8. 热力学响应函数与数学关系

8.1 热响应函数

热膨胀系数：

$\alpha_p = \frac{1}{V} \left( \frac{\partial V}{\partial T} \right)_p$

理想气体： $\alpha_p = \frac{1}{T}$
等温压缩率：

$\kappa_T = -\frac{1}{V} \left( \frac{\partial V}{\partial P} \right)_T$

理想气体： $\kappa_T = \frac{1}{P}$
热容关系：

$C_p - C_v = TV \frac{\alpha_p^2}{\kappa_T} = N k_B$

其中 $k_B \approx 1.4 \times 10^{-23}~J/K$ 为玻尔兹曼常数

$\eta = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} \leq 1$

$S(B) - S(A) = \int_A^B \frac{dQ_{\text{rev}}}{T}$

微分形式：

$dS = \frac{dQ_{\text{rev}}}{T}$

$dE = T dS - P dV + \mu dN$

$S dT - V dP + N d\mu = 0$

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

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

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

$\mu_w = \mu_s$

其中 $\mu_w$ 和 $\mu_s$ 分别为水和蒸汽的化学势

$E(\lambda S, \lambda V, \lambda N) = \lambda E(S, V, N)$

$E = TS - PV + \mu N$

$\left( \frac{\partial \mu}{\partial T} \right)_V = -\left( \frac{\partial S}{\partial N} \right)_{T,V}$

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

$\left( \frac{\partial S}{\partial B} \right)_T = \left( \frac{\partial M}{\partial T} \right)_B$

其中 $B$ 为磁场强度， $M$ 为磁化强度