# 吉布斯-杜安方程

$\sum_{i=1}^I N_i\mathrm{d}\mu_i = - S\mathrm{d}T + V\mathrm{d}p \,$

## 推导

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

$\mathrm{d}G =V \mathrm{d}p-S \mathrm{d}T +\sum_{i=1}^I \mu_i \mathrm{d}N_i \,$

$G = \sum_{i=1}^I \mu_i N_i \,$.

$\mathrm{d}G = \sum_{i=1}^I \mu_i \mathrm{d}N_i + \sum_{i=1}^I N_i \mathrm{d}\mu_i \,$

$\sum_{i=1}^I N_i\mathrm{d}\mu_i = - S\mathrm{d}T + V\mathrm{d}p \,$

## 应用

$0= N_1\mathrm{d}\mu_1 + N_2\mathrm{d}\mu_2 \,$

$x_1 \left. \frac{\mathrm{d}ln \gamma_1}{\mathrm{d}x_1} \right |_{p,T} =x_2 \left. \frac{\mathrm{d}ln \gamma_2}{\mathrm{d}x_2} \right |_{p,T} \,$

## 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. ^ 3.0 3.1 3.2 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） （English）.
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