# 麦克斯韦关系式

${\displaystyle {\frac {\partial }{\partial x_{j}}}\left({\frac {\partial \Phi }{\partial x_{i}}}\right)={\frac {\partial }{\partial x_{i}}}\left({\frac {\partial \Phi }{\partial x_{j}}}\right)}$

## 四个最常见的麦克斯韦关系式

${\displaystyle \left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial p}{\partial S}}\right)_{V}\qquad ={\frac {\partial ^{2}U}{\partial S\partial V}}}$
${\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{p}\qquad ={\frac {\partial ^{2}H}{\partial S\partial p}}}$
${\displaystyle \left({\frac {\partial S}{\partial V}}\right)_{T}=+\left({\frac {\partial p}{\partial T}}\right)_{V}\qquad =-{\frac {\partial ^{2}A}{\partial T\partial V}}}$
${\displaystyle -\left({\frac {\partial S}{\partial p}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{p}\qquad ={\frac {\partial ^{2}G}{\partial T\partial p}}}$

${\displaystyle U(S,V)\,}$内能
${\displaystyle H(S,p)\,}$
${\displaystyle A(T,V)\,}$亥姆霍兹自由能
${\displaystyle G(T,p)\,}$吉布斯能

## 麦克斯韦关系式的推导

${\displaystyle dU=TdS-pdV\,}$
${\displaystyle dH=TdS+Vdp\,}$
${\displaystyle dA=-SdT-pdV\,}$
${\displaystyle dG=-SdT+Vdp\,}$

${\displaystyle dz=\left({\frac {\partial z}{\partial x}}\right)_{y}\!dx+\left({\frac {\partial z}{\partial y}}\right)_{x}\!dy}$

${\displaystyle dz=Mdx+Ndy\,}$

${\displaystyle M=\left({\frac {\partial z}{\partial x}}\right)_{y},\quad N=\left({\frac {\partial z}{\partial y}}\right)_{x}}$

${\displaystyle T=\left({\frac {\partial H}{\partial S}}\right)_{p},\quad V=\left({\frac {\partial H}{\partial p}}\right)_{S}}$

${\displaystyle {\frac {\partial }{\partial y}}\left({\frac {\partial z}{\partial x}}\right)_{y}={\frac {\partial }{\partial x}}\left({\frac {\partial z}{\partial y}}\right)_{x}={\frac {\partial ^{2}z}{\partial y\partial x}}={\frac {\partial ^{2}z}{\partial x\partial y}}}$

${\displaystyle {\frac {\partial }{\partial p}}\left({\frac {\partial H}{\partial S}}\right)_{p}={\frac {\partial }{\partial S}}\left({\frac {\partial H}{\partial p}}\right)_{S}}$

${\displaystyle \left({\frac {\partial T}{\partial p}}\right)_{S}=\left({\frac {\partial V}{\partial S}}\right)_{p}}$

## 一般的麦克斯韦关系式

${\displaystyle \left({\frac {\partial \mu }{\partial p}}\right)_{S,N}=\left({\frac {\partial V}{\partial N}}\right)_{S,p}\qquad ={\frac {\partial ^{2}H}{\partial p\partial N}}}$

${\displaystyle \left({\frac {\partial y}{\partial x}}\right)_{z}=1\left/\left({\frac {\partial x}{\partial y}}\right)_{z}\right.}$