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. 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,

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

where $\mathrm {d} P/\mathrm {d} T$ is the slope of the tangent to the coexistence curve at any point, $L$ is the specific latent heat, $T$ is the temperature, $\Delta v$ is the specific volume change of the phase transition, and $\Delta s$ is the specific entropy change of the phase transition.

Derivations

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

Using the state postulate, take the specific entropy $s$ for a homogeneous substance to be a function of specific volume $v$ and temperature $T$.^[3]:508

$\mathrm {d} s=\left({\frac {\partial s}{\partial v}}\right)_{T}\mathrm {d} v+\left({\frac {\partial s}{\partial T}}\right)_{v}\mathrm {d} T.$

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]:508

$\mathrm{d} s = \left(\frac{\partial s}{\partial v}\right)_T \mathrm{d} v.$

Using the appropriate Maxwell relation gives^[3]:508

$\mathrm{d} s = \left(\frac{\partial P}{\partial T}\right)_v \mathrm{d} v.$

where $P$ is 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 & 671} Therefore, the partial derivative of specific entropy may be changed into a total derivative

${\Delta s}={\frac {{\mathrm {d}}P}{{\mathrm {d}}T}}{\Delta v}$

and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase $\alpha$ to a final phase $\beta$,^[3]:508 to obtain

${\frac {dP}{dT}}={\frac {\Delta s}{\Delta v}}$

where $\Delta s\equiv s_{\beta}-s_{\alpha}$ and $\Delta v\equiv v_{\beta}-v_{\alpha}$ 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

${\displaystyle \mathrm {d} u=\delta q+\delta w=T\;\mathrm {d} s-P\;\mathrm {d} v}$

where $u$ is the internal energy of the system. Given constant pressure and temperature (during a phase change) and the definition of specific enthalpy $h$, we obtain

$dh={\mathrm {d}}u+P\;{\mathrm {d}}v$

$dh=T\;{\mathrm {d}}s$

${\mathrm {d}}s={\frac {{\mathrm {d}}h}{T}}$

Given constant pressure and temperature (during a phase change), we obtain^[3]:508

$\Delta s={\frac {\Delta h}{T}}$

Substituting the definition of specific latent heat $L=\Delta h$ gives

$\Delta s={\frac {L}{T}}$

Substituting this result into the pressure derivative given above (${\mathrm {d}}P/{\mathrm {d}}T={\mathrm {\Delta s}}/{\mathrm {\Delta v}}$), we obtain^[3]:508[6]

$\frac{\mathrm{d} P}{\mathrm{d} T} = \frac {L}{T \Delta v}.$

This result (also known as the Clapeyron equation) equates the slope of the tangent to the coexistence curve $\mathrm {d} P/\mathrm {d} T$, at any given point on the curve, to the function ${L}/{T{\Delta v}}$ of the specific latent heat $L$, the temperature $T$, and the change in specific volume $\Delta v$.

Derivation from Gibbs–Duhem relation

Suppose two phases, $\alpha$ and $\beta$, are in contact and at equilibrium with each other. Their chemical potentials are related by

$\mu_{\alpha} = \mu_{\beta}.$

Furthermore, along the coexistence curve,

$\mathrm{d}\mu_{\alpha} = \mathrm{d}\mu_{\beta}.$

One may therefore use the Gibbs–Duhem relation

$\mathrm{d}\mu = M(-s\mathrm{d}T + v\mathrm{d}P)$

(where $s$ is the specific entropy, $v$ is the specific volume, and $M$ is the molar mass) to obtain

$-(s_{{\beta }}-s_{{\alpha }}){\mathrm {d}}T+(v_{{\beta }}-v_{{\alpha }}){\mathrm {d}}P=0$

Rearrangement gives

${\frac {{\mathrm {d}}P}{{\mathrm {d}}T}}={\frac {s_{{\beta }}-s_{{\alpha }}}{v_{{\beta }}-v_{{\alpha }}}}={\frac {\Delta s}{\Delta v}}$

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 $v_{\mathrm{g}}$ greatly exceeds that of the condensed phase $v_{\mathrm{c}}$. Therefore, one may approximate

$\Delta v =v_{\mathrm{g}}\left(1-\tfrac{v_{\mathrm{c}}}{v_{\mathrm{g}}}\right)\approx v_{\mathrm{g}}$

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

$v_{{{\mathrm {g}}}}=RT/P$

where $P$ is the pressure, $R$ is the specific gas constant, and $T$ is the temperature. Substituting into the Clapeyron equation

${\frac {{\mathrm {d}}P}{{\mathrm {d}}T}}={\frac {\Delta s}{\Delta v}}$

we can obtain the Clausius–Clapeyron equation^[3]:509

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {PL}{T^{2}R}}}$

for low temperatures and pressures,^[3]:509 where $L$ is the specific latent heat of the substance.

Let $(P_1,T_1)$ and $(P_2,T_2)$ be any two points along the coexistence curve between two phases $\alpha$ and $\beta$. In general, $L$ varies between any two such points, as a function of temperature. But if $L$ is constant,

$\frac {\mathrm{d} P}{P} = \frac {L}{R} \frac {\mathrm{d}T}{T^2},$

${\displaystyle \int _{P_{1}}^{P_{2}}{\frac {\mathrm {d} P}{P}}={\frac {L}{R}}\int _{T_{1}}^{T_{2}}{\frac {\mathrm {d} T}{T^{2}}}}$

$\left. \ln P\right|_{P=P_1}^{P_2} = -\frac{L}{R} \cdot \left.\frac{1}{T}\right|_{T=T_1}^{T_2}$

or^[5]:672

$\ln {\frac {P_{1}}{P_{2}}}=-{\frac {L}{R}}\left({\frac {1}{T_{1}}}-{\frac {1}{T_{2}}}\right)$

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

$\ln P=-{\frac {L}{R}}\left({\frac {1}{T}}\right)+c$

where $c$ is a constant. For a liquid-gas transition, $L$ is the specific latent heat (or specific enthalpy) of vaporization; for a solid-gas transition, $L$ 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 $\ln P$ and $1/T$ 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

${\frac {{\mathrm {d}}e_{s}}{{\mathrm {d}}T}}={\frac {L_{v}(T)e_{s}}{R_{v}T^{2}}}$

where:

• $e_s$ is saturation vapor pressure
• $T$ is temperature
• $L_v$ is the specific latent heat of evaporation of water
• $R_v$ is the gas constant of water vapor

The temperature dependence of the latent heat $L_{v}(T)$, and therefore of the saturation vapor pressure $e_{s}(T)$, 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:

$e_{s}(T)=6.1094\exp \left({\frac {17.625T}{T+243.04}}\right)$ ^[7][8]

(This is also sometimes called the Magnus or Magnus-Tetens approximation, though this attribution is historically inaccurate.^[9])

Under typical atmospheric conditions, the denominator of the exponent depends weakly on $T$ (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.^[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 temperature ${\Delta T}$ 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 fusion for water),

$T$ = 273 K (absolute temperature), and

$\Delta v$ = −9.05×10⁻⁵ m³/kg (change in specific 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^[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]

${\displaystyle {\begin{aligned}{\frac {\mathrm {d} ^{2}P}{\mathrm {d} T^{2}}}={\frac {1}{v_{2}-v_{1}}}\left[{\frac {c_{p2}-c_{p1}}{T}}-2(v_{2}\alpha _{2}-v_{1}\alpha _{1}){\frac {\mathrm {d} P}{\mathrm {d} T}}\right]+\\{\frac {1}{v_{2}-v_{1}}}\left[(v_{2}\kappa _{T2}-v_{1}\kappa _{T1})\left({\frac {\mathrm {d} P}{\mathrm {d} T}}\right)^{2}\right],\end{aligned}}}$

where subscripts 1 and 2 denote the different phases, $c_{p}$ is the specific heat capacity at constant pressure, $\alpha = (1/v)(\mathrm{d}v/\mathrm{d}T)_P$ is the thermal expansion coefficient, and $\kappa_T = -(1/v)(\mathrm{d}v/\mathrm{d}P)_T$ is the isothermal compressibility.

See also

• Van 't Hoff equation
• Antoine equation
• Lee–Kesler method

References

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: 500–524. Bibcode:1850AnP...155..500C. doi:10.1002/andp.18501550403.
2. 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.
3. Wark, Kenneth (1988) [1966]. "Generalized Thermodynamic Relationships". Thermodynamics (5th ed.). New York, NY: McGraw-Hill, Inc. ISBN 0-07-068286-0.
4. Carl Rod Nave (2006). "PvT Surface for a Substance which Contracts Upon Freezing". HyperPhysics. Georgia State University. Retrieved 2007-10-16.
5. 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.
6. Salzman, William R. (2001-08-21). "Clapeyron and Clausius–Clapeyron Equations". Chemical Thermodynamics. University of Arizona. Retrieved 2007-10-11.
7. Alduchov, Oleg; Eskridge, Robert (1997-11-01), "Improved Magnus' Form Approximation of Saturation Vapor Pressure", NOAA Missing or empty |title= (help) - Equation 25 provides these coefficients
8. Alduchov, Oleg; Eskridge, Robert (April 1996), "Improved Magnus' Form Approximation of Saturation Vapor Pressure", Journal of Applied Meteorology Missing or empty |title= (help) - Equation 21 provides these coefficients
9. 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
10. 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
11. Zorina, Yana (2000). "Mass of a Car". The Physics Factbook.
12. 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: 301–305. Bibcode:2014AmJPh..82..301K. doi:10.1119/1.4858403.

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