第二章

Mr.Hope2020年12月28日...大约 2 分钟

四个热力学函数

\begin{aligned} H = & U + p V \\ F = & U - T S \\ G = & U - T S + p V \end{aligned}

微分形式:

U = T d S - p d V

F = - S d T - p d V

H = T d S + V d p

G = - S d T + V d p

H = G - T \frac{\partial G}{\partial T}

U = F - T \frac{\partial F}{\partial T} = G - T \frac{\partial G}{\partial T} - p \frac{\partial G}{\partial p}

麦克斯韦关系

(\frac{\partial T}{\partial V})_{S} = - (\frac{\partial p}{\partial S})_{V}

(\frac{\partial T}{\partial p})_{S} = (\frac{\partial V}{\partial S})_{p}

(\frac{\partial S}{\partial V})_{T} = (\frac{\partial p}{\partial T})_{V}

(\frac{\partial S}{\partial p})_{T} = - (\frac{\partial V}{\partial T})_{p}

一些扩展:

C_{v} = (\frac{\partial U}{\partial T})_{V} = T (\frac{\partial S}{\partial T})_{V}

C_{p} = (\frac{\partial U}{\partial T})_{p} + p (\frac{\partial V}{\partial T})_{p} = (\frac{\partial H}{\partial T})_{p} = T (\frac{\partial S}{\partial T})_{p}

(\frac{\partial U}{\partial V})_{T} = T (\frac{\partial p}{\partial T})_{V} - p = \frac{T}{p} β - p

(\frac{\partial H}{\partial p})_{T} = V - T (\frac{\partial V}{\partial T})_{p} = V - \frac{T}{V} α

节流过程

气体在节流过程中焓不变。

$μ = (\frac{\partial T}{\partial p})_{H} = \frac{V}{C_{p}} (T α - 1) = \frac{1}{C_{p}} [T (\frac{\partial V}{\partial T})_{p} - V]$ 称为焦汤系数。

可以利用节流过程中 $μ > 0$ 一侧制冷区，利用节流过程使得液体降温而液化。

内能与焓的积分形式

U = \int {C_{v} d T + [T (\frac{\partial p}{\partial T})_{V} - p] d V} + U_{0}

S = \int [\frac{C_{v}}{T} d T + (\frac{\partial p}{\partial T} d V)] + S_{0}

热辐射

辐射压强 $p = \frac{1}{3} u$ ，而能态密度 $u = a T^{4}$ 。

$S = \frac{4}{3} a T^{3} V$ (可逆绝热下有 $T^{3} V$ 常数)

辐射通量密度 $J_{u} = \frac{1}{4} C U$

磁介质

$m = M V$ 是介质的总磁矩

所做的功为 $d W = μ_{0} H d m$

磁介质的内能满足 $d U = T d S + μ_{0} h d m$

磁介质的吉布斯函数满足 $d G = - S d T - μ_{0} m d H$

磁介质的热容 $C_{H} = T (\frac{\partial S}{\partial T})_{H}$ 则 $(\frac{\partial T}{\partial H})_{S} = - \frac{μ_{0} T}{C_{H}} (\frac{\partial m}{\partial T})_{M}$

居里定律: $m = \frac{C_{V}}{T} H$ 可以得出: $(\frac{\partial T}{\partial H})_{S} = \frac{C_{V}}{C_{H} T} μ_{0} H$ 和 $T d S = C_{V} d T + T \frac{α}{κ_{T}} d V$

重要习题

证明 $\frac{κ_{s}}{κ_{T}} = \frac{C_{v}}{C_{p}}$
证明 $C_{p} - C_{V} = - T \frac{(\frac{\partial p}{\partial T})_{V}^{2}}{(\frac{\partial p}{\partial V})_{T}}$

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