The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent.
- 1 Definition
- 2 Derivations
- 3 Applications
- 4 Example
- 5 Second derivative
- 6 See also
- 7 References
- 8 Bibliography
On a pressure–temperature (P–T) diagram, the line separating the two phases is known as the coexistence curve. The Clausius–Clapeyron relation 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.
Derivation from state postulate
where is the pressure. Since pressure and temperature are constant, by definition 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 tangent to the coexistence curve , at any given point on the curve, to the function of the specific latent heat , the temperature , and the change in specific volume .
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
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 constant,
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 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 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
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 therefore of the saturation vapor pressure , cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation, using pressure in hPa and temperature in Celsius:
(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 
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- 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.
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- 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.
- 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.
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- 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.
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