# 克劳修斯-克拉佩龙方程

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\,\Delta V}}}$

## 推导

### 从状态假设出发进行的推导

${\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\mathrm {d} v+\left({\frac {\partial s}{\partial T}}\right)_{v}\mathrm {d} T.}$

${\displaystyle \mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\mathrm {d} v}$

${\displaystyle \mathrm {d} s=\left({\frac {\partial P}{\partial T}}\right)_{v}\mathrm {d} v}$

${\displaystyle s_{\beta }-s_{\alpha }={\frac {\mathrm {d} P}{\mathrm {d} T}}(v_{\beta }-v_{\alpha }),}$
${\displaystyle {\frac {dP}{dT}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}={\frac {\Delta s}{\Delta v}}}$

${\displaystyle \mathrm {d} u=\delta q+\delta w=T\;\mathrm {d} s-P\;\mathrm {d} v.\,}$

${\displaystyle \mathrm {d} u+P\;\mathrm {d} v=dh=T\;\mathrm {d} s\Rightarrow \mathrm {d} s={\frac {\mathrm {d} h}{T}}\Rightarrow \Delta s={\frac {\Delta h}{T}}={\frac {L}{T}}}$

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {L}{T\Delta v}}}$

### 从吉布斯-杜亥姆方程进行推导

${\displaystyle -(s_{\beta }-s_{\alpha })\mathrm {d} T+(v_{\beta }-v_{\alpha })\mathrm {d} P=0.\,}$

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}}$

### 使用理想气体状态方程近似

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}}}$

${\displaystyle {\frac {\mathrm {d} P}{P}}={\frac {L}{R}}{\frac {\mathrm {d} T}{T^{2}}},}$
${\displaystyle \int _{P_{1}}^{P_{2}}{\frac {\mathrm {d} P}{P}}={\frac {L}{R}}\int {\frac {\mathrm {d} T}{T^{2}}}}$
${\displaystyle \left.\ln P\right|_{P=P_{1}}^{P_{2}}=-{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}}}$

${\displaystyle \ln {\frac {P_{2}}{P_{1}}}={\frac {L}{R}}\left({\frac {1}{T_{1}}}-{\frac {1}{T_{2}}}\right)}$

## 参考文献

1. ^ Clausius, R. Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen. Annalen der Physik, 155: 500–524 (1850). doi:10.1002/andp.18501550403
2. ^ Clapeyron, M. C. Mémoire sur la puissance motrice de la chaleur.页面存档备份，存于互联网档案馆） Journal de l'École polytechnique 23: 153–190 (1834). ark:/12148/bpt6k4336791/f157
3. Wark, Kenneth. Generalized Thermodynamic Relationships. Thermodynamics 5th. New York, NY: McGraw-Hill, Inc. 1988 [1966]. ISBN 0-07-068286-0.
4. ^ Carl Rod Nave. PvT Surface for a Substance which Contracts Upon Freezing. HyperPhysics. Georgia State University. 2006 [2007-10-16]. （原始内容存档于2007-10-29）.
5. Çengel, Yunus A.; Boles, Michael A. Thermodynamics – An Engineering Approach. McGraw-Hill Series in Mechanical Engineering 3rd. Boston, MA.: McGraw-Hill. 1998 [1989]. ISBN 0-07-011927-9.
6. ^ Salzman, William R. Clapeyron and Clausius–Clapeyron Equations. Chemical Thermodynamics. University of Arizona. 2001-08-21 [2007-10-11]. （原始内容存档于2007-06-07）.