The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. Its relevance to climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature.
where is the slope of the tangent to the coexistence curve at any point, is the specific latent heat, is the temperature, is the specific volume change of the phase transition, and is the specific entropy change of the phase transition. The Clausius–Clapeyron equation: 509
expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.
Derivation from state postulate
where is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change.: 57, 62 & 671 Therefore, the partial derivative of specific entropy may be changed into a total derivative
where and are respectively the change in specific entropy and specific volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds
Given constant pressure and temperature (during a phase change), we obtain: 508
Substituting the definition of specific latent heat gives
This result (also known as the Clapeyron equation) equates the slope of the coexistence curve to the function of the specific latent heat , the temperature , and the change in specific volume . Instead of the specific, corresponding molar values may also be used.
Derivation from Gibbs–Duhem relation
Suppose two phases, and , are in contact and at equilibrium with each other. Their chemical potentials are related by
Furthermore, along the coexistence curve,
One may therefore use the Gibbs–Duhem relation
from which the derivation of the Clapeyron equation continues as in the previous section.
Ideal gas approximation at low temperatures
When the phase transition of a substance is between a gas phase and a condensed phase (liquid or solid), and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase greatly exceeds that of the condensed phase . Therefore, one may approximate
where is the pressure, is the specific gas constant, and is the temperature. Substituting into the Clapeyron equation
we can obtain the Clausius–Clapeyron equation: 509
for low temperatures and pressures,: 509 where is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e. in kJ/mol and R = 8.31 J mol-1 K-1) may also be used.
Let and be any two points along the coexistence curve between two phases and . In general, varies between any two such points, as a function of temperature. But if is approximated as constant,
These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the latent heat of the phase change, without requiring specific volume data. For instance, for water near its normal boiling point, with a molar enthalpy of vaporization of 40.7 kJ/mol and R = 8.31 J mol-1 K-1,
In the original work by Clapeyron, the following argument is advanced. Clapyeron considered a Carnot process of wet steam with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water, , and an infinitesimal difference in temperature, the amount of heat absorbed is
and the amount of work done is given by
- , where is the difference in volume between the volumes of the boiling water and of the saturated vapour.
The ratio of these quantities is the efficiency of the carnot engine, .[a] Substituting and rearranging gives
Chemistry and chemical engineering
For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as:
where, is the pressure in bar, is the specific gas constant (i.e., the gas constant R divided by the molar mass), T, the absolute temperature and is a constant. For a liquid-gas transition, is the specific latent heat (or specific enthalpy) of vaporization; for a solid-gas transition, is the specific latent heat of sublimation. If the latent heat is known, then knowledge of one point on the coexistence curve, for instance (1 bar, 373 K) for water, determines the rest of the curve. Conversely, the relationship between and is linear, and so linear regression is used to estimate the latent heat.
Meteorology and climatology
This section is missing information about name and links of persons: which Magnus, which Roche?.(January 2021)
Atmospheric water vapor drives many important meteorologic phenomena (notably precipitation), motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions (near standard temperature and pressure) is
- is saturation vapor pressure
- is temperature
- is the specific latent heat of evaporation of water
- is the gas constant of water vapor
The temperature dependence of the latent heat (and of the saturation vapor pressure ) cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation:
In the above expression, is in hPa and is in Celsius, whereas everywhere else on this page, is an absolute temperature (e.g. in Kelvin). (This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate.) But see also this discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.
Under typical atmospheric conditions, the denominator of the exponent depends weakly on (for which the unit is Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximately exponentially with temperature under typical atmospheric conditions, and hence the water-holding capacity of the atmosphere increases by about 7% for every 1 °C rise in temperature.
One 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 below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume
and substituting in
- (latent heat of fusion for water),
- K (absolute temperature), and
- (change in specific volume from solid to liquid),
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) on a thimble (area = 1 cm2).
While the Clausius–Clapeyron relation gives the slope of the coexistence curve, it does not provide any information about its curvature or second derivative. The second derivative of the coexistence curve of phases 1 and 2 is given by 
- Clausius, R. (1850). "Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen" [On the motive power of heat and the laws which can be deduced therefrom regarding the theory of heat]. Annalen der Physik (in German). 155 (4): 500–524. Bibcode:1850AnP...155..500C. doi:10.1002/andp.18501550403. hdl:2027/uc1.$b242250.
- Clapeyron, M. C. (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l'École polytechnique (in French). 23: 153–190. ark:/12148/bpt6k4336791/f157.
- Wark, Kenneth (1988) . "Generalized Thermodynamic Relationships". Thermodynamics (5th ed.). New York, NY: McGraw-Hill, Inc. ISBN 978-0-07-068286-3.
- Çengel, Yunus A.; Boles, Michael A. (1998) . Thermodynamics – An Engineering Approach. McGraw-Hill Series in Mechanical Engineering (3rd ed.). Boston, MA.: McGraw-Hill. ISBN 978-0-07-011927-7.
- Salzman, William R. (2001-08-21). "Clapeyron and Clausius–Clapeyron Equations". Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-06-07. Retrieved 2007-10-11.
- Masterton, William L.; Hurley, Cecile N. (2008). Chemistry : principles and reactions (6th ed.). Cengage Learning. p. 230. ISBN 9780495126713. Retrieved 3 April 2020.
- Clapeyron, E (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l ́École Polytechnique. Vol XIV: 153–190.
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- Alduchov, Oleg; Eskridge, Robert (1997-11-01), Improved Magnus' Form Approximation of Saturation Vapor Pressure, NOAA, doi:10.2172/548871 — Equation 25 provides these coefficients.
- Alduchov, Oleg A.; Eskridge, Robert E. (1996). "Improved Magnus Form Approximation of Saturation Vapor Pressure". Journal of Applied Meteorology. 35 (4): 601–9. Bibcode:1996JApMe..35..601A. doi:10.1175/1520-0450(1996)035<0601:IMFAOS>2.0.CO;2. Equation 21 provides these coefficients.
- Lawrence, M. G. (2005). "The Relationship between Relative Humidity and the Dewpoint Temperature in Moist Air: A Simple Conversion and Applications" (PDF). Bulletin of the American Meteorological Society. 86 (2): 225–233. Bibcode:2005BAMS...86..225L. doi:10.1175/BAMS-86-2-225.
- IPCC, Climate Change 2007: Working Group I: The Physical Science Basis, "FAQ 3.2 How is Precipitation Changing ?", URL http://www.ipcc.ch/publications_and_data/ar4/wg1/en/faq-3-2.html Archived 2018-11-02 at the Wayback Machine
- Zorina, Yana (2000). "Mass of a Car". The Physics Factbook.
- Krafcik, Matthew; Sánchez Velasco, Eduardo (2014). "Beyond Clausius–Clapeyron: Determining the second derivative of a first-order phase transition line". American Journal of Physics. 82 (4): 301–305. Bibcode:2014AmJPh..82..301K. doi:10.1119/1.4858403.
- Yau, M.K.; Rogers, R.R. (1989). Short Course in Cloud Physics (3rd ed.). Butterworth–Heinemann. ISBN 978-0-7506-3215-7.
- Iribarne, J.V.; Godson, W.L. (2013). "4. Water-Air systems § 4.8 Clausius–Clapeyron Equation". Atmospheric Thermodynamics. Springer. pp. 60–. ISBN 978-94-010-2642-0.
- Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics. Wiley. ISBN 978-0-471-86256-7.
- In the original work, was simply known as the Carnot function, and was not known in this form to Clapeyron. Clausius determined this form 30 years later, and lent his name to the epynomous Clausius Clapeyron relation