Everipedia is now IQ.wiki - Join the IQ Brainlist and our Discord for early access to editing on the new platform and to participate in the beta testing.

Clausius–Clapeyron relation

The Clausius–Clapeyron relation, named after Rudolf Clausius[1] and Benoît Paul Émile Clapeyron,[2] is a way of characterizing a discontinuous phase transition between two phases of matter of a single constituent.

Definition

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,

whereis the slope of the tangent to the coexistence curve at any point,is the specificlatent heat,is thetemperature,is thespecific volumechange of the phase transition, andis thespecific entropychange of the phase transition.

Derivations

Derivation from state postulate

Using thestate postulate, take thespecific entropyfor ahomogeneoussubstance to be a function ofspecific volumeandtemperature.[3]:508

The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change, during which temperature and pressure are constant by definition. Therefore,[3] []

Using the appropriate Maxwell relation gives[3] []

whereis the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change.[4][5]:57, 62 & 671Therefore, thepartial derivativeof specific entropy may be changed into atotal derivative
and the total derivative of pressure with respect to temperature may befactored outwhenintegratingfrom an initial phaseto a final phase,[3]:508to obtain
whereandare respectively the change in specific entropy and specific volume. Given that a phase change is an internallyreversible process, and that our system is closed, thefirst law of thermodynamicsholds
whereis theinternal energyof the system. Given constant pressure and temperature (during a phase change) and the definition ofspecific enthalpy, we obtain

Given constant pressure and temperature (during a phase change), we obtain[3] []

Substituting the definition ofspecific latent heatgives
Substituting this result into the pressure derivative given above (), we obtain[3]:508[6]
This result (also known as the Clapeyron equation) equates the slope of the tangent to thecoexistence curve, at any given point on the curve, to the functionof 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

(whereis the specificentropy,is thespecific volume, andis themolar 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

When thephase transitionof a substance is between agas phaseand a condensed phase (liquidorsolid), and occurs at temperatures much lower than thecritical temperatureof that substance, thespecific volumeof the gas phasegreatly 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

whereis the pressure,is thespecific gas constant, andis the temperature. Substituting into the Clapeyron equation

we can obtain the Clausius–Clapeyron equation[3] []

for low temperatures and pressures,[3]:509whereis thespecific latent heatof the substance.
Letandbe any two points along thecoexistence curvebetween two phasesand. In general,varies between any two such points, as a function of temperature. But ifis 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.

Applications

Chemistry and chemical engineering

For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as

whereis a constant. For a liquid-gas transition,is thespecific latent heat(orspecific enthalpy) ofvaporization; for a solid-gas transition,is the specific latent heat ofsublimation. If the latent heat is known, then knowledge of one point on thecoexistence curvedetermines the rest of the curve. Conversely, the relationship betweenandis linear, and solinear regressionis 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

where:

• 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 inhPaand temperature inCelsius:

(This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate.)[9] But see also this discussion of the accuracy of different approximating formulae for saturation vapour pressure of water.

Under typical atmospheric conditions, thedenominatorof theexponentdepends weakly on(for which the unit is Celsius). Therefore, the August–Roche–Magnus equation implies that saturation water vapor pressure changes approximatelyexponentiallywith 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.[10]

Example

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 temperaturebelow 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),

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[11]) on a thimble (area = 1 cm²).

Second derivative

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 [12]

where subscripts 1 and 2 denote the different phases,is the specificheat capacityat constant pressure,is thethermal expansion coefficient, andis theisothermal compressibility.

• Van 't Hoff equation

• Antoine equation

• Lee–Kesler method

References

[1]
Citation Link//doi.org/10.1002%2Fandp.18501550403Clausius, 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.
Sep 26, 2019, 12:16 AM
[2]
Citation Linkgallica.bnf.frClapeyron, 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.
Sep 26, 2019, 12:16 AM
[3]
Citation Linkopenlibrary.orgWark, Kenneth (1988) [1966]. "Generalized Thermodynamic Relationships". Thermodynamics (5th ed.). New York, NY: McGraw-Hill, Inc. ISBN 978-0-07-068286-3.
Sep 26, 2019, 12:16 AM
[4]
Citation Linkhyperphysics.phy-astr.gsu.eduCarl Rod Nave (2006). "PvT Surface for a Substance which Contracts Upon Freezing". HyperPhysics. Georgia State University. Retrieved 2007-10-16.
Sep 26, 2019, 12:16 AM
[5]
Citation Linkopenlibrary.orgÇengel, 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.
Sep 26, 2019, 12:16 AM
[6]
Citation Linkwww.chem.arizona.eduSalzman, 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.
Sep 26, 2019, 12:16 AM
[7]
Citation Link//doi.org/10.2172%2F548871Alduchov, Oleg; Eskridge, Robert (1997-11-01), Improved Magnus' Form Approximation of Saturation Vapor Pressure, NOAA, doi:10.2172/548871 — Equation 25 provides these coefficients.
Sep 26, 2019, 12:16 AM
[8]
Citation Link//doi.org/10.1175%2F1520-0450%281996%29035%3C0601%3AIMFAOS%3E2.0.CO%3B2Alduchov, 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.
Sep 26, 2019, 12:16 AM
[9]
Citation Link//doi.org/10.1175%2FBAMS-86-2-225Lawrence, 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.
Sep 26, 2019, 12:16 AM
[10]
Citation Linkweb.archive.orgIPCC, 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
Sep 26, 2019, 12:16 AM
[11]
Citation Linkhypertextbook.comZorina, Yana (2000). "Mass of a Car". The Physics Factbook.
Sep 26, 2019, 12:16 AM
[12]
Citation Link//doi.org/10.1119%2F1.4858403Krafcik, 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.
Sep 26, 2019, 12:16 AM
[13]
Sep 26, 2019, 12:16 AM
[14]
Citation Linkgallica.bnf.fr"Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus für die Wärmelehre selbst ableiten lassen"
Sep 26, 2019, 12:16 AM
[15]
Sep 26, 2019, 12:16 AM
[16]