均匀体系的热力学性质

均匀体系要求体系内部各个状态量处处相等. 即要求体系较小.

1. 麦克斯韦关系

由热力学第一定律: \[ \mathrm{d}U = T\mathrm{d}S- p\mathrm{d}V \] 同样有: \[ \mathrm{d}F = -S\mathrm{d}T - p\mathrm{d}V;\] \[ \mathrm{d}G = -S\mathrm{d}T - V\mathrm{d}p;\] \[ \mathrm{d}H = -T\mathrm{d}S - V\mathrm{d}p.\] 故而得到上述量关于几个独立变量关于各个态函数的微商关系. e.g., \[ \frac{\partial U}{\partial S}\Big|_V = T \] \[ \frac{\partial U}{\partial V}\Big|_S = -p \] 按照完全微分条件: \[ \frac{\partial^{2} U}{\partial V \partial S} = \frac{\partial^{2} U}{\partial S \partial V} \] 即

\[ \frac{\partial T}{\partial V}\Big|_S = -\frac{\partial p}{\partial S}\Big|_V. \]

由勒让德变换可以接着得出:

\[ \frac{\partial p}{\partial T}\Big|_V = \frac{\partial S}{\partial V}\Big|_T. \]
\[ \frac{\partial S}{\partial V}\Big|_T = -\frac{\partial p}{\partial T}\Big|_V. \]
\[ \frac{\partial T}{\partial P}\Big|_S = \frac{\partial V}{\partial S}\Big|_p. \]

e.g.: 由 \(\bar{d}Q = T\mathrm{d}S\): \[ C_i = T \frac{\partial S}{\partial T}\Big|_i, i=V, p \] \[ \begin{aligned} C_p - C_V &= T \frac{\partial S}{\partial T}|_p - T \frac{\partial S}{\partial T}|_V\\ &= T ( \frac{\partial S}{\partial T}|_V + \frac{\partial S}{\partial V}|_T \frac{\partial V}{\partial T}|_p) - T \frac{\partial S}{\partial T}|_V\\ &= T \frac{\partial S}{\partial V}|_T \frac{\partial V}{\partial T}|_p = T \frac{\partial p}{\partial T}|_V \frac{\partial V}{\partial T}|_p\\ &= pVT\times \beta\alpha = TV \frac{\alpha^2}{\kappa_T}. \end{aligned} \] 应用到理想气体即 \(C_p - C_v = NR\).

Another e.g.: 从 \(\mathrm{d}U = T\mathrm{d}S - p\mathrm{d}V\) 出发, 展开 \(\mathrm{d}S\): \[\mathrm{d}U = \left( T \frac{\partial S}{\partial V}|_T -p \right)\mathrm{d}V + \frac{\partial S}{\partial T}\mathrm{d}T. \] 故 \[ \frac{\partial U}{\partial V}|_T = T \frac{\partial S}{\partial V}|_T -p. \] 进而有: \[ \frac{\partial U}{\partial V}|_T = T \frac{\partial p}{\partial T}|_V -p. \] 对于理想气体 \(\partial p/\partial T |_V = p/T\), 故 \[ \frac{\partial U}{\partial V}|_T = 0. \]

2. 热力学函数

2.1. 理想气体

易得 \(U = U_0 + \int C_V(T)\mathrm{d}T\) \[ S(T,V) = S_0 + \int^T \frac{C_V(T^{\prime})}{T^{\prime}}\mathrm{d}T^{\prime} + NR\ln V. \] 其他热力学函数如 \(H = U + PV, F = U - TS, G = U - TS + pV\) 可以立即得到.

2.2. 范德瓦耳斯气体

容易求得 1 mol 范德瓦耳斯气体的内能与熵为: \[\begin{aligned} u(T, v) &= \int c_v \mathrm{d}T - \frac{a}{v} + u_0,\\ s(T, v) &= \int c_v \frac{\mathrm{d}T}{T} + R\ln (v-b) + s_0. \end{aligned}\]

2.3. 构造的一般规则

\[ \mathrm{d}S = \frac{1}{T}C_V(T, V)\mathrm{d}T + \frac{\partial p}{\partial T}\Big|_V \mathrm{d}V\] \[ \mathrm{d}U = C_V(T,V)\mathrm{d}T + \left[ \frac{\partial p}{\partial T}\Big|_V - p(T,V)\right]\mathrm{d}V \] 由 \[ S = \int \mathrm{d}S, U = \int \mathrm{d}U \] 得出基本的热力学函数.

3. 特性函数

由 \(F(T,V) = U -TS, \mathrm{d}F = -S\mathrm{d}T - p\mathrm{d}V\) 得出 \[ S = - \frac{\partial F}{\partial T}\Big|_V = S(T, V)\] 同理 \[ p = - \frac{\partial F}{\partial V}\Big|_T = p(T, V)\] 从 \(F(T, V)\) 出发可以导出三个基本热力学函数, 故 \(F(T, V)\) 是特性函数.

同理可证 \(G(T, p)\) 也是特性函数.

3.1. 吉布斯-亥姆霍兹方程

\[ \begin{aligned} U(T, V) &= F + TS \\ &= F + T \frac{\partial F}{\partial T}\Big|_V\\ &= -T^2 \frac{\partial}{\partial T}\left( \frac{F}{T} \right)\\ &= -T^2 \left[ \frac{\partial}{\partial T}\left( \frac{F(T,V)}{T} \right) \right] \end{aligned}\]

4. 制冷方式

卡诺制冷机
等温压缩
节流过程焦耳汤姆森系数: \[ \mu\equiv \frac{\partial T}{\partial p}\Big|_H = \frac{1}{C_p} \left[ V \frac{\partial V}{\partial T}\Big|_p - V \right] \]
绝热去磁设有一均匀各向同性顺磁固体, 在外磁场下有: \[ \bar{d}W = -p\mathrm{d}V + \mu_0 V \mathcal{H}\mathrm{d}\mathcal{M} \] 忽略磁介质的体积变化, 基本微分方程有: \[ \mathrm{d}U = T\mathrm{d}S + \mu_0\mathcal{H}\mathrm{d}\mathcal{M} \] 体系热力学函数有:
- 物态方程: \( \mathcal{M} = CH/T \)
- 比热: \( C_{\mathcal{M}}(T, \mathcal{M}=0) = b/T^2 \)
最终解得: \[ S(T, H) = -\frac{1}{2}\frac{b}{T^2} -\frac{1}{2}\mu_0 C \frac{H^2}{T^2} +S_0 \] \[ U(T, M) = -\frac{b}{T} + U_0 \] 绝热可逆过程熵不变, \(S = \text{const}\). 等温加磁场, 绝热去磁场降温.