# Clausius–Clapeyron relation

# 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

`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]}:508The 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,`

or^{[5]} ^{[]}

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:^{[7]}

^{[8]}

(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.`

See also

Van 't Hoff equation

Antoine equation

Lee–Kesler method

## References

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**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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