# 吉布斯-杜安方程

${\displaystyle \sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}=-S\mathrm {d} T+V\mathrm {d} p\,}$

## 推导

${\displaystyle \mathrm {d} G=\left.{\frac {\partial G}{\partial p}}\right|_{T,N}\mathrm {d} p+\left.{\frac {\partial G}{\partial T}}\right|_{p,N}\mathrm {d} T+\sum _{i=1}^{I}\left.{\frac {\partial G}{\partial N_{i}}}\right|_{p,T,N_{j\neq i}}\mathrm {d} N_{i}\,}$.

${\displaystyle \mathrm {d} G=V\mathrm {d} p-S\mathrm {d} T+\sum _{i=1}^{I}\mu _{i}\mathrm {d} N_{i}\,}$

${\displaystyle G=\sum _{i=1}^{I}\mu _{i}N_{i}\,}$.

${\displaystyle \mathrm {d} G=\sum _{i=1}^{I}\mu _{i}\mathrm {d} N_{i}+\sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}\,}$

${\displaystyle \sum _{i=1}^{I}N_{i}\mathrm {d} \mu _{i}=-S\mathrm {d} T+V\mathrm {d} p\,}$

## 应用

${\displaystyle 0=N_{1}\mathrm {d} \mu _{1}+N_{2}\mathrm {d} \mu _{2}\,}$

${\displaystyle x_{1}\left.{\frac {\mathrm {d} ln\gamma _{1}}{\mathrm {d} x_{1}}}\right|_{p,T}=x_{2}\left.{\frac {\mathrm {d} ln\gamma _{2}}{\mathrm {d} x_{2}}}\right|_{p,T}\,}$

## 参考文献

1. ^ A to Z of Thermodynamics Pierre Perrot ISBN 0-19-856556-9
2. ^ Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 538 ISBN 0-471-07681-3
3. Salzman, William R. Open Systems. Chemical Thermodynamics. University of Arizona. 2001-08-21 [2007-10-11]. （原始内容存档于2007-07-07） （英语）.
4. ^ Fundamentals of Engineering Thermodynamics, 3rd Edition Michael J. Moran and Howard N. Shapiro, p. 710 ISBN 0-471-07681-3
5. ^ The Properties of Gases and Liquids, 5th Edition Poling, Prausnitz and O'Connell, p. 8.13, ISBN 0-07-011682-2