Clausius-Clapeyron equation

The Clausius-Clapeyron equation was developed by Émile Clapeyron in 1834 and later derived from the theories of thermodynamics by Rudolf Clausius . It is a special form of the Clapeyron equation (derivation there). The course of the boiling point curve can be calculated using the Clausius-Clapeyron equation , i.e. H. the phase boundary line of a phase diagram between the liquid and the gaseous phase of a substance .

Thermodynamically correct equation

The thermodynamically correct version of the equation is

${\displaystyle {\frac {\mathrm {d} p}{\mathrm {d} T}}={\frac {\Delta _{\mathrm {vap} }H}{\Delta _{\mathrm {vap} }V\cdot T}}}$

With

• ${\displaystyle p}$- vapor pressure
• ${\displaystyle T}$- temperature in  K
• ${\displaystyle \Delta _{\mathrm {vap} }H}$- Molar enthalpy of evaporation (index  for evaporation or English vapor = vapor ) and ${\displaystyle \mathrm {vap} }$
• ${\displaystyle \Delta _{\mathrm {vap} }V=V_{\mathrm {m(g)} }-V_{\mathrm {m(fl)} }}$- Change in molar volume between gaseous and liquid phase.

Approximation in the case of an ideal gas

As a rule, the Clausius-Clapeyron equation is the approximately valid equation

${\displaystyle {\frac {1}{p}}\,\mathrm {d} p={\frac {\Delta _{\mathrm {vap} }H}{R\cdot T^{2}}}\,\mathrm {d} T}$

With

• ${\displaystyle R=8{,}314\,462\;\mathrm {J\,mol^{-1}\,K^{-1}} }$- universal gas constant .

Derivation:
Since for most purposes the molar volume of the gas is significantly larger than that of the liquid:

${\displaystyle V_{\mathrm {m(g)} }\gg V_{\mathrm {m(fl)} },}$

the volume difference was expressed by the molar volume of  the gas compared to the thermodynamically correct equation  : ${\displaystyle \Delta _{\mathrm {vap} }V}$ ${\displaystyle V_{\mathrm {m(g)} }}$

${\displaystyle \Rightarrow \Delta _{\mathrm {vap} }V\approx V_{\mathrm {m(g)} }.}$

In addition, an ideal gas was assumed for the gaseous phase , for which the following equation of state applies:

${\displaystyle V_{\mathrm {m(g)} }={\frac {RT}{p}}.}$

Integrated form

If the enthalpy of vaporization of a substance is considered to be constant over a small temperature range ( up to ), the Clausius-Clapeyron equation can be integrated over this temperature range. Then: ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$

${\displaystyle \ln {\frac {p_{2}}{p_{1}}}={\frac {\Delta _{\mathrm {vap} }H}{R}}\cdot \left({\frac {1}{T_{1}}}-{\frac {1}{T_{2}}}\right)}$

With

• the known saturation vapor pressure  and the temperature of  the initial state, ${\displaystyle p_{1}}$ ${\displaystyle T_{1}}$
• the pressure  and temperature of  the state to be calculated. ${\displaystyle p_{2}}$ ${\displaystyle T_{2}}$

Web links

• Video: Vapor pressure of pure substances according to Clausius-Clapeyron - How comfortable does a component feel in a phase? . Jakob Günter Lauth (SciFox) 2013, made available by the Technical Information Library (TIB), doi : 10.5446 / 15670 .

literature

• MK Yau, RR Rogers: Short Course in Cloud Physics, Third Edition , Butterworth-Heinemann, January 1989, 304 pages. ISBN 0-7506-3215-1 .
• Gerd Wedler : Textbook of Physical Chemistry: Fifth, completely revised and updated edition , Wiley-VCH Verlag GmbH & Co. KGaA, August 2004, 1102 pages. ISBN 3527310665