Clausius–Clapeyron relation#Meteorology and climatology

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{{cite web |last1=Koziol |first1=Andrea |last2=Perkins |first2=Dexter |title=Teaching Phase Equilibria |url=https://serc.carleton.edu/research_education/equilibria/clapeyronequation.html |website=serc.carleton.edu |publisher=Carleton University |access-date=1 February 2023}}{{cite web | title=Clausius-Clapeyron Equation | website=Chemistry LibreTexts | date=2014-06-01 | url=https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_%28Physical_and_Theoretical_Chemistry%29/Physical_Properties_of_Matter/States_of_Matter/Phase_Transitions/Clausius-Clapeyron_Equation | access-date=2024-10-21}} gives the slope of the tangents to this curve. Mathematically,

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

where $\backslash mathrm\{d\}P/\backslash mathrm\{d\}T$ is the slope of the tangent to the coexistence curve at any point, $L$ is the molar change in enthalpy (latent heat, the amount of energy absorbed in the transformation), $T$ is the temperature, $\backslash Delta\; v$ is the molar volume change of the phase transition, and $\backslash Delta\; s$ is the molar entropy change of the phase transition. Alternatively, the specific values may be used instead of the molar ones.

The Clausius–Clapeyron equation{{rp|p=509}} applies to vaporization of liquids where vapor follows ideal gas law using the ideal gas constant $R$ and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.{{cite web |last1=Clausius |last2=Clapeyron |title=The Clausius-Clapeyron Equation |url=https://chemed.chem.purdue.edu/genchem/topicreview/bp/ch14/clausius.php |website=Bodner Research Web |publisher=Purdue University |access-date=1 February 2023}}

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

$$v\; =\; \backslash frac\{V\}\{n\}\; =\; \backslash frac\{RT\}\{P\}.$$

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

File:Phase-diag2.svg. 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.]]

Using the state postulate, take the molar entropy $s$ for a homogeneous substance to be a function of molar volume $v$ and temperature $T$.{{cite book |last=Wark |first=Kenneth |title=Thermodynamics |orig-year=1966 |edition=5th |year=1988 |publisher=McGraw-Hill, Inc. |location=New York, NY |isbn=978-0-07-068286-3 |chapter=Generalized Thermodynamic Relationships}}{{rp|p=508}}

$$\backslash mathrm\{d\}\; s\; =\; \backslash left(\backslash frac\{\backslash partial\; s\}\{\backslash partial\; v\}\backslash right)\_T\; \backslash ,\; \backslash mathrm\{d\}\; v\; +\; \backslash left(\backslash frac\{\backslash partial\; s\}\{\backslash partial\; T\}\backslash right)\_v\; \backslash ,\; \backslash mathrm\{d\}\; T.$$

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,{{rp|p=508}}

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

Using the appropriate Maxwell relation gives{{rp|p=508}}

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

where $P$ is the pressure. Since pressure and temperature are constant, the derivative of pressure with respect to temperature does not change.{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/pvtsur.html |title=PvT Surface for a Substance which Contracts Upon Freezing |access-date=2007-10-16 |author=Carl Rod Nave |year=2006 |work=HyperPhysics |publisher=Georgia State University }}{{cite book |last=Çengel |first=Yunus A. |author2=Boles, Michael A. |title=Thermodynamics – An Engineering Approach |orig-year=1989 |edition=3rd |series=McGraw-Hill Series in Mechanical Engineering |year=1998 |publisher=McGraw-Hill |location=Boston, MA. |isbn=978-0-07-011927-7}}{{rp|pp=57, 62, 671}} Therefore, the partial derivative of molar entropy may be changed into a total derivative

$$\backslash mathrm\{d\}\; s\; =\; \backslash frac\{\backslash mathrm\{d\}\; P\}\{\backslash mathrm\{d\}\; T\}\; \backslash ,\; \backslash mathrm\{d\}\; v,$$

and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase $\backslash alpha$ to a final phase $\backslash beta$,{{rp|p=508}} to obtain

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

where $\backslash Delta\; s\; \backslash equiv\; s\_\backslash beta\; -\; s\_\backslash alpha$ and $\backslash Delta\; v\; \backslash equiv\; v\_\backslash beta\; -\; v\_\backslash alpha$ are respectively the change in molar entropy and molar volume. Given that a phase change is an internally reversible process, and that our system is closed, the first law of thermodynamics holds:

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

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

$$\backslash mathrm\{d\}\; h\; =\; T\; \backslash ,\backslash mathrm\{d\}\; s\; +\; v\; \backslash ,\backslash mathrm\{d\}\; P,$$

$$\backslash mathrm\{d\}\; h\; =\; T\backslash ,\backslash mathrm\{d\}s,$$

$$\backslash mathrm\{d\}s\; =\; \backslash frac\{\backslash mathrm\{d\}\; h\}\{T\}.$$

Given constant pressure and temperature (during a phase change), we obtain{{rp|p=508}}

$$\backslash Delta\; s\; =\; \backslash frac\; \{\backslash Delta\; h\}\{T\}.$$

Substituting the definition of molar latent heat $L\; =\; \backslash Delta\; h$ gives

$$\backslash Delta\; s\; =\; \backslash frac\{L\}\{T\}.$$

Substituting this result into the pressure derivative given above ($\backslash mathrm\{d\}P/\backslash mathrm\{d\}T\; =\; \backslash Delta\; s\; /\; \backslash Delta\; v$), we obtain{{rp|p=508}}{{cite web |url=http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html |title=Clapeyron and Clausius–Clapeyron Equations |access-date=2007-10-11 |last=Salzman |first=William R. |date=2001-08-21 |work=Chemical Thermodynamics |publisher=University of Arizona |archive-url=https://web.archive.org/web/20070607143600/http://www.chem.arizona.edu/~salzmanr/480a/480ants/clapeyro/clapeyro.html |archive-date=2007-06-07 |url-status=dead }}

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

This result (also known as the Clapeyron equation) equates the slope $\backslash mathrm\{d\}P/\backslash mathrm\{d\}T$ of the coexistence curve $P(T)$ to the function $L/(T\; \backslash ,\; \backslash Delta\; v)$ of the molar latent heat $L$, the temperature $T$, and the change in molar volume $\backslash Delta\; v$. Instead of the molar values, corresponding specific values may also be used.

Suppose two phases, $\backslash alpha$ and $\backslash beta$, 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,

$$\backslash mathrm\{d\}\backslash mu\_\backslash alpha\; =\; \backslash mathrm\{d\}\backslash mu\_\backslash beta.$$

One may therefore use the Gibbs–Duhem relation

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

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

$$-(s\_\backslash beta\; -\; s\_\backslash alpha)\; \backslash ,\; \backslash mathrm\{d\}T\; +\; (v\_\backslash beta\; -\; v\_\backslash alpha)\; \backslash ,\; \backslash mathrm\{d\}P\; =\; 0.$$

Rearrangement gives

$$\backslash frac\{\backslash mathrm\{d\}P\}\{\backslash mathrm\{d\}T\}\; =\; \backslash frac\{s\_\backslash beta\; -\; s\_\backslash alpha\}\{v\_\backslash beta\; -\; v\_\backslash alpha\}\; =\; \backslash frac\{\backslash Delta\; s\}\{\backslash Delta\; v\},$$

from which the derivation of the Clapeyron equation continues as in the previous section.

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 $v\_\backslash text\{g\}$ greatly exceeds that of the condensed phase $v\_\backslash text\{c\}$. Therefore, one may approximate

$$\backslash Delta\; v\; =\; v\_\backslash text\{g\}\; \backslash left(1\; -\; \backslash frac\{v\_\backslash text\{c\}\}\{v\_\backslash text\{g\}\}\backslash right)\; \backslash approx\; v\_\backslash text\{g\}$$

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

$$v\_\backslash text\{g\}\; =\; \backslash frac\{RT\}\{P\},$$

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

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

we can obtain the Clausius–Clapeyron equation{{rp|p=509}}

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

for low temperatures and pressures,{{rp|p=509}} where $L$ is the specific latent heat of the substance. Instead of the specific, corresponding molar values (i.e. $L$ in kJ/mol and {{mvar|R}} = 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 $\backslash alpha$ and $\backslash beta$. In general, $L$ varies between any two such points, as a function of temperature. But if $L$ is approximated as constant,

$$\backslash frac\{\backslash mathrm\{d\}\; P\}\{P\}\; \backslash cong\; \backslash frac\; \{L\}\{R\}\; \backslash frac\; \{\backslash mathrm\{d\}T\}\{T^2\},$$

$$\backslash int\_\{P\_1\}^\{P\_2\}\; \backslash frac\{\backslash mathrm\{d\}P\}\{P\}\; \backslash cong\; \backslash frac\{L\}\{R\}\; \backslash int\_\{T\_1\}^\{T\_2\}\; \backslash frac\{\backslash mathrm\{d\}\; T\}\{T^2\},$$

$$\backslash ln\; P\backslash Big|\_\{P=P\_1\}^\{P\_2\}\; \backslash cong\; -\backslash frac\{L\}\{R\}\; \backslash cdot\; \backslash left.\backslash frac\{1\}\{T\}\backslash right|\_\{T=T\_1\}^\{T\_2\},$$

or{{rp|p=672}}{{cite book |last1=Masterton |first1=William L. |last2=Hurley |first2=Cecile N. |title=Chemistry : principles and reactions |date=2008 |publisher=Cengage Learning |isbn=9780495126713 |page=230 |edition=6th |url=https://books.google.com/books?id=teubNK-b2bsC&q=clapeyron%20equation%20boiling |access-date=3 April 2020}}

$$\backslash ln\; \backslash frac\{P\_2\}\{P\_1\}\; \backslash cong\; -\backslash frac\{L\}\{R\}\; \backslash left(\; \backslash frac\{1\}\{T\_2\}\; -\; \backslash frac\; \{1\}\{T\_1\}\; \backslash right).$$

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 {{mvar|R}} = 8.31 J/(mol⋅K),

$$P\_\backslash text\{vap\}(T)\; \backslash cong\; 1~\backslash text\{bar\}\; \backslash cdot\; \backslash exp\backslash left[-\backslash frac\{40\backslash ,700~\backslash text\{K\}\}\{8.31\}\; \backslash left(\; \backslash frac\{1\}\{T\}\; -\; \backslash frac\{1\}\{373~\backslash text\{K\}\}\; \backslash right)\; \backslash right].$$

In the original work by Clapeyron, the following argument is advanced.{{Cite journal |last=Clapeyron |first=E |date=1834 |title=Mémoire sur la puissance motrice de la chaleur |journal= Journal de l ́École Polytechnique |volume=XIV |pages=153–190}}

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, $\backslash mathrm\{d\}x$, and an infinitesimal difference in temperature $\backslash mathrm\{d\}T$, the heat absorbed is

$$Q\; =\; L\backslash ,\backslash mathrm\{d\}x,$$

and the corresponding work is

$$W\; =\; \backslash frac\{\backslash mathrm\{d\}p\}\{\backslash mathrm\{d\}T\}\backslash ,\backslash mathrm\{d\}T(V\text{'}\text{'}\; -\; V\text{'}),$$

where $V\text{'}\text{'}\; -\; V\text{'}$ is the difference between the volumes of $\backslash mathrm\{d\}x$ in the liquid phase and vapor phases.

The ratio $W/Q$ is the efficiency of the Carnot engine, $\backslash mathrm\{d\}T/T$.{{efn|In the original work, $1/T$ was simply called the Carnot function and was not known in this form. Clausius determined the form 30 years later and added his name to the eponymous Clausius–Clapeyron relation.}} Substituting and rearranging gives

$$\backslash frac\{\backslash mathrm\{d\}p\}\{\backslash mathrm\{d\}T\}\; =\; \backslash frac\{L\}\{T(v\text{'}\text{'}\; -\; v\text{'})\},$$

where lowercase $v$ - v' denotes the change in specific volume'' during the transition.

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

$$\backslash ln\; \backslash left(\; \backslash frac\{P\_1\}\{P\_0\}\; \backslash right)\; =\; \backslash frac\{L\}\{R\}\; \backslash left(\; \backslash frac\{1\}\{T\_0\}-\backslash frac\{1\}\{T\_1\}\; \backslash right)$$

where $P\_0,\; P\_1$ are the pressures at temperatures $T\_0,\; T\_1$ respectively and $R$ is the ideal gas constant. For a liquid–gas transition, $L$ is the molar latent heat (or molar enthalpy) of vaporization; for a solid–gas transition, $L$ is the molar 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 $\backslash ln\; P$ and $1/T$ is linear, and so linear regression is used to estimate the latent heat.

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

$$\backslash frac\{\backslash mathrm\{d\}e\_s\}\{\backslash mathrm\{d\}T\}\; =\; \backslash frac\{L\_v(T)\; e\_s\}\{R\_v\; T^2\},$$

where

{{unbulleted list | style = padding-left: 1.5em

| $e\_s$ 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 $L\_v(T)$ can be neglected in this application. The {{Anchor|August–Roche–Magnus formula|August–Roche–Magnus approximation}}August–Roche–Magnus formula provides a solution under that approximation:{{Citation

|last1 = Alduchov |first1 = Oleg

|last2 = Eskridge |first2 = Robert

|date = 1997-11-01

|title = Improved Magnus' Form Approximation of Saturation Vapor Pressure

|url = http://www.osti.gov/scitech/servlets/purl/548871

|publisher = NOAA

| doi = 10.2172/548871

| doi-access = free

}} Equation 21 provides these coefficients.{{cite journal |last1=Alduchov |first1=Oleg A. |last2=Eskridge |first2=Robert E. |title=Improved Magnus Form Approximation of Saturation Vapor Pressure |journal=Journal of Applied Meteorology |date=1996 |volume=35 |issue=4 |pages=601–609 |doi=10.1175/1520-0450(1996)035<0601:IMFAOS>2.0.CO;2 |bibcode=1996JApMe..35..601A |url=https://digital.library.unt.edu/ark:/67531/metadc693874/ |doi-access=free}} Equation 25 provides these coefficients.

$$e\_s(T)\; =\; 6.1094\; \backslash exp\; \backslash left(\; \backslash frac\{17.625T\}\{T\; +\; 243.04\}\; \backslash right),$$

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.{{cite journal |author1-link=Mark G. Lawrence |last1=Lawrence |first1=M. G. |title=The Relationship between Relative Humidity and the Dewpoint Temperature in Moist Air: A Simple Conversion and Applications |journal=Bulletin of the American Meteorological Society |date=2005 |volume=86 |issue=2 |pages=225–233 |doi=10.1175/BAMS-86-2-225 |url=http://climate.envsci.rutgers.edu/pdf/LawrenceRHdewpointBAMS.pdf |bibcode=2005BAMS...86..225L}} 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.IPCC, Climate Change 2007: Working Group I: The Physical Science Basis, [http://www.ipcc.ch/publications_and_data/ar4/wg1/en/faq-3-2.html "FAQ 3.2 How is Precipitation Changing?"]. {{Webarchive|url=https://web.archive.org/web/20181102212322/http://www.ipcc.ch/publications_and_data/ar4/wg1/en/faq-3-2.html |date=2018-11-02 }}.

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 $\{\backslash Delta\; T\}$ below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume

$$\backslash Delta\; P\; =\; \backslash frac\{L\}\{T\backslash ,\backslash Delta\; v\}\; \backslash ,\; \backslash Delta\; T,$$

and substituting in

{{unbulleted list | style = padding-left: 1.6em

| $L\; =\; 3.34\; \backslash times\; 10^5~\backslash mathrm\{J/kg\}$ (latent heat of fusion for water),

| $T\; =\; 273\; \backslash ,\; \backslash mathrm\{K\}$ (absolute temperature in kelvins),

| $\backslash Delta\; v\; =\; -9.05\; \backslash times\; 10^\{-5\}~\backslash mathrm\{m^3\; /\; kg\}$ (change in specific volume from solid to liquid),

}}

we obtain

$$\backslash frac\{\backslash Delta\; P\}\{\backslash Delta\; T\}\; =\; -13.5~\backslash text\{MPa\}/\backslash text\{K\}.$$

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{{cite web |url=http://hypertextbook.com/facts/2000/YanaZorina.shtml |title=Mass of a Car |last=Zorina |first=Yana |year=2000 |work=The Physics Factbook}}) 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.{{Cite journal |last1=Liefferink |first1=Rinse W. |last2=Hsia |first2=Feng-Chun |last3=Weber |first3=Bart |last4=Bonn |first4=Daniel |date=2021-02-08 |title=Friction on Ice: How Temperature, Pressure, and Speed Control the Slipperiness of Ice |journal=Physical Review X |volume=11 |issue=1 |pages=011025 |doi=10.1103/PhysRevX.11.011025|doi-access=free |hdl=11245.1/a901a712-0a68-42ca-a80f-732e541288d2 |hdl-access=free }}

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{{cite journal

| last1 = Krafcik | first1 = Matthew

| last2 = Sánchez Velasco | first2 = Eduardo

| title = Beyond Clausius–Clapeyron: Determining the second derivative of a first-order phase transition line

| journal = American Journal of Physics

| volume = 82

| issue = 4

| pages = 301–305

| date = 2014

| doi = 10.1119/1.4858403|bibcode = 2014AmJPh..82..301K }}

$$\backslash begin\{align\}$$

\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{align}

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

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

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{{refbegin}}

• {{cite book |first1=M. K. |last1=Yau |first2=R. R. |last2=Rogers |title=Short Course in Cloud Physics |publisher=Butterworth–Heinemann |edition=3rd |year=1989 |isbn=978-0-7506-3215-7 }}
• {{cite book |first1=J. V. |last1=Iribarne |first2=W. L. |last2=Godson |chapter=4. Water-Air systems § 4.8 Clausius–Clapeyron Equation |title=Atmospheric Thermodynamics |chapter-url=https://books.google.com/books?id=UHHyCAAAQBAJ&pg=PA60 |date=2013 |publisher=Springer |isbn=978-94-010-2642-0 |pages=60–}}
• {{cite book |first=H. B. |last=Callen |title=Thermodynamics and an Introduction to Thermostatistics |publisher=Wiley |year=1985 |isbn=978-0-471-86256-7 }}

{{refend}}

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