偏微分关系

倒易关系：$(\dfrac{\partial x}{\partial y})_z(\dfrac{\partial y}{\partial x})_z=1$
循环关系：$(\dfrac{\partial x}{\partial y})_z(\dfrac{\partial y}{\partial z})_x(\dfrac{\partial z}{\partial x})_y=-1$
复合关系1：$(\dfrac{\partial x}{\partial y})_w(\dfrac{\partial y}{\partial z})_w=(\dfrac{\partial x}{\partial z})_w$
复合关系2：$(\dfrac{\partial x}{\partial y})_w=(\dfrac{\partial x}{\partial y})_z+(\dfrac{\partial x}{\partial z})_y(\dfrac{\partial z}{\partial y})_w$

热力学基本函数与方程

$H=U+pV$ $F=U-TS$ $G=H-TS$ $dU=TdS-pdV$ $dH=TdS+Vdp$ $dF=-SdT-pdV$ $dG=-SdT+Vdp$

麦克斯韦关系

图示

$\begin{matrix} G&p&H\\ T&&S\\ F&-V&U \end{matrix}$

一阶关系

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

二阶关系

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

一阶导处理方法

对于自由度为2的系统，有自变量$a,b$和因变量$X,Y,Z$，均可以做一下预处理。

三阶$(\dfrac{\partial X}{\partial Y})_Z$ $(\dfrac{\partial X}{\partial Y})_Z=(\dfrac{\partial X}{\partial a})_Z(\dfrac{\partial a}{\partial Y})_Z$
二阶$(\dfrac{\partial X}{\partial Y})_a$ $(\dfrac{\partial X}{\partial Y})_a=(\dfrac{\partial X}{\partial b})_a(\dfrac{\partial b}{\partial Y})_a$
二阶$(\dfrac{\partial X}{\partial a})_Y$ $(\dfrac{\partial X}{\partial a})_Y=(\dfrac{\partial X}{\partial a})_b+(\dfrac{\partial X}{\partial b})_a(\dfrac{\partial b}{\partial a})_Y$
一阶$(\dfrac{\partial b}{\partial a})_X$ $(\dfrac{\partial b}{\partial a})_X=-(\dfrac{\partial b}{\partial X})_a(\dfrac{\partial X}{\partial a})_b$

通过这样的办法，可以将所有的式子转换为一阶表达式

alt text

二阶导处理

$(\dfrac{\partial C_V}{\partial V})_T=T(\dfrac{\partial^2 S}{\partial V\partial T})=T(\dfrac{\partial^2 S}{\partial T\partial V})=T(\dfrac{\partial^2 p}{\partial T^2})_V$ $(\dfrac{\partial C_P}{\partial p})_T=T(\dfrac{\partial^2 S}{\partial p\partial T})=T(\dfrac{\partial^2 S}{\partial T\partial p})=-T(\dfrac{\partial^2 V}{\partial T^2})_p$

所有结果一览

一般在化简的时候有技巧可循：如果你的形式是热力学函数偏两个自变量，那么四个热力学函数中总有对应的自变量组：

$U\rightarrow S,V$ $H\rightarrow S,p$ $F\rightarrow T,V$ $G\rightarrow T,p$

另一种方法是采用全微分的办法；或者用偏微分关系。

常数

体积膨胀系数：$\alpha=\frac1V\left(\frac{\partial V}{\partial T}\right)_p$
压强系数：$\beta=\frac1p\left(\frac{\partial p}{\partial T}\right)_V$
等温压缩系数：$\kappa_T=-\frac1V\left(\frac{\partial V}{\partial P}\right)_T$
等容热容：$C_V=(\dfrac{\partial U}{\partial T})_V(definition)=T(\dfrac{\partial S}{\partial T})_V$
等压热容：$C_P=(\dfrac{\partial H}{\partial T})_p(definition)=T(\dfrac{\partial S}{\partial T})_p$

三者满足关系：
$\left(\frac{\partial V}{\partial T}\right)_p\left(\frac{\partial p}{\partial T}\right)_V\left(\frac{\partial V}{\partial P}\right)_T=-1\Rightarrow \alpha=\kappa_T\beta p$
二者满足的关系：
$C_P-C_V=T(\dfrac{\partial S}{\partial T})_p-T(\dfrac{\partial S}{\partial T})_V=T(\dfrac{\partial p}{\partial T})_V(\dfrac{\partial V}{\partial T})_p=-T(\dfrac{\partial p}{\partial V})_T$

$S$

由于热力学函数对S的偏导可以全部转化为TPV对S的偏导，这里整合到下面。

$\begin{aligned} (\frac{\partial S}{\partial P})_T&=-(\frac{\partial V}{\partial T})_P\\ (\frac{\partial S}{\partial V})_T&=(\frac{\partial P}{\partial T})_V\\ (\frac{\partial S}{\partial T})_V&=(\frac{\partial S}{\partial U})_V(\frac{\partial U}{\partial T})_V=\frac{C_V}{T}\\ (\frac{\partial S}{\partial P})_V&=(\frac{\partial S}{\partial T})_V(\frac{\partial T}{\partial P})_V=\frac{C_V}{T}(\frac{\partial T}{\partial P})_V\\ (\frac{\partial S}{\partial T})_P&=(\frac{\partial S}{\partial H})_P(\frac{\partial H}{\partial T})_P=\frac{C_P}{T}\\ (\frac{\partial S}{\partial V})_P&=(\frac{\partial S}{\partial T})_P(\frac{\partial T}{\partial V})_P=\frac{C_P(\frac{\partial T}{\partial V})_P}{T}\\ \end{aligned}$

$U$

$\begin{aligned} (\frac{\partial U}{\partial T})_V&=C_V\\ (\frac{\partial U}{\partial V})_T&=(\frac{\partial [F+TS]}{\partial V})_T=T(\frac{\partial P}{\partial T})_V-P\\ (\frac{\partial U}{\partial T})_P&=(\frac{\partial [H-PV]}{\partial V})_T=C_P-P(\frac{\partial V}{\partial T})_P\\ (\frac{\partial U}{\partial P})_T&=(\frac{\partial [G+TS-PV]}{\partial V})_T=-P(\frac{\partial V}{\partial P})_T-T(\frac{\partial V}{\partial T})_P\\ (\frac{\partial U}{\partial P})_V&=(\frac{\partial U}{\partial T})_V(\frac{\partial T}{\partial P})_V=C_V(\frac{\partial T}{\partial P})_V\\ (\frac{\partial U}{\partial V})_P&=(\frac{\partial [H-PV]}{\partial T})_P(\frac{\partial T}{\partial V})_P=C_P(\frac{\partial T}{\partial V})_P-P\\ \end{aligned}$

当然可以配合$\alpha,\beta,\kappa_T$进一步简写，不过就没什么必要了。

$H$

H和U对偶，相当于把$C_V$和$C_P$，$V$和$P$对换，不过注意符号。

$\begin{aligned} (\frac{\partial H}{\partial T})_P&=C_P\\ (\frac{\partial H}{\partial P})_T&=-T(\frac{\partial V}{\partial T})_P+V\\ (\frac{\partial H}{\partial T})_V&=C_V+V(\frac{\partial P}{\partial T})_V\\ (\frac{\partial H}{\partial V})_T&=V(\frac{\partial P}{\partial V})_T+T(\frac{\partial P}{\partial T})_V\\ (\frac{\partial H}{\partial V})_P&=C_P(\frac{\partial T}{\partial V})_P\\ (\frac{\partial H}{\partial P})_V&=C_V(\frac{\partial T}{\partial P})_V+V\\ \end{aligned}$