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,
Derivation from state postulate
During a phase change, the temperature is constant, so:508
Since temperature and pressure are constant during a phase transition, the derivative of pressure with respect to temperature is not a function of the specific volume.:57, 62 & 671 Thus the partial derivative may be changed into a total derivative and be factored out when taking an integral from one phase to another,:508
Here and are respectively the change in specific entropy and specific volume from the initial phase to the final phase .
For a closed system undergoing an internally reversible process, the first law is
This last equation is the Clapeyron equation.
Derivation from Gibbs–Duhem relation
Suppose two phases, and , are in contact and at equilibrium with each other. Then the chemical potentials are related by Along the coexistence curve, we also have We now use the Gibbs–Duhem relation , where and are, respectively, the specific entropy and specific volume and is the molar mass, to obtain
Hence, rearranging, we have
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 greatly exceeds that of the condensed phase , and so one may approximate . Furthermore, at low pressures, the gas may be approximated by the ideal gas law, so that where R is the specific gas constant. Thus,:509
This is the Clausius–Clapeyron equation.:509 In general, varies along the coexistence curve as a function of temperature. However, if is a constant, then
Here and 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 – coexistence curve, whereas in previous equations the subscripts and label the phases on either side of the coexistence curve.
Chemistry and chemical engineering
For transitions between a gas and a condensed phase with the approximations described above, the expression may be rewritten as
For a liquid-gas transition, is the specific latent heat (or specific enthalpy) of vaporization, whereas 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
The Clausius–Clapeyron equation for water vapor in typical atmospheric conditions is
- is saturation vapor pressure,
- is a temperature,
- is the specific latent heat of evaporation,
- 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):
is the equilibrium or saturation vapor pressure in hPa as a function of temperature 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 .
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.
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
- = 3.34×105 J/kg (latent heat of fusion for water),
- = 273 K (absolute temperature), and
- = −9.05×10−5 m³/kg (change in specific volume from solid to liquid),
- = −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).
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