Clausius–Clapeyron relation


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.

Definition

On a pressure–temperature 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,
where is the slope of the tangent to the coexistence curve at any point, is the specific latent heat, is the temperature, is the specific volume change of the phase transition, and is the specific entropy change of the phase transition.

Derivations

Derivation from state postulate

Using the state postulate, take the specific entropy for a homogeneous substance to be a function of specific volume and temperature.
The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change, during which temperature and pressure are constant by definition. Therefore,
Using the appropriate Maxwell relation gives
where is the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change. Therefore, the partial derivative of specific entropy may be changed into a total derivative
and the total derivative of pressure with respect to temperature may be factored out when integrating from an initial phase to a final phase, to obtain
where and 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
where is the internal energy of the system. Given constant pressure and temperature and the definition of specific enthalpy, we obtain
Given constant pressure and temperature, we obtain
Substituting the definition of specific latent heat gives
Substituting this result into the pressure derivative given above, we obtain
This result equates the slope of the tangent to the coexistence curve, at any given point on the curve, to the function of the specific latent heat, the temperature, and the change in specific volume.

Derivation from Gibbs–Duhem relation

Suppose two phases, and, are in contact and at equilibrium with each other. Their chemical potentials are related by
Furthermore, along the coexistence curve,
One may therefore use the Gibbs–Duhem relation
to obtain
Rearrangement gives
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, and occurs at temperatures much lower than the critical temperature of that substance, the specific volume of the gas phase greatly exceeds that of the condensed phase. Therefore, one may approximate
at low temperatures. If pressure is also low, the gas may be approximated by the ideal gas law, so that
where is the pressure, is the specific gas constant, and is the temperature. Substituting into the Clapeyron equation
we can obtain the Clausius–Clapeyron equation
for low temperatures and pressures, where is the specific latent heat of the substance.
Let and be any two points along the coexistence curve between two phases and. In general, varies between any two such points, as a function of temperature. But if is constant,
or
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
where is a constant. For a liquid-gas transition, is the specific latent heat of vaporization; for a solid-gas transition, 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 and is linear, and so linear regression is used to estimate the latent heat.

Meteorology and climatology

drives many important meteorologic phenomena, motivating interest in its dynamics. The Clausius–Clapeyron equation for water vapor under typical atmospheric conditions is
where:
The temperature dependence of the latent heat cannot be neglected in this application. Fortunately, the August–Roche–Magnus formula provides a very good approximation:
In the above expression, is in hPa and is in Celsius, whereas everywhere else on this page, is an absolute temperature.
But see also this 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 . 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.

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 below 0 °C. Note that water is unusual in that its change in volume upon melting is negative. We can assume
and substituting in
we obtain
To provide a rough example of how much pressure this is, to melt ice at −7 °C would require balancing a small car on a thimble.

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
where subscripts 1 and 2 denote the different phases, is the specific heat capacity at constant pressure, is the thermal expansion coefficient, and is the isothermal compressibility.