# 三乘积法则

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

${\displaystyle \left({\frac {\partial p}{\partial T}}\right)_{V}\left({\frac {\partial V}{\partial p}}\right)_{T}\left({\frac {\partial T}{\partial V}}\right)_{p}=-1.}$

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

## 推导

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

dz = 0的轨迹上，xy之间满足

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

${\displaystyle 0=\left({\frac {\partial z}{\partial x}}\right)_{y}\,dx+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial x}}\right)_{z}\,dx}$

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

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

## 参考资料

• Elliott, JR, and Lira, CT. Introductory Chemical Engineering Thermodynamics, 1st Ed., Prentice Hall PTR, 1999. p. 184.
• Carter, Ashley H. Classical and Statistical Thermodynamics, Prentice Hall, 2001, p. 392.