Clausius–Clapeyron relation

The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, who defined it sometime after 1834, is a way of characterizing a discontinuous phase transition between two phases of matter. 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 this curve. Mathematically,

$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{L}{T\,\Delta V}$

where dP / dT is the slope of the coexistence curve, L is the latent heat, T is the temperature, and ΔV is the volume change of the phase transition.

Disambiguation

The generalized equation given in the opening of this article is sometimes called the Clapeyron equation, while a less general form is sometimes called the Clausius–Clapeyron equation. The less general form neglects the magnitude of the specific volume of the liquid (or solid) state relative to that of the gas state and also approximates the specific volume of the gas state via the ideal gas law.^[1]:509

Derivations

A typical phase diagram. The dotted green line gives the anomalous behavior of water. The Clausius–Clapeyron relation can be used to (numerically) find the relationships between pressure and temperature for the phase change boundaries. Entropy and volume changes (due to phase change) are orthogonal to the plane of this drawing

Derivation from state postulate

Using the state postulate, take the specific entropy, s, for a homogeneous substance to be a function of specific volume, v, and temperature, T.^[1]:508

$d s = \frac{\partial s}{\partial v} d v + \frac {\partial s}{\partial T} d T.$

During a phase change, the temperature is constant, so^[1]:508

$d s = \frac{\partial s}{\partial v} d v.$

Using the appropriate Maxwell relation gives^[1]:508

$d s = \frac{\partial P}{\partial T} d v.$

Since temperature and pressure are constant during a phase change, the derivative of pressure with respect to temperature is not a function of the specific volume.^{[2][3]:57, 62 & 671} Thus the partial derivative may be changed into a total derivative and be factored out when taking an integral from one phase to another,^[1]:508

$s_2 - s_1 = \frac{d P}{d T} (v_2 - v_1),$

$\frac{d P}{d T} = \frac {s_2 - s_1}{v_2 - v_1} = \frac {\Delta s}{\Delta v}.$

Δ is used as an operator to represent the change in the variable that follows it—final (2) minus initial (1)

For a closed system undergoing an internally reversible process, the first law is

$d u = \delta q + \delta w = T d s - P d v.\,$

Using the definition of specific enthalpy, h, and the fact that the temperature and pressure are constant, we have^[1]:508

$d u + P d v = d h = T ds \Rightarrow ds = \frac {d h}{T} \Rightarrow \Delta s = \frac {\Delta h}{T}.$

After substitution of this result into the derivative of the pressure, one finds^[1]:508[4]

$\frac{d P}{d T} = \frac {\Delta h}{T \Delta v} = \frac {\Delta H}{T \Delta V} = \frac {L}{T \Delta V},$

where the shift to capital letters indicates a shift to extensive variables. This last equation is called the Clausius–Clapeyron equation, though some thermodynamics texts just call it the Clapeyron equation, possibly to distinguish it from the approximation below.

When the transition is to a gas phase, the final specific volume can be many times the size of the initial specific volume. A natural approximation would be to replace Δv with v₂. Furthermore, at low pressures, the gas phase may be approximated by the ideal gas law, so that v₂ = v_gas = RT / P, where R is the mass specific gas constant (forcing h and v to be mass specific). Thus,^[1]:509

$\frac{d P}{d T} = \frac {P \Delta h}{T^2 R}.$

This leads to a version of the Clausius–Clapeyron equation that is simpler to integrate:^[1]:509

$\frac {d P}{P} = \frac {\Delta h}{R} \frac {dT}{T^2},$

$\ln P = - \frac {\Delta h}{R} \frac {1}{T} + C,$ or^[3]:672

$\ln \frac {P_1}{P_2} = \frac {\Delta h}{R} \left ( \frac {1}{T_2} - \frac {1}{T_1} \right ).$

C is a constant of integration.

These last equations are useful because they relate equilibrium or saturation vapor pressure and temperature to the enthalpy of phase change, without requiring specific volume data. Note that in this last equation, the subscripts 1 and 2 correspond to different locations on the pressure versus temperature phase lines. In earlier equations, they corresponded to different specific volumes and entropies at the same saturation pressure and temperature.

Derivation from Gibbs–Duhem relation

Suppose two phases, I and II, are in contact and at equilibrium with each other. Then the chemical potentials are related by μ_I = μ_II. Along the coexistence curve, we also have dμ_I = dμ_II. We now use the Gibbs–Duhem relation dμ = − sdT + vdP, where s and v are, respectively, the entropy and volume per particle, to obtain

$-(s_I-s_{II}) \mathrm{d}T + (v_I-v_{II}) \mathrm{d}P = 0. \,$

Hence, rearranging, we have

$\frac{\mathrm{d}P}{\mathrm{d}T} = \frac{s_I-s_{II}}{v_I-v_{II}}.$

From the relation between heat and change of entropy in a reversible process δQ = T dS, we have that the quantity of heat added in the transformation is

$L= T (s_I-s_{II}). \,$

Combining the last two equations we obtain the standard relation.

Derivation assuming equilibrium

Suppose we have a system in equilibrium, then:

$\mu^\alpha = \mu^\beta \,$

Then assume that p and T are changed, but in such a way that the system is still kept in equilibrium:

$\mathrm{d}\mu^\alpha = \mathrm{d}\mu^\beta \,$

Remembering that

$dG = d \mu = VdP - SdT \,$

$V^\alpha dP - S^\alpha dT = V^\beta dP - S^\beta dT\,$

$(V^\alpha - V^\beta)dP = (S^\alpha - S^\beta)dT\,$

$\frac{dP}{dT} = \frac{\Delta S_{Trs}}{\Delta V_{Trs}}$

By substituting :$\Delta S_{Trs} = \frac {\Delta H_{Trs}}{T_{Trs}}$ we get:

$\frac{dP}{dT} = \frac{\Delta H_{Trs}}{T \Delta V_{Trs}}$

which is the Clapeyron equation.

The Clausius-Clapeyron equation is now obtained by inserting the molar volume of an ideal gas into the equation:

$\frac{dP}{PdT} =\frac{d \ln P}{dT} = \frac{\Delta H_{vap}}{RT^2}$

Approximations

An analytical solution to the Clausius-Clapeyron equation, accounting for the weak dependence of the enthalpy of vaporization (or "latent heat") on the temperature, is complex and difficult to use. Fortunately approximations have been developed.

August-Roche-Magnus approximation

A very good approximation can usually be made using the August-Roche-Magnus formula (usually called the Magnus or Magnus-Tetens approximation, though this is historically inaccurate^[5]):

$e_s(T)= 6.1094 \exp \left( \frac{17.625T}{T+243.04} \right)$

e_s(T) is the equilibrium or saturation vapor pressure in hPa, which is a function of temperature; T is in Celsius. Since there is only a weak dependence on temperature of the denominator of the exponent, this equation shows that saturation water vapor pressure changes approximately exponentially with T.

Applications

Chemistry and chemical engineering

The Clausius–Clapeyron equation for the liquid–vapor boundary may be used in either of two equivalent forms.

$\ln \left(\frac{P_2}{P_1} \right) = \frac{\Delta H_\mathrm{vap}}{R}\left(\frac{1}{T_1} - \frac{1}{T_2} \right),$

where

• T₁ and P₁ are a corresponding temperature (in kelvins or other absolute temperature units) and vapor pressure
• T₂ and P₂ are the corresponding temperature and pressure at another point
• ΔH_vap is the molar enthalpy of vaporization
• R is the gas constant (8.314 J mol⁻¹K⁻¹)

This can be used to predict the temperature at a certain pressure, given the temperature at another pressure, or vice versa. Alternatively, if the corresponding temperature and pressure is known at two points, the enthalpy of vaporization can be determined.

The equivalent formulation, in which the values associated with one P,T point are combined into a constant (the constant of integration as above), is

$\ln P = -\frac{\Delta H_\mathrm{vap}}{RT}+C.$

For instance, if the p,T values are known for a series of data points along the phase boundary, then the enthalpy of vaporization may be determined from a plot of ln P against 1 / T.

Notes:

• As in the derivation above, the enthalpy of vaporization is assumed to be constant over the pressure/temperature range considered
• Equivalent expressions for the solid–vapor boundary are found by replacing the molar enthalpy of vaporization by the molar enthalpy of sublimation, ΔH_sub

Meteorology and climatology

Clausius–Clapeyron equations is given for typical atmospheric conditions as

$\frac{\mathrm{d}e_s}{\mathrm{d}T} = \frac{L_v e_s}{R_v T^2}$

where:

• e_s is saturation water vapor pressure,
• T is a temperature,
• L_v is latent heat of evaporation,
• R_v is water vapor gas constant.

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 temperature ΔT below 0°C. Note that water is unusual in that its change in volume upon melting is negative. We can assume

${\Delta P} = \frac{L}{T\,\Delta V} {\Delta T}$

and substituting in

L = 3.34×10⁵ J/kg (latent heat of water),

T = 273 K (absolute temperature), and

ΔV = −9.05×10⁻⁵ m³/kg (change in volume from solid to liquid),

we obtain

$\frac{\Delta P}{\Delta T}$ = −13.5 MPa/K.

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^[6]) on a thimble (area = 1 cm²).

See also

• Van't Hoff equation
• Antoine equation

References

1. ^ ^{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 0-07-068286-0. 
2. ^ Carl Rod Nave (2006). "PvT Surface for a Substance which Contracts Upon Freezing". HyperPhysics. Georgia State University. http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/pvtsur.html. Retrieved 2007-10-16. 
3. ^ ^{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 0-07-011927-9. 
4. ^ Salzman, William R. (2001-08-21). "Clapeyron and Clausius–Clapeyron Equations". Chemical Thermodynamics. University of Arizona. Archived from the original on 2007-07-07. http://web.archive.org/web/20070607143600/http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html. Retrieved 2007-10-11. 
5. ^ M. G. Lawrence, "The relationship between relative humidity and the dew point temperature in moist air: A simple conversion and applications", Bull. Am. Meteorol. Soc., 86, 225-233, 2005
6. ^ Zorina, Yana (2000). "Mass of a Car". The Physics Factbook. http://hypertextbook.com/facts/2000/YanaZorina.shtml. 

Bibliography

• M.K. Yau and R.R. Rogers, Short Course in Cloud Physics, Third Edition, published by Butterworth–Heinemann, January 1, 1989, 304 pages. EAN 9780750632157 ISBN 0-7506-3215-1
• J.V. Iribarne and W.L. Godson, Atmospheric Thermodynamics, published by D. Reidel Publishing Company, Dordrecht, Holland, 1973, 222 pages
• H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, published by Wiley, 1985. ISBN 0-471-86256-8