Clausius–Clapeyron relation explained

The Clausius–Clapeyron relation, in chemical thermodynamics, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. It is named after Rudolf Clausius^[1] and Benoît Paul Émile Clapeyron.^[2] However, this relation was in fact originally derived by Sadi Carnot in his Reflections on the Motive Power of Fire, which was published in 1824 but largely ignored until it was rediscovered by Clausius, Clapeyron, and Lord Kelvin decades later.^[3] Kelvin said of Carnot's argument that "nothing in the whole range of Natural Philosophy is more remarkable than the establishment of general laws by such a process of reasoning."^[4]

Kelvin and his brother James Thomson confirmed the relation experimentally in 1849–50, and it was historically important as a very early successful application of theoretical thermodynamics.^[5] Its relevance to meteorology and climatology is the increase of the water-holding capacity of the atmosphere by about 7% for every 1 °C (1.8 °F) rise in temperature.

Definition

Exact Clapeyron equation

On a pressure–temperature (P–T) diagram, for any phase change the line separating the two phases is known as the coexistence curve. The Clapeyron relation^[6] gives the slope of the tangents to this curve. Mathematically,$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac,$$where

d P/ d T is the slope of the tangent to the coexistence curve at any point, L is the specific latent heat (the amout of energy absorbed in the transformation), 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.

Clausius–Clapeyron equation

R and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.^[7]

$$\backslash frac\; =\; \backslash frac,$$

$$V\; =\; \backslash frac.$$

The equation expresses this in a more convenient form just in terms of the latent heat, for moderate temperatures and pressures.

Derivations

Derivation from state postulate

s for a homogeneous substance to be a function of specific volume v and temperature T .^[8] $$\backslash mathrm\; s\; =\; \backslash left(\backslash frac\backslash right)\_T\; \backslash ,\; \backslash mathrm\; v\; +\; \backslash left(\backslash frac\backslash right)\_v\; \backslash ,\; \backslash mathrm\; T.$$

The Clausius–Clapeyron relation describes a Phase transition in a closed system composed of two contiguous phases, condensed matter and ideal gas, of a single substance, in mutual thermodynamic equilibrium, at constant temperature and pressure. Therefore,^[8] $$\backslash mathrm\; s\; =\; \backslash left(\backslash frac\backslash right)\_T\; \backslash ,\backslash mathrm\; v.$$

Using the appropriate Maxwell relation gives^[8] $$\backslash mathrm\; s\; =\; \backslash left(\backslash frac\backslash right)\_v\; \backslash ,\backslash mathrm\; v,$$where

P is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change.^{[9] [10]} Therefore, the partial derivative of specific entropy may be changed into a total derivative$$\backslash mathrm\; s\; =\; \backslash frac\; \backslash ,\; \backslash mathrm\; v,$$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 ,^[8] to obtain$$\backslash frac\; =\; \backslash frac,$$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:$$\backslash mathrm\; u\; =\; \backslash delta\; q\; +\; \backslash delta\; w\; =\; T\backslash ,\backslash mathrm\; s\; -\; P\backslash ,\backslash mathrm\; v,$$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$$\backslash mathrm\; h\; =\; T\; \backslash ,\backslash mathrm\; s\; +\; v\; \backslash ,\backslash mathrm\; P,$$$$\backslash mathrm\; h\; =\; T\backslash ,\backslash mathrms,$$$$\backslash mathrms\; =\; \backslash frac.$$

Given constant pressure and temperature (during a phase change), we obtain^[8] $$\backslash Delta\; s\; =\; \backslash frac\; .$$

L = \Delta h gives$$\backslash Delta\; s\; =\; \backslash frac.$$

Substituting this result into the pressure derivative given above (

d P/ d T = \Delta s / \Delta v ), we obtain^{[8] [11]} $$\backslash frac\; =\; \backslash frac\; .$$

This result (also known as the Clapeyron equation) equates the slope

d P/ d T of the coexistence curve P(T) to the function L/(T \Delta v) of the specific latent heat L , the temperature T , and the change in specific volume \Delta v . Instead of the specific, corresponding molar values may also be used.

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$$\backslash mu\_\backslash alpha\; =\; \backslash mu\_\backslash beta.$$

Furthermore, along the coexistence curve,$$\backslash mathrm\backslash mu\_\backslash alpha\; =\; \backslash mathrm\backslash mu\_\backslash beta.$$

One may therefore use the Gibbs–Duhem relation$$\backslash mathrm\backslash mu\; =\; M(-s\; \backslash ,\; \backslash mathrmT\; +\; v\; \backslash ,\; \backslash mathrmP)$$(where

s is the specific entropy, v is the specific volume, and M is the molar mass) to obtain$$-(s\_\backslash beta\; -\; s\_\backslash alpha)\; \backslash ,\; \backslash mathrmT\; +\; (v\_\backslash beta\; -\; v\_\backslash alpha)\; \backslash ,\; \backslash mathrmP\; =\; 0.$$

Rearrangement gives$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac,$$

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 _{g} greatly exceeds that of the condensed phase v _{c} . Therefore, one may approximate$$\backslash Delta\; v\; =\; v\_\backslash text\; \backslash left(1\; -\; \backslash frac\backslash right)\; \backslash approx\; v\_\backslash text$$at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that$$v\_\backslash text\; =\; \backslash frac,$$

where

P is the pressure, R is the specific gas constant, and T is the temperature. Substituting into the Clapeyron equation$$\backslash frac\; =\; \backslash frac,$$we can obtain the Clausius–Clapeyron equation^[8] $$\backslash frac\; =\; \backslash frac$$for low temperatures and pressures,^[8] where L is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e. L in kJ/mol and = 8.31 J/(mol⋅K)) may also be used.

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 approximated as constant,$$\backslash frac\; \backslash cong\; \backslash frac\; \backslash frac,$$$$\backslash int\_^\; \backslash frac\; \backslash cong\; \backslash frac\; \backslash int\_^\; \backslash frac,$$$$\backslash ln\; P\backslash Big|\_^\; \backslash cong\; -\backslash frac\; \backslash cdot\; \backslash left.\backslash frac\backslash right|\_^,$$or^{[10] [12]} $$\backslash ln\; \backslash frac\; \backslash cong\; -\backslash frac\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right).$$

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. For instance, for water near its normal boiling point, with a molar enthalpy of vaporization of 40.7 kJ/mol and = 8.31 J/(mol⋅K),$$P\_\backslash text(T)\; \backslash cong\; 1~\backslash text\; \backslash cdot\; \backslash exp\backslash left[-\backslash frac\{40\backslash ,700~\backslash text\{K\}\}\{8.31\}\; \backslash left(\backslash frac\{1\}\{T\}\; -\; \backslash frac\{1\}\{373~\backslash text\{K\}\}\; \backslash right)\; \backslash right].$$

Clapeyron's derivation

In the original work by Clapeyron, the following argument is advanced.^[13] Clapeyron considered a Carnot process of saturated water vapor with horizontal isobars. As the pressure is a function of temperature alone, the isobars are also isotherms. If the process involves an infinitesimal amount of water,

d x , and an infinitesimal difference in temperature d T , the heat absorbed is$$Q\; =\; L\backslash ,\backslash mathrmx,$$and the corresponding work is$$W\; =\; \backslash frac\backslash ,\backslash mathrmT(V$$ - V'),where V'' - V' is the difference between the volumes of d x in the liquid phase and vapor phases.The ratio W/Q is the efficiency of the Carnot engine, d T/T . Substituting and rearranging gives$$\backslash frac\; =\; \backslash frac,$$where lowercase v'' - v' denotes the change in specific volume during the transition.

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$$\backslash ln\; P\; =\; -\backslash frac\; \backslash frac\; +\; c,$$where

P is the pressure, R is the specific gas constant (i.e., the gas constant divided by the molar mass), T is the absolute temperature, and 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, for instance (1 bar, 373 K) for water, 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

$$\backslash frac\; =\; \backslash frac,$$

whereThe temperature dependence of the latent heat

L _{v(T)} cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation:^{[14] [15]} $$e\_s(T)\; =\; 6.1094\; \backslash exp\; \backslash left(\backslash frac\; \backslash right),$$where e _{s} is in hPa, and T is in degrees Celsius (whereas everywhere else on this page, T is an absolute temperature, e.g. in kelvins).

This is also sometimes called the Magnus or Magnus–Tetens approximation, though this attribution is historically inaccurate.^[16] But see also the 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

T (for which the unit is degree 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.^[17]

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$$\backslash Delta\; P\; =\; \backslash frac\; \backslash ,\; \backslash Delta\; T,$$and substituting inwe obtain$$\backslash frac\; =\; -13.5~\backslash text/\backslash text.$$

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^[18]) on a thimble (area ~ 1 cm²). This shows that ice skating cannot be simply explained by pressure-caused melting point depression, and in fact the mechanism is quite complex.^[19]

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^[20] where subscripts 1 and 2 denote the different phases,

c _{p} is the specific heat capacity at constant pressure, \alpha = ( 1 /v)( d v/ d T) _{P} is the thermal expansion coefficient, and \kappa _{T} = - ( 1 /v)( d v/ d P) _{T} is the isothermal compressibility.

See also

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

Bibliography

• Book: M. K. . Yau . R. R. . Rogers . Short Course in Cloud Physics . Butterworth–Heinemann . 3rd . 1989 . 978-0-7506-3215-7 .
• Book: J. V. . Iribarne . W. L. . Godson . 4. Water-Air systems § 4.8 Clausius–Clapeyron Equation . Atmospheric Thermodynamics . https://books.google.com/books?id=UHHyCAAAQBAJ&pg=PA60 . 2013 . Springer . 978-94-010-2642-0 . 60–.
• Book: Callen, H. B. . Thermodynamics and an Introduction to Thermostatistics . Wiley . 1985 . 978-0-471-86256-7 .

Notes and 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 . de . 155 . 4 . 500–524 . 1850AnP...155..500C . 10.1002/andp.18501550403 . free . 2027/uc1.$b242250.
2. Clapeyron . M. C. . 1834 . Mémoire sur la puissance motrice de la chaleur . . fr . 23 . 153–190 . ark:/12148/bpt6k4336791/f157.
3. Web site: Illustrations of Thermodynamics . Feynman . Richard . Richard Feynman . 1963 . The Feynman Lectures on Physics . California Institute of Technology . 13 December 2023 . This relationship was deduced by Carnot, but it is called the Clausius-Clapeyron equation..
4. Thomson . William . 1849 . An Account of Carnot's Theory of the Motive Power of Heat; with Numerical Results deduced from Regnault's Experiments on Steam . Transactions of the Edinburgh Royal Society . 16 . 5 . 541–574 . 10.1017/S0080456800022481 . Lord Kelvin.
5. Book: Pippard, Alfred B. . Elements of classical thermodynamics: for advanced students of physics . 1981 . Univ. Pr . 978-0-521-09101-5 . Repr . Cambridge . 116.
6. Web site: Koziol . Andrea . Perkins . Dexter . Teaching Phase Equilibria . serc.carleton.edu . Carleton University . 1 February 2023.
7. Web site: Clausius . Clapeyron . The Clausius-Clapeyron Equation . Bodner Research Web . Purdue University . 1 February 2023.
8. Book: Wark, Kenneth . Thermodynamics . 1966 . 5th . 1988 . McGraw-Hill, Inc. . New York, NY . 978-0-07-068286-3 . Generalized Thermodynamic Relationships.
9. Web site: PvT Surface for a Substance which Contracts Upon Freezing . 2007-10-16 . Carl Rod Nave . 2006 . HyperPhysics . Georgia State University .
10. Book: Çengel, Yunus A. . Boles, Michael A. . Thermodynamics – An Engineering Approach . 1989 . 3rd . . 1998 . McGraw-Hill . Boston, MA. . 978-0-07-011927-7.
11. Web site: Clapeyron and Clausius–Clapeyron Equations . 2007-10-11 . Salzman . William R. . 2001-08-21 . Chemical Thermodynamics . University of Arizona . https://web.archive.org/web/20070607143600/http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html . 2007-06-07 . dead .
12. Book: Masterton . William L. . Hurley . Cecile N. . Chemistry : principles and reactions . 2008 . Cengage Learning . 9780495126713 . 230 . 6th . 3 April 2020.
13. Clapeyron . E . 1834 . Mémoire sur la puissance motrice de la chaleur . Journal de l ́École Polytechnique . XIV . 153–190.
14. Equation 21 provides these coefficients.
15. Alduchov . Oleg A. . Eskridge . Robert E. . Improved Magnus Form Approximation of Saturation Vapor Pressure . Journal of Applied Meteorology . 1996 . 35 . 4 . 601–609 . 10.1175/1520-0450(1996)035<0601:IMFAOS>2.0.CO;2 . 1996JApMe..35..601A . free. Equation 25 provides these coefficients.
16. Mark G. Lawrence . Lawrence . M. G. . The Relationship between Relative Humidity and the Dewpoint Temperature in Moist Air: A Simple Conversion and Applications . Bulletin of the American Meteorological Society . 2005 . 86 . 2 . 225–233 . 10.1175/BAMS-86-2-225 . 2005BAMS...86..225L.
17. IPCC, Climate Change 2007: Working Group I: The Physical Science Basis, "FAQ 3.2 How is Precipitation Changing?". .
18. Web site: Mass of a Car . Zorina . Yana . 2000 . The Physics Factbook.
19. Liefferink . Rinse W. . Hsia . Feng-Chun . Weber . Bart . Bonn . Daniel . 2021-02-08 . Friction on Ice: How Temperature, Pressure, and Speed Control the Slipperiness of Ice . Physical Review X . 11 . 1 . 011025 . 10.1103/PhysRevX.11.011025. free .
20. Krafcik . Matthew . Sánchez Velasco . Eduardo . Beyond Clausius–Clapeyron: Determining the second derivative of a first-order phase transition line . American Journal of Physics . 82 . 4 . 301–305 . 2014 . 10.1119/1.4858403. 2014AmJPh..82..301K .