Clausius–Clapeyron relation

"Clapeyron equation" and "Clapeyron's equation" redirect here. For a state equation, see ideal gas law.

The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, 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,

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

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, and $\Delta v$ is the specific volume 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 $T$.

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

During a phase change, the temperature is constant, so

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

Using the appropriate Maxwell relation gives

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

Since temperature and pressure are constant during a phase transition, the derivative of pressure with respect to temperature does not change as the specific volume varies. Thus the partial derivative may be changed into a total derivative and be factored out when taking an integral from one phase to another,

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

${\frac {dP}{dT}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}={\frac {\Delta s}{\Delta v}}.$

Here $\Delta s\equiv s_{\beta }-s_{\alpha }$ and $\Delta v\equiv v_{\beta }-v_{\alpha }$ are respectively the change in specific entropy and specific volume from the initial phase $\alpha$ to the final phase $\beta$.

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

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

Using the definition of specific enthalpy, $h,$ and the fact that the temperature and pressure are constant, we have

$\mathrm {d} u+P\;\mathrm {d} v=dh=T\;\mathrm {d} s\Rightarrow \mathrm {d} s={\frac {\mathrm {d} h}{T}}\Rightarrow \Delta s={\frac {\Delta h}{T}}={\frac {L}{T}}.$

After substitution of this result into the derivative of the pressure, one finds

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

This last equation is the Clapeyron equation.

Derivation from Gibbs–Duhem relation

Suppose two phases, $\alpha$ and $\beta$, are in contact and at equilibrium with each other. Then the chemical potentials are related by $\mu _{\alpha }=\mu _{\beta }.$ Along the coexistence curve, we also have $\mathrm {d} \mu _{\alpha }=\mathrm {d} \mu _{\beta }.$ We now use the Gibbs–Duhem relation $\mathrm {d} \mu =M(-s\mathrm {d} T+v\mathrm {d} P)$, where $s$ and $v$ are, respectively, the specific entropy and specific volume and $M$ is the molar mass, to obtain

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

Hence, rearranging, we have

${\frac {\mathrm {d} P}{\mathrm {d} T}}={\frac {s_{\beta }-s_{\alpha }}{v_{\beta }-v_{\alpha }}}.$

From which the derivation continues as above.

Ideal gas approximation at low temperatures

When the transition is to a gas phase at temperatures much lower than the critical temperature, the specific volume of the gas phase $v_{\mathrm {g} }$ greatly exceeds that of the condensed phase $v_{\mathrm {c} }$, and so one may approximate $\Delta v=v_{\mathrm {g} }\left(1-{\tfrac {v_{\mathrm {c} }}{v_{\mathrm {g} }}}\right)\approx v_{\mathrm {g} }$. Furthermore, at low pressures, the gas may be approximated by the ideal gas law, so that $v_{\mathrm {g} }=RT/P,$ where R is the specific gas constant. Thus,

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

This is the Clausius–Clapeyron equation for low temperatures and pressures. In general, $L$ varies along the coexistence curve as a function of temperature. However, if $L$ is a constant, then

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

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

$\left.\ln P\right|_{P=P_{1}}^{P_{2}}=-{\frac {L}{R}}\cdot \left.{\frac {1}{T}}\right|_{T=T_{1}}^{T_{2}}$ or

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

Here $(P_{1},T_{1})$ and $(P_{2},T_{2})$ are two points along the coexistence curve. 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. Note that in this last equation, the subscripts 1 and 2 label different locations on the $P$–$T$ coexistence curve, whereas in previous equations the subscripts $\alpha$ and $\beta$ label the phases on either side of the coexistence curve.

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

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

For a liquid-gas transition, $L$ is the specific latent heat (or specific enthalpy) of vaporization, whereas 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

The Clausius–Clapeyron equation for water vapor in typical atmospheric conditions 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 a temperature,
• $L_{v}$ is the specific latent heat of evaporation,
• $R_{v}$ is water vapor gas constant.

The temperature dependence of the latent heat cannot be neglected in this application (see dew point). In this context, a very good approximation can usually be made using the August-Roche-Magnus formula (usually called the Magnus or Magnus-Tetens approximation, though this attribution is historically inaccurate):

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

$e_{s}(T)$ is the equilibrium or saturation vapor pressure in hPa as a function of temperature $T$ on the Celsius scale. 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 $T$.

In practical terms, this equation determines that the water-holding capacity of the atmosphere increases by about 7% for every 1°C rise in temperature.

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) 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

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

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

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