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## Clausius-Clapeyron relationThe where d ## Additional recommended knowledge
## DisambiguationThe generalized equation given in the opening of this article is sometimes called the Clapeyron equation, while a less general form is sometimes called the Clausius-Clapeyron equation. The less general form neglects the magnitude of the specific volume of the liquid (or solid) state relative to that of the gas state and also approximates the specific volume of the gas state via the ideal gas law. ## Derivation
Using the state postulate, take the specific entropy, During a phase change, the temperature is constant, so - .
Using the appropriate Maxwell relation gives - .
Since temperature and pressure are constant - ,
- .
- Δ is used as an operator to represent the change in the variable that follows it—final (2) minus initial (1)
For a closed system undergoing an internally reversible process, the first law is - .
Using the definition of specific enthalpy, - .
After substitution of this result into the derivative of the pressure, one finds - ,
where the shift to capital letters indicates a shift to extensive variables. This last equation is called the Clausius-Clapeyron equation, though some thermodynamics texts just call it the Clapeyron equation, possibly to distinguish it from the approximation below. When the transition is to a gas phase, the final specific volume can be many times the size of the initial specific volume. A natural approximation would be to replace Δ - .
This leads to a version of the Clausius-Clapeyron equation that is simpler to integrate: - ,
- , or
^{[3]} - .
*C*is a constant of integration
These last equations are useful because they relate saturation pressure and saturation temperature to the enthalpy of phase change, ## Other derivationSuppose two phases, I and II, are in contact and at equilibrium with each other. Then the chemical potentials are related by μ Hence, rearranging, we have From the relation between heat and change of entropy in a reversible process δ Combining the last two equations we obtain the standard relation. ## Applications## Chemistry and chemical engineeringThe Clausius-Clapeyron equation for the liquid-vapor boundary may be used in either of two equivalent forms. where *T*_{1}and*P*_{1}are a corresponding temperature (in kelvin or other absolute temperature units) and vapor pressure*T*_{2}and*P*_{2}are the corresponding temperature and pressure at another point- Δ
_{vap}*H*is the molar enthalpy of vaporization *R*is the gas constant (8.314 J mol^{-1}K^{-1})
This can be used to predict the temperature at a certain pressure, given the temperature at another pressure, or vice versa. Alternatively, if the corresponding temperature and pressure is known at two points, the enthalpy of vaporization can be determined. The equivalent formulation, in which the values associated with one For instance, if the Notes: - As in the derivation above, the enthalpy of vaporization is assumed to be constant over the pressure/temperature range considered
- Equivalent expressions for the solid-vapor boundary are found by replacing the molar enthalpy of vaporization by the molar enthalpy of sublimation, Δ
_{sub}*H*
## Meteorology
In meteorology, a specific derivation of the Clausius-Clapeyron equation is used to describe dependence of saturated water vapor pressure on temperature. This is similar to its use in chemistry and chemical engineering. It plays a crucial role in the current debate on climate change because its solution predicts exponential behavior of saturation water vapor pressure (and, therefore water vapor concentration) as a function of temperature. In turn, because water vapor is a greenhouse gas, it might lead to further increase in the sea surface temperature leading to runaway greenhouse effect. Debate on iris hypothesis and intensity of tropical cyclones dependence on temperature depends in part on “Clausius-Clapeyron” solution. Clausius-Clapeyron equations is given for typical atmospheric conditions as where: *e*_{s}is saturation water vapor pressure*T*is a temperature*L*_{v}is latent heat of evaporation*R*_{v}is water vapor gas constant.
One can solve this equation to give where: *e*_{s}(*T*) is in hPa (mbar)*T*is in degrees Celsius.
Thus, neglecting the weak variation of ( ## ExampleOne of the uses of this equation is to determine if a phase transition will occur in a given situation. Consider the question of how much pressure is needed to melt ice at a temperature Δ and substituting in -
*L*= 3.34×10^{5}J/kg (latent heat of water), -
*T*= 273 K (absolute temperature), and - Δ
*V*= -9.05×10^{-5}m³/kg (change in volume from solid to liquid),
we obtain - = -13.1 MPa/°C.
To provide a rough example of how much pressure this is, to melt ice at -7 °C (the temperature many ice skating rinks are set at) would require balancing a small car (mass = 1000 kg ## References- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}^{i}Wark, Kenneth [1966] (1988). "Generalized Thermodynamic Relationships",*Thermodynamics*, 5th (in English), New York, NY: McGraw-Hill, Inc..__ISBN 0-07-068286-0__. - ^
^{a}^{b}Çengel, Yunus A.; Boles, Michael A. [1989] (1998).*Thermodynamics - An Engineering Approach*, 3rd, McGraw-Hill Series in Mechanical Engineering (in English), Boston, MA.: McGraw-Hill.__ISBN 0-07-011927-9__. **^**Salzman, William R. (2001-08-21). Clapeyron and Clausius-Clapeyron Equations (English).*Chemical Thermodynamics*. University of Arizona. Archived from the original on 2007-07-07. Retrieved on 2007-10-11.**^**American Meteorological Society - The Computation of Equivalent Potential Temperature**^**Zorina, Yana (2000). Mass of a Car.*The Physics Factbook*.
## Bibliography- M.K. Yau and R.R. Rogers,
*Short Course in Cloud Physics, Third Edition*, published by Butterworth-Heinemann, January 1, 1989, 304 pages. EAN 9780750632157__ISBN 0-7506-3215-1__
- J.V. Iribarne and W.L. Godson,
*Atmospheric Thermodynamics*, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages
- H.B. Callen,
*Thermodynamics and an Introduction to Thermostatistics*, published by Wiley, 1985.__ISBN 0-471-86256-8__
Categories: Thermodynamics | Atmospheric thermodynamics | Chemical engineering |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Clausius-Clapeyron_relation". A list of authors is available in Wikipedia. |