha V$$
$$\left(\frac{\partial S}{\partial p}\right)_U = -\frac{\alpha V}{\gamma(\beta T - 1) + 1}$$
$$\left(\frac{\partial S}{\partial p}\right)_H = -\frac{V}{T}$$
$$\left(\frac{\partial S}{\partial p}\right)_F = \alpha V\left(\frac{C_p}{\alpha\beta p T V + \beta S T} - 1\right)$$
$$\left(\frac{\partial S}{\partial p}\right)_G = \frac{V C_p}{S T} - \alpha V$$

2.4.2 对 $V$ 求偏导

$$\left(\frac{\partial S}{\partial V}\right)_p = \frac{\beta\gamma p}{\gamma - 1}$$
$$\left(\frac{\partial S}{\partial V}\right)_T = \beta p$$
$$\left(\frac{\partial S}{\partial V}\right)_U = \frac{p}{T}$$
$$\left(\frac{\partial S}{\partial V}\right)_H = \frac{\beta\gamma p}{\gamma + \alpha T - 1}$$
$$\left(\frac{\partial S}{\partial V}\right)_F = p\left(\beta - \frac{C_v}{S T}\right)$$
$$\left(\frac{\partial S}{\partial V}\right)_G = \frac{\alpha}{\kappa} + \frac{C_v}{\alpha T V - \kappa S T}$$

2.4.3 对 $T$ 求偏导

$$\left(\frac{\partial S}{\partial T}\right)_p = \frac{C_p}{T}$$
$$\left(\frac{\partial S}{\partial T}\right)_V = \frac{C_v}{T}$$
$$\left(\frac{\partial S}{\partial T}\right)_U = \frac{C_v}{T(1 - \beta T)}$$
$$\left(\frac{\partial S}{\partial T}\right)_H = \frac{C_p}{T(1 - \alpha T)}$$
$$\left(\frac{\partial S}{\partial T}\right)_F = \frac{C_v}{T} - \beta S$$
$$\left(\frac{\partial S}{\partial T}\right)_G = \frac{C_p}{T} - \alpha S$$

2.4.4 对 $U$ 求偏导

$$\left(\frac{\partial S}{\partial U}\right)_p = \frac{\beta\gamma}{\gamma(\beta T - 1) + 1}$$
$$\left(\frac{\partial S}{\partial U}\right)_V = \frac{1}{T}$$
$$\left(\frac{\partial S}{\partial U}\right)_T = \frac{\beta}{\beta T - 1}$$
$$\left(\frac{\partial S}{\partial U}\right)_H = \frac{\beta\gamma}{1 - \gamma - \alpha T + \beta\gamma T}$$
$$\left(\frac{\partial S}{\partial U}\right)_F = -\frac{\beta S T - C_v}{T C_v - \beta S T^2 + S T}$$
$$\left(\frac{\partial S}{\partial U}\right)_G = \frac{\alpha S T - C_p}{T(-C_p - \kappa p S + \alpha p V + \alpha S T)}$$

2.4.5 对 $H$ 求偏导

$$\left(\frac{\partial S}{\partial H}\right)_p = \frac{1}{T}$$
$$\left(\frac{\partial S}{\partial H}\right)_V = \frac{\alpha}{\gamma + \alpha T - 1}$$
$$\left(\frac{\partial S}{\partial H}\right)_T = \frac{\alpha}{\alpha T - 1}$$
$$\left(\frac{\partial S}{\partial H}\right)_U = \frac{\alpha}{\gamma + T(\alpha - \beta\gamma) - 1}$$
$$\left(\frac{\partial S}{\partial H}\right)_F = \frac{\alpha(\alpha p V - \gamma S + S)}{\alpha p V(\gamma + \alpha T - 1) - (\gamma - 1)S(\alpha T - 1)}$$
$$\left(\frac{\partial S}{\partial H}\right)_G = -\frac{\alpha S T - C_p}{T C_p - \alpha S T^2 + S T}$$

2.4.6 对 $F$ 求偏导

$$\left(\frac{\partial S}{\partial F}\right)_p = -\frac{C_p}{\alpha p T V + S T}$$
$$\left(\frac{\partial S}{\partial F}\right)_V = -\frac{C_v}{S T}$$
$$\left(\frac{\partial S}{\partial F}\right)_T = -\beta$$
$$\left(\frac{\partial S}{\partial F}\right)_U = -\frac{C_v}{T C_v - \beta S T^2 + S T}$$
$$\left(\frac{\partial S}{\partial F}\right)_H = \frac{C_p}{T((\alpha T - 1)(\alpha p V + S) - \kappa p C_p)}$$
$$\left(\frac{\partial S}{\partial F}\right)_G = \frac{\alpha S T - C_p}{-\kappa p S T + \alpha p T V + S T}$$

2.4.7 对 $G$ 求偏导

$$\left(\frac{\partial S}{\partial G}\right)_p = -\frac{C_p}{S T}$$
$$\left(\frac{\partial S}{\partial G}\right)_V = -\frac{C_v}{S T - \beta p T V}$$
$$\left(\frac{\partial S}{\partial G}\right)_T = -\alpha$$
$$\left(\frac{\partial S}{\partial G}\right)_U = \frac{\alpha C_v}{T(\alpha(\beta T - 1)(S - \beta p V) - \beta C_v)}$$
$$\left(\frac{\partial S}{\partial G}\right)_H = -\frac{C_p}{T C_p - \alpha S T^2 + S T}$$
$$\left(\frac{\partial S}{\partial G}\right)_F = \frac{\alpha(\alpha p V - \gamma S + S)}{(\gamma - 1)(\kappa(-p)S + \alpha p V + S)}$$

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2.5 $U$ 开头（42个）

2.5.1 对 $p$ 求偏导

$$\left(\frac{\partial U}{\partial p}\right)_V = \frac{\kappa C_v}{\alpha}$$
$$\left(\frac{\partial U}{\partial p}\right)_T = \alpha V\left(\frac{1}{\beta} - T\right)$$
$$\left(\frac{\partial U}{\partial p}\right)_S = \frac{\kappa p V}{\gamma}$$
$$\left(\frac{\partial U}{\partial p}\right)_H = \frac{V(\gamma + T(\alpha - \beta\gamma) - 1)}{\beta\gamma T}$$
$$\left(\frac{\partial U}{\partial p}\right)_F = \frac{\alpha V(C_v + S - \beta S T)}{\beta(\alpha p V + S)}$$
$$\left(\frac{\partial U}{\partial p}\right)_G = \frac{V(C_p + \kappa p S - \alpha p V - \alpha S T)}{S}$$

2.5.2 对 $V$ 求偏导

$$\left(\frac{\partial U}{\partial V}\right)_p = \frac{C_p}{\alpha V} - p$$
$$\left(\frac{\partial U}{\partial V}\right)_T = p(\beta T - 1)$$
$$\left(\frac{\partial U}{\partial V}\right)_S = -p$$
$$\left(\frac{\partial U}{\partial V}\right)_H = -\frac{p(\gamma + T(\alpha - \beta\gamma) - 1)}{\gamma + \alpha T - 1}$$
$$\left(\frac{\partial U}{\partial V}\right)_F = p\left(-\frac{C_v}{S} + \beta T - 1\right)$$
$$\left(\frac{\partial U}{\partial V}\right)_G = p(\beta T - 1) - \frac{C_v}{\kappa S - \beta\kappa p V}$$

2.5.3 对 $T$ 求偏导

$$\left(\frac{\partial U}{\partial T}\right)_p = C_p - \alpha p V$$
$$\left(\frac{\partial U}{\partial T}\right)_V = C_v$$
$$\left(\frac{\partial U}{\partial T}\right)_S = \frac{\alpha p V}{\gamma - 1}$$
$$\left(\frac{\partial U}{\partial T}\right)_H = \frac{\alpha p V(\gamma + T(\alpha - \beta\gamma) - 1)}{(\gamma - 1)(\alpha T - 1)}$$
$$\left(\frac{\partial U}{\partial T}\right)_F = C_v + S - \beta S T$$
$$\left(\frac{\partial U}{\partial T}\right)_G = C_p + \kappa p S - \alpha p V - \alpha S T$$

2.5.4 对 $S$ 求偏导

$$\left(\frac{\partial U}{\partial S}\right)_p = T - \frac{\alpha p T V}{C_p}$$
$$\left(\frac{\partial U}{\partial S}\right)_V = T$$
$$\left(\frac{\partial U}{\partial S}\right)_T = T - \frac{1}{\beta}$$
$$\left(\frac{\partial U}{\partial S}\right)_H = \frac{1 - \gamma - \alpha T + \beta\gamma T}{\beta\gamma}$$
$$\left(\frac{\partial U}{\partial S}\right)_F = \frac{T(S(\beta T - 1) - C_v)}{\beta S T - C_v}$$
$$\left(\frac{\partial U}{\partial S}\right)_G = \frac{(\gamma - 1)\kappa p(\kappa S - \alpha V)}{\alpha(\alpha\gamma V - (\gamma - 1)\kappa S)} + T$$

2.5.5 对 $H$ 求偏导

$$\left(\frac{\partial U}{\partial H}\right)_p = 1 - \frac{\alpha p V}{C_p}$$
$$\left(\frac{\partial U}{\partial H}\right)_V = \frac{\alpha T}{\gamma + \alpha T - 1}$$
$$\left(\frac{\partial U}{\partial H}\right)_T = \frac{\alpha T - \kappa p}{\alpha T - 1}$$
$$\left(\frac{\partial U}{\partial H}\right)_S = \frac{\kappa p}{\gamma}$$
$$\left(\frac{\partial U}{\partial H}\right)_F = \frac{\alpha(C_v + S - \beta S T)}{\alpha C_v + \beta(\alpha p V + S - \alpha S T)}$$
$$\left(\frac{\partial U}{\partial H}\right)_G = -\frac{-C_p - \kappa p S + \alpha p V + \alpha S T}{C_p + S - \alpha S T}$$

2.5.6 对 $F$ 求偏导

$$\left(\frac{\partial U}{\partial F}\right)_p = \frac{\alpha p V - C_p}{\alpha p V + S}$$
$$\left(\frac{\partial U}{\partial F}\right)_V = -\frac{C_v}{S}$$
$$\left(\frac{\partial U}{\partial F}\right)_T = 1 - \beta T$$
$$\left(\frac{\partial U}{\partial F}\right)_S = \frac{\alpha p V}{\alpha p V - \gamma S + S}$$
$$\left(\frac{\partial U}{\partial F}\right)_H = \frac{\alpha p V(\gamma + T(\alpha - \beta\gamma) - 1)}{\alpha p V(\gamma + \alpha T - 1) - (\gamma - 1)S(\alpha T - 1)}$$
$$\left(\frac{\partial U}{\partial F}\right)_G = \frac{-C_p - \kappa p S + \alpha p V + \alpha S T}{\kappa(-p)S + \alpha p V + S}$$

2.5.7 对 $G$ 求偏导

$$\left(\frac{\partial U}{\partial G}\right)_p = \frac{\alpha p V - C_p}{S}$$
$$\left(\frac{\partial U}{\partial G}\right)_V = \frac{C_v}{\beta p V - S}$$
$$\left(\frac{\partial U}{\partial G}\right)_T = \kappa p - \alpha T$$
$$\left(\frac{\partial U}{\partial G}\right)_S = \frac{\alpha p V}{\beta\gamma p V - \gamma S + S}$$
$$\left(\frac{\partial U}{\partial G}\right)_H = \frac{C_p(\kappa p - 1) + \alpha p V(1 - \alpha T)}{C_p + S - \alpha S T}$$
$$\left(\frac{\partial U}{\partial G}\right)_F = \frac{\alpha(C_v + S - \beta S T)}{\alpha\beta p V + S(\beta - \alpha)}$$

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2.6 $H$ 开头（42个）

2.6.1 对 $p$ 求偏导

$$\left(\frac{\partial H}{\partial p}\right)_V = \frac{\kappa C_v}{\alpha} + V$$
$$\left(\frac{\partial H}{\partial p}\right)_T = V - \alpha T V$$
$$\left(\frac{\partial H}{\partial p}\right)_S = V$$
$$\left(\frac{\partial H}{\partial p}\right)_U = -\frac{V(\gamma + T(\alpha - \beta\gamma) - 1)}{\gamma(\beta T - 1) + 1}$$
$$\left(\frac{\partial H}{\partial p}\right)_F = \frac{\kappa p V C_p}{\alpha p V + S} - \alpha T V + V$$
$$\left(\frac{\partial H}{\partial p}\right)_G = \frac{V C_p}{S} - \alpha T V + V$$

2.6.2 对 $V$ 求偏导

$$\left(\frac{\partial H}{\partial V}\right)_p = \frac{\beta\gamma p T}{\gamma - 1}$$
$$\left(\frac{\partial H}{\partial V}\right)_T = \frac{\alpha T - 1}{\kappa}$$
$$\left(\frac{\partial H}{\partial V}\right)_S = -\frac{\gamma}{\kappa}$$
$$\left(\frac{\partial H}{\partial V}\right)_U = \frac{p(\gamma + T(\alpha - \beta\gamma) - 1)}{\alpha T}$$
$$\left(\frac{\partial H}{\partial V}\right)_F = -\frac{\kappa p C_v + \alpha p V + S - \alpha S T}{\kappa S}$$
$$\left(\frac{\partial H}{\partial V}\right)_G = \frac{C_p + S - \alpha S T}{\alpha V - \kappa S}$$

2.6.3 对 $T$ 求偏导

$$\left(\frac{\partial H}{\partial T}\right)_p = C_p$$
$$\left(\frac{\partial H}{\partial T}\right)_V = C_v + \beta p V$$
$$\left(\frac{\partial H}{\partial T}\right)_S = \frac{\beta\gamma p V}{\gamma - 1}$$
$$\left(\frac{\partial H}{\partial T}\right)_U = \frac{(\beta - \alpha)C_v}{\alpha(\beta T - 1)} + \beta p V$$
$$\left(\frac{\partial H}{\partial T}\right)_F = C_v + \beta p V + \beta S\left(\frac{1}{\alpha} - T\right)$$
$$\left(\frac{\partial H}{\partial T}\right)_G = C_p + S - \alpha S T$$

2.6.4 对 $S$ 求偏导

$$\left(\frac{\partial H}{\partial S}\right)_p = T$$
$$\left(\frac{\partial H}{\partial S}\right)_V = \frac{\gamma + \alpha T - 1}{\alpha}$$
$$\left(\frac{\partial H}{\partial S}\right)_T = T - \frac{1}{\alpha}$$
$$\left(\frac{\partial H}{\partial S}\right)_U = \frac{\gamma + T(\alpha - \beta\gamma) - 1}{\alpha}$$
$$\left(\frac{\partial H}{\partial S}\right)_F = \frac{T(-\kappa p C_v - \alpha p V + S(\alpha T - 1))}{\alpha S T - \kappa p C_v}$$
$$\left(\frac{\partial H}{\partial S}\right)_G = \frac{(\gamma - 1)\kappa S}{\alpha(\alpha\gamma V - (\gamma - 1)\kappa S)} + T$$

2.6.5 对 $U$ 求偏导

$$\left(\frac{\partial H}{\partial U}\right)_p = \frac{\beta\gamma T}{\gamma(\beta T - 1) + 1}$$
$$\left(\frac{\partial H}{\partial U}\right)_V = 1 + \frac{\beta p V}{C_v}$$
$$\left(\frac{\partial H}{\partial U}\right)_T = \frac{\alpha T - 1}{\alpha T - \kappa p}$$
$$\left(\frac{\partial H}{\partial U}\right)_S = \frac{\beta\gamma}{\alpha}$$
$$\left(\frac{\partial H}{\partial U}\right)_F = \frac{\alpha C_v + \beta(\alpha p V + S - \alpha S T)}{\alpha(C_v + S - \beta S T)}$$
$$\left(\frac{\partial H}{\partial U}\right)_G = \frac{C_p + S - \alpha S T}{C_p + \kappa p S - \alpha p V - \alpha S T}$$

2.6.6 对 $F$ 求偏导

$$\left(\frac{\partial H}{\partial F}\right)_p = -\frac{C_p}{\alpha p V + S}$$
$$\left(\frac{\partial H}{\partial F}\right)_V = -\frac{C_v + \beta p V}{S}$$
$$\left(\frac{\partial H}{\partial F}\right)_T = \beta\left(\frac{1}{\alpha} - T\right)$$
$$\left(\frac{\partial H}{\partial F}\right)_S = \frac{\beta\gamma p V}{\alpha p V - \gamma S + S}$$
$$\left(\frac{\partial H}{\partial F}\right)_U = \frac{\beta p V(-\gamma + \beta\gamma T - \alpha T + 1)}{\alpha\beta p T V - (\gamma - 1)S(\beta T - 1)}$$
$$\left(\frac{\partial H}{\partial F}\right)_G = -\frac{C_p + S - \alpha S T}{\kappa(-p)S + \alpha p V + S}$$

2.6.7 对 $G$ 求偏导

$$\left(\frac{\partial H}{\partial G}\right)_p = -\frac{C_p}{S}$$
$$\left(\frac{\partial H}{\partial G}\right)_V = \frac{C_v + \beta p V}{\beta p V - S}$$
$$\left(\frac{\partial H}{\partial G}\right)_T = 1 - \alpha T$$
$$\left(\frac{\partial H}{\partial G}\right)_S = -\frac{C_p}{\alpha S T - C_p}$$
$$\left(\frac{\partial H}{\partial G}\right)_U = \frac{C_p(1 - \kappa p) + \alpha p V(\alpha T - 1)}{C_p + \kappa p S - \alpha p V - \alpha S T}$$
$$\left(\frac{\partial H}{\partial G}\right)_F = \frac{\kappa p C_v + \alpha p V + S - \alpha S T}{\kappa(-p)S + \alpha p V + S}$$

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2.7 $F$ 开头（42个）

2.7.1 对 $p$ 求偏导

$$\left(\frac{\partial F}{\partial p}\right)_V = -\frac{\kappa S}{\alpha}$$
$$\left(\frac{\partial F}{\partial p}\right)_T = \kappa p V$$
$$\left(\frac{\partial F}{\partial p}\right)_S = \frac{\kappa(\alpha p V - \gamma S + S)}{\alpha\gamma}$$
$$\left(\frac{\partial F}{\partial p}\right)_U = \frac{\alpha V(C_v + S - \beta S T)}{\beta(C_v + \alpha p V(\beta T - 1))}$$
$$\left(\frac{\partial F}{\partial p}\right)_H = \kappa p V - \frac{V(\alpha T - 1)(\alpha p V + S)}{C_p}$$
$$\left(\frac{\partial F}{\partial p}\right)_G = V\left(\frac{\alpha}{\beta} - \frac{\alpha p V}{S} - 1\right)$$

2.7.2 对 $V$ 求偏导

$$\left(\frac{\partial F}{\partial V}\right)_p = -\frac{\alpha p V + S}{\alpha V}$$
$$\left(\frac{\partial F}{\partial V}\right)_T = -p$$
$$\left(\frac{\partial F}{\partial V}\right)_S = p\left(\frac{\beta S T}{C_v} - 1\right)$$
$$\left(\frac{\partial F}{\partial V}\right)_U = p\left(\frac{S(\beta T - 1)}{C_v} - 1\right)$$
$$\left(\frac{\partial F}{\partial V}\right)_H = \frac{(\gamma - 1)S(\alpha T - 1)}{\alpha V(\gamma + \alpha T - 1)} - p$$
$$\left(\frac{\partial F}{\partial V}\right)_G = \frac{\kappa(-p)S + \alpha p V + S}{\kappa S - \alpha V}$$

2.7.3 对 $T$ 求偏导

$$\left(\frac{\partial F}{\partial T}\right)_p = -\alpha p V - S$$
$$\left(\frac{\partial F}{\partial T}\right)_V = -S$$
$$\left(\frac{\partial F}{\partial T}\right)_S = \frac{C_v}{\beta T} - S$$
$$\left(\frac{\partial F}{\partial T}\right)_U = \frac{C_v + S - \beta S T}{\beta T - 1}$$
$$\left(\frac{\partial F}{\partial T}\right)_H = \frac{\kappa p C_p}{\alpha T - 1} - \alpha p V - S$$
$$\left(\frac{\partial F}{\partial T}\right)_G = S(\kappa p - 1) - \alpha p V$$

2.7.4 对 $S$ 求偏导

$$\left(\frac{\partial F}{\partial S}\right)_p = -\frac{T(\alpha p V + S)}{C_p}$$
$$\left(\frac{\partial F}{\partial S}\right)_V = -\frac{S T}{C_v}$$
$$\left(\frac{\partial F}{\partial S}\right)_T = -\frac{1}{\beta}$$
$$\left(\frac{\partial F}{\partial S}\right)_U = T\left(\frac{S(\beta T - 1)}{C_v} - 1\right)$$
$$\left(\frac{\partial F}{\partial S}\right)_H = \frac{T(\alpha T - 1)(\alpha p V + S)}{C_p} - \kappa p T$$
$$\left(\frac{\partial F}{\partial S}\right)_G = \frac{T(\kappa(-p)S + \alpha p V + S)}{\alpha S T - C_p}$$

2.7.5 对 $U$ 求偏导

$$\left(\frac{\partial F}{\partial U}\right)_p = \frac{\alpha p V + S}{\alpha p V - C_p}$$
$$\left(\frac{\partial F}{\partial U}\right)_V = -\frac{S}{C_v}$$
$$\left(\frac{\partial F}{\partial U}\right)_T = \frac{1}{1 - \beta T}$$
$$\left(\frac{\partial F}{\partial U}\right)_S = 1 - \frac{\beta S T}{C_v}$$
$$\left(\frac{\partial F}{\partial U}\right)_H = \frac{\beta(\alpha T - 1)(\alpha p V + S) - \alpha C_p}{(\beta - \alpha)C_p + \alpha\beta p V(\alpha T - 1)}$$
$$\left(\frac{\partial F}{\partial U}\right)_G = \frac{\kappa(-p)S + \alpha p V + S}{-C_p - \kappa p S + \alpha p V + \alpha S T}$$

2.7.6 对 $H$ 求偏导

$$\left(\frac{\partial F}{\partial H}\right)_p = -\frac{\alpha p V + S}{C_p}$$
$$\left(\frac{\partial F}{\partial H}\right)_V = -\frac{S}{C_v + \beta p V}$$
$$\left(\frac{\partial F}{\partial H}\right)_T = \frac{\alpha}{\beta(1 - \alpha T)}$$
$$\left(\frac{\partial F}{\partial H}\right)_S = \kappa p - \frac{\alpha T(\alpha p V + S)}{C_p}$$
$$\left(\frac{\partial F}{\partial H}\right)_U = \frac{\alpha(C_v + S - \beta S T)}{(\beta - \alpha)C_v + \alpha\beta p V(\beta T - 1)}$$
$$\left(\frac{\partial F}{\partial H}\right)_G = -\frac{\kappa(-p)S + \alpha p V + S}{C_p + S - \alpha S T}$$

2.7.7 对 $G$ 求偏导

$$\left(\frac{\partial F}{\partial G}\right)_p = \frac{\alpha p V}{S} + 1$$
$$\left(\frac{\partial F}{\partial G}\right)_V = \frac{S}{S - \beta p V}$$
$$\left(\frac{\partial F}{\partial G}\right)_T = \kappa p$$
$$\left(\frac{\partial F}{\partial G}\right)_S = \frac{\alpha p V - \gamma S + S}{\beta\gamma p V - \gamma S + S}$$
$$\left(\frac{\partial F}{\partial G}\right)_U = \frac{\kappa p(C_v + S - \beta S T)}{C_v + \kappa p(\beta T - 1)(\beta p V - S)}$$
$$\left(\frac{\partial F}{\partial G}\right)_H = \frac{(\alpha T - 1)(\alpha p V + S) - \kappa p C_p}{S(\alpha T - 1) - C_p}$$

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2.8 $G$ 开头（42个）

2.8.1 对 $p$ 求偏导

$$\left(\frac{\partial G}{\partial p}\right)_T = V$$
$$\left(\frac{\partial G}{\partial p}\right)_S = V - \frac{\alpha S T V}{C_p}$$
$$\left(\frac{\partial G}{\partial p}\right)_U = V - \frac{(\gamma - 1)S(\beta T - 1)}{\beta p(\gamma(\beta T - 1) + 1)}$$
$$\left(\frac{\partial G}{\partial p}\right)_H = \frac{S V(1 - \alpha T)}{C_p} + V$$
$$\left(\frac{\partial G}{\partial p}\right)_F = \frac{V(\kappa(-p)S + \alpha p V + S)}{\alpha p V + S}$$

2.8.2 对 $V$ 求偏导

$$\left(\frac{\partial G}{\partial V}\right)_p = -\frac{S}{\alpha V}$$
$$\left(\frac{\partial G}{\partial V}\right)_T = -\frac{1}{\kappa}$$
$$\left(\frac{\partial G}{\partial V}\right)_S = \frac{(\gamma - 1)S}{\alpha V} - \frac{\gamma}{\kappa}$$
$$\left(\frac{\partial G}{\partial V}\right)_U = \frac{p(\beta T - 1)(S - \beta p V)}{C_v} - \frac{1}{\kappa}$$
$$\left(\frac{\partial G}{\partial V}\right)_H = \frac{C_p + S - \alpha S T}{\alpha V(\alpha T - 1) - \kappa C_p}$$
$$\left(\frac{\partial G}{\partial V}\right)_F = -\frac{1}{\kappa} - \frac{\beta p^2 V}{S} + p$$

2.8.3 对 $T$ 求偏导

$$\left(\frac{\partial G}{\partial T}\right)_p = -S$$
$$\left(\frac{\partial G}{\partial T}\right)_V = \beta p V - S$$
$$\left(\frac{\partial G}{\partial T}\right)_S = \frac{C_p}{\alpha T} - S$$
$$\left(\frac{\partial G}{\partial T}\right)_U = \frac{\beta C_v}{\alpha(\beta T - 1)} + \beta p V - S$$
$$\left(\frac{\partial G}{\partial T}\right)_H = \frac{C_p + S - \alpha S T}{\alpha T - 1}$$
$$\left(\frac{\partial G}{\partial T}\right)_F = \beta p V + S\left(\frac{\beta}{\alpha} - 1\right)$$

2.8.4 对 $S$ 求偏导

$$\left(\frac{\partial G}{\partial S}\right)_p = -\frac{S T}{C_p}$$
$$\left(\frac{\partial G}{\partial S}\right)_V = -\frac{T(S - \beta p V)}{C_v}$$
$$\left(\frac{\partial G}{\partial S}\right)_T = -\frac{1}{\alpha}$$
$$\left(\frac{\partial G}{\partial S}\right)_U = \frac{T(\beta T - 1)(S - \beta p V)}{C_v} - \frac{\beta T}{\alpha}$$
$$\left(\frac{\partial G}{\partial S}\right)_H = T\left(\frac{S(\alpha T - 1)}{C_p} - 1\right)$$
$$\left(\frac{\partial G}{\partial S}\right)_F = -\frac{T(\kappa(-p)S + \alpha p V + S)}{\alpha S T - \kappa p C_v}$$

2.8.5 对 $U$ 求偏导

$$\left(\frac{\partial G}{\partial U}\right)_p = \frac{S}{\alpha p V - C_p}$$
$$\left(\frac{\partial G}{\partial U}\right)_V = \frac{\beta p V - S}{C_v}$$
$$\left(\frac{\partial G}{\partial U}\right)_T = \frac{1}{\kappa p - \alpha T}$$
$$\left(\frac{\partial G}{\partial U}\right)_S = \frac{\beta\gamma p V - \gamma S + S}{\alpha p V}$$
$$\left(\frac{\partial G}{\partial U}\right)_H = \frac{C_p + S - \alpha S T}{C_p(\kappa p - 1) + \alpha p V(1 - \alpha T)}$$
$$\left(\frac{\partial G}{\partial U}\right)_F = \frac{\kappa(-p)S + \alpha p V + S}{\kappa p C_v + \kappa p S - \alpha S T}$$

2.8.6 对 $H$ 求偏导

$$\left(\frac{\partial G}{\partial H}\right)_p = -\frac{S}{C_p}$$
$$\left(\frac{\partial G}{\partial H}\right)_V = \frac{\beta p V - S}{C_v + \beta p V}$$
$$\left(\frac{\partial G}{\partial H}\right)_T = \frac{1}{1 - \alpha T}$$
$$\left(\frac{\partial G}{\partial H}\right)_S = 1 - \frac{\alpha S T}{C_p}$$
$$\left(\frac{\partial G}{\partial H}\right)_U = \frac{C_p + \kappa p S - \alpha p V - \alpha S T}{C_p(1 - \kappa p) + \alpha p V(\alpha T - 1)}$$
$$\left(\frac{\partial G}{\partial H}\right)_F = \frac{\kappa(-p)S + \alpha p V + S}{\kappa p C_v + \alpha p V + S - \alpha S T}$$

2.8.7 对 $F$ 求偏导

$$\left(\frac{\partial G}{\partial F}\right)_p = \frac{S}{\alpha p V + S}$$
$$\left(\frac{\partial G}{\partial F}\right)_V = 1 - \frac{\beta p V}{S}$$
$$\left(\frac{\partial G}{\partial F}\right)_T = \frac{\beta}{\alpha}$$
$$\left(\frac{\partial G}{\partial F}\right)_S = \frac{\beta\gamma p V - \gamma S + S}{\alpha p V - \gamma S + S}$$
$$\left(\frac{\partial G}{\partial F}\right)_U = \frac{\alpha(\beta T - 1)(S - \beta p V) - \beta C_v}{\alpha(S(\beta T - 1) - C_v)}$$
$$\left(\frac{\partial G}{\partial F}\right)_H = \frac{S(\alpha T - 1) - C_p}{(\alpha T - 1)(\alpha p V + S) - \kappa p C_p}$$

---

三、热力学势的凹凸性与平衡稳定性

3.1 稳定性条件公式汇总

对于粒子数不变的系统（省略下标 $N$），进行勒让德变换后，对于变换的那一对变量，凹凸性改变：

内能 $U(S,V)$：
$$
\left(\frac{\partial^2 U}{\partial S^2}\right)_V \geq 0,\qquad
\left(\frac{\partial^2 U}{\partial V^2}\right)_S \geq 0
$$

亥姆霍兹自由能 $A(T,V)$：
$$
\left(\frac{\partial^2 A}{\partial T^2}\right)_V \leq 0,\qquad
\left(\frac{\partial^2 A}{\partial V^2}\right)_T \geq 0
$$

焓 $H(S,p)$：
$$
\left(\frac{\partial^2 H}{\partial S^2}\right)_p \geq 0,\qquad
\left(\frac{\partial^2 H}{\partial p^2}\right)_S \leq 0
$$

吉布斯自由能 $G(T,p)$：
$$
\left(\frac{\partial^2 G}{\partial T^2}\right)_p \leq 0,\qquad
\left(\frac{\partial^2 G}{\partial p^2}\right)_T \leq 0
$$

---

3.2 内能 $U$ 凸性的证明

熵最大原理（公理）：假设系统分为两部分，两部分之间可进行内能交换。平衡时熵最大，即把第一部分的内能移动到第二部分，总熵变小：

$$
S(U+\Delta U, V, N) + S(U-\Delta U, V, N) \leq 2S(U, V, N)
$$

这对应凹性定义，所以 $S$ 对 $U$ 的二阶偏导小于0：

$$
\left(\frac{\partial^2 S}{\partial U^2}\right)_{V,N} \leq 0
$$

同理，两部分之间进行体积交换：

$$
\left(\frac{\partial^2 S}{\partial V^2}\right)_{U,N} \leq 0
$$

更一般的结论（二维凹函数，Hessian 矩阵负半定）：

$$
S(U+\Delta U, V+\Delta V, N) + S(U-\Delta U, V-\Delta V, N) \leq 2S(U, V, N)
$$

能量最小原理（由熵最大原理推出）：

$$
U(S+\Delta S, V+\Delta V, N) + U(S-\Delta S, V-\Delta V, N) \geq 2U(S, V, N)
$$

即 $U$ 是凸函数，Hessian 矩阵正半定，所以：

$$
\left(\frac{\partial^2 U}{\partial S^2}\right)_V \geq 0,\qquad
\left(\frac{\partial^2 U}{\partial V^2}\right)_S \geq 0
$$

---

3.3 亥姆霍兹自由能 $A$ 凹凸性的证明

$A$ 是 $U$ 的勒让德变换：$A(T,V) = U(S,V) - TS$

记：
$$
U_{VV} \equiv \frac{\partial^2 U}{\partial V^2},\qquad
U_{VS} \equiv \frac{\partial^2 U}{\partial V \partial S},\qquad
U_{SS} \equiv \frac{\partial^2 U}{\partial S^2}
$$

3.3.1 $A$ 对 $V$ 的二阶偏导

Schur 补形式：

$$
\left(\frac{\partial^2 A}{\partial V^2}\right)_T = \frac{U_{SS}U_{VV} - U_{SV}^2}{U_{SS}} \geq 0
$$

分子是能量最小原理得出的 Hessian 矩阵正半定表达式（行列式），分母 $U_{SS} \geq 0$。

证明：

由 $A(T,V) = U(S,V) - TS$，热力学基本关系推论：

$$
\left(\frac{\partial A}{\partial V}\right)_T = \left(\frac{\partial U}{\partial V}\right)_S = -p \equiv U_V
$$

固定 $T$，对 $V$ 再求导：

$$
\left(\frac{\partial^2 A}{\partial V^2}\right)_T = \left(\frac{\partial U_V}{\partial V}\right)_T
$$

由复合关系2：

$$
\left(\frac{\partial U_V}{\partial V}\right)_T = \left(\frac{\partial U_V}{\partial V}\right)_S + \left(\frac{\partial U_V}{\partial S}\right)_V\left(\frac{\partial S}{\partial V}\right)_T = U_{VV} + U_{VS}\left(\frac{\partial S}{\partial V}\right)_T \tag{*}
$$

由温度定义 $T = U_S(S,V)$，固定 $T$ 对 $V$ 求偏导：

$$
0 = \left(\frac{\partial U_S}{\partial V}\right)_T = \left(\frac{\partial U_S}{\partial V}\right)_S + \left(\frac{\partial U_S}{\partial S}\right)_V\left(\frac{\partial S}{\partial V}\right)_T = U_{VS} + U_{SS}\left(\frac{\partial S}{\partial V}\right)_T
$$

因此：

$$
\left(\frac{\partial S}{\partial V}\right)_T = -\frac{U_{VS}}{U_{SS}}
$$

代回 (*) 式：

$$
\left(\frac{\partial^2 A}{\partial V^2}\right)_T = U_{VV} - \frac{U_{VS}^2}{U_{SS}} = \frac{U_{SS}U_{VV} - U_{SV}^2}{U_{SS}} \geq 0
$$

证毕。

3.3.2 $A$ 对 $T$ 的二阶偏导

$$
\left(\frac{\partial^2 A}{\partial T^2}\right)_V = -\frac{1}{U_{SS}} \leq 0
$$

证明：

由热力学基本关系推论：

$$
\left(\frac{\partial A}{\partial T}\right)_V = -S
$$

两边固定 $V$ 对 $T$ 再求偏导：

$$
\left(\frac{\partial^2 A}{\partial T^2}\right)_V = -\left(\frac{\partial S}{\partial T}\right)_V \tag{**}
$$

由温度定义 $T = U_S(S,V)$，固定 $V$ 对 $T$ 求偏导：

$$
1 = \left(\frac{\partial U_S}{\partial T}\right)_V = U_{SS}\left(\frac{\partial S}{\partial T}\right)_V
$$

所以：

$$
\left(\frac{\partial S}{\partial T}\right)_V = \frac{1}{U_{SS}}
$$

代回 (**) 式：

$$
\left(\frac{\partial^2 A}{\partial T^2}\right)_V = -\frac{1}{U_{SS}} \leq 0
$$

证毕。

---

3.4 焓 $H$ 与吉布斯自由能 $G$ 的凹凸性

$H$ 和 $G$ 都是 $U$ 的勒让德变换，参考亥姆霍兹自由能的证明即可。核心结论：进行勒让德变换后，对于变换的那一对变量，热力学势对其的凹凸性改变。

• $H(S,p) = U + pV$：变换 $V \to p$，所以对 $p$ 二阶导 $\leq 0$，对 $S$ 二阶导 $\geq 0$。
• $G(T,p) = U - TS + pV$：同时变换 $S\to T$ 和 $V\to p$，所以两个方向二阶导均 $\leq 0$。

从凸分析角度看，这是 Legendre–Fenchel 变换与 Schur 补的关系。

---

四、结语

本文系统给出了热力学8个状态量之间全部336个条件偏导数的推导方法与完整分类结果。核心要点：

1. 所有偏导数均可通过雅可比行列式方法，由8个量对 $(p,T)$ 的16个基础偏导数推出；
2. 纸笔推导可归约为30个最简一阶表达式，利用倒易、循环、复合关系和麦克斯韦关系进行化简；
3. 全部336个结果按被求导量分为8大类，每类42个，均可用 $\alpha, \kappa, \beta, C_p, C_v, \gamma, p, V, T, S$ 表示；
4. 热力学势的凹凸性由熵最大原理导出，勒让德变换翻转共轭变量方向的凹凸性，对应系统的平衡稳定性条件。

这一体系为物理化学、工程热力学的科研与教学提供了完整的偏微分查阅工具，借助易测量物理量可间接推导难以直接观测的热力学特性。