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Clausius–Clapeyron relation

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The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. It is named after Rudolf Clausius[1] and Benoît Paul Émile Clapeyron.[2] However, this relation was in fact originally derived by Sadi Carnot in his Reflections on the Motive Power of Fire, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and Lord Kelvin decades later.[3] Kelvin said of Carnot's argument that "nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning."[4]

Kelvin and his brother James Thomson confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics.[5] Its relevance to meteorology and 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.


Exact Clapeyron equation[edit]

On a pressuretemperature (PT) diagram, for any phase change the line separating the two phases is known as the coexistence curve. The Clapeyron relation[6] gives the slope of the tangents to this curve. Mathematically, 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.

Clausius–Clapeyron equation[edit]

The Clausius–Clapeyron equation[7]: 509  applies to vaporization of liquids where vapor follows ideal gas law using the specific gas constant and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.[8]

The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.


A typical phase diagram. The dotted green line gives the anomalous behavior of water. The Clausius–Clapeyron relation can be used to find the relationship between pressure and temperature along phase boundaries.

Derivation from state postulate[edit]

Using the state postulate, take the specific entropy for a homogeneous substance to be a function of specific volume and temperature .[7]: 508 

The Clausius–Clapeyron relation describes a Phase transition in a closed system composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and pressure. Therefore,[7]: 508 

Using the appropriate Maxwell relation gives[7]: 508  where is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change.[9][10]: 57, 62, 671  Therefore, the partial derivative of specific entropy may be changed into a total derivative and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase to a final phase ,[7]: 508  to obtain 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: where is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of specific enthalpy , we obtain

Given constant pressure and temperature (during a phase change), we obtain[7]: 508 

Substituting the definition of specific latent heat gives

Substituting this result into the pressure derivative given above (), we obtain[7]: 508 [11]

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[edit]

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 (where is the specific entropy, is the specific volume, and is the molar mass) to obtain

Rearrangement gives

from which the derivation of the Clapeyron equation continues as in the previous section.

Ideal gas approximation at low temperatures[edit]

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 at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that

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[7]: 509  for low temperatures and pressures,[7]: 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⋅K)) 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, or[10]: 672 [12]

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⋅K),

Clapeyron's derivation[edit]

In the original work by Clapeyron, the following argument is advanced.[13] Clapeyron considered a Carnot process of saturated water vapor 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 heat absorbed is and the corresponding work is where is the difference between the volumes of in the liquid phase and vapor phases. The ratio is the efficiency of the Carnot engine, .[a] Substituting and rearranging gives where lowercase denotes the change in specific volume during the transition.


Chemistry and chemical engineering[edit]

For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as where is the pressure, is the specific gas constant (i.e., the gas constant R divided by the molar mass), is 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[edit]

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


The temperature dependence of the latent heat cannot be neglected in this application. Fortunately, the AugustRocheMagnus formula provides a very good approximation:[14][15] where is in hPa, and is in degrees Celsius (whereas everywhere else on this page, is an absolute temperature, e.g. in kelvins).

This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate.[16] But see also the 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 degree 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.[17]


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),
  • (absolute temperature in kelvins),
  • (change in specific volume from solid to liquid),

we obtain

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[18]) on a thimble (area ~ 1 cm2). This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex.[19]

Second derivative[edit]

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[20] where subscripts 1 and 2 denote the different phases, is the specific heat capacity at constant pressure, is the thermal expansion coefficient, and is the isothermal compressibility.

See also[edit]


  1. ^ 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.
  2. ^ Clapeyron, M. C. (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l'École polytechnique [fr] (in French). 23: 153–190. ark:/12148/bpt6k4336791/f157.
  3. ^ Feynman, Richard (1963). "Illustrations of Thermodynamics". The Feynman Lectures on Physics. California Institute of Technology. Retrieved 13 December 2023. This relationship was deduced by Carnot, but it is called the Clausius-Clapeyron equation.
  4. ^ Thomson, William (1849). "An Account of Carnot's Theory of the Motive Power of Heat; with Numerical Results deduced from Regnault's Experiments on Steam". Transactions of the Edinburgh Royal Society. 16 (5): 541–574. doi:10.1017/S0080456800022481.
  5. ^ Pippard, Alfred B. (1981). Elements of classical thermodynamics: for advanced students of physics (Repr ed.). Cambridge: Univ. Pr. p. 116. ISBN 978-0-521-09101-5.
  6. ^ Koziol, Andrea; Perkins, Dexter. "Teaching Phase Equilibria". serc.carleton.edu. Carleton University. Retrieved 1 February 2023.
  7. ^ a b c d e f g h i Wark, Kenneth (1988) [1966]. "Generalized Thermodynamic Relationships". Thermodynamics (5th ed.). New York, NY: McGraw-Hill, Inc. ISBN 978-0-07-068286-3.
  8. ^ Clausius; Clapeyron. "The Clausius-Clapeyron Equation". Bodner Research Web. Purdue University. Retrieved 1 February 2023.
  9. ^ Carl Rod Nave (2006). "PvT Surface for a Substance which Contracts Upon Freezing". HyperPhysics. Georgia State University. Retrieved 2007-10-16.
  10. ^ a b Çengel, Yunus A.; Boles, Michael A. (1998) [1989]. Thermodynamics – An Engineering Approach. McGraw-Hill Series in Mechanical Engineering (3rd ed.). Boston, MA.: McGraw-Hill. ISBN 978-0-07-011927-7.
  11. ^ 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.
  12. ^ Masterton, William L.; Hurley, Cecile N. (2008). Chemistry : principles and reactions (6th ed.). Cengage Learning. p. 230. ISBN 9780495126713. Retrieved 3 April 2020.
  13. ^ Clapeyron, E (1834). "Mémoire sur la puissance motrice de la chaleur". Journal de l ́École Polytechnique. XIV: 153–190.
  14. ^ 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.
  15. ^ Alduchov, Oleg A.; Eskridge, Robert E. (1996). "Improved Magnus Form Approximation of Saturation Vapor Pressure". Journal of Applied Meteorology. 35 (4): 601–609. Bibcode:1996JApMe..35..601A. doi:10.1175/1520-0450(1996)035<0601:IMFAOS>2.0.CO;2. Equation 21 provides these coefficients.
  16. ^ 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.
  17. ^ IPCC, Climate Change 2007: Working Group I: The Physical Science Basis, "FAQ 3.2 How is Precipitation Changing?". Archived 2018-11-02 at the Wayback Machine.
  18. ^ Zorina, Yana (2000). "Mass of a Car". The Physics Factbook.
  19. ^ Liefferink, Rinse W.; Hsia, Feng-Chun; Weber, Bart; Bonn, Daniel (2021-02-08). "Friction on Ice: How Temperature, Pressure, and Speed Control the Slipperiness of Ice". Physical Review X. 11 (1): 011025. doi:10.1103/PhysRevX.11.011025.
  20. ^ 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.



  1. ^ In the original work, was simply called the Carnot function and was not known in this form. Clausius determined the form 30 years later and added his name to the eponymous Clausius–Clapeyron relation.