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's named after Rudolf Clausius and Benoît Paul Émile Clapeyron. 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.
James Thomson and William Thomson confirmed the relation experimentally in 1849-50, and it was historically important as a very early successful application of theoretical thermodynamics.
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 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.
Clausius–Clapeyron equation
The Clausius–Clapeyron equation applies to vaporization of liquids where vapor follows ideal gas law using the ideal gas constant and liquid volume is neglected as being much smaller than vapor volume V. It is often used to calculate vapor pressure of a liquid.
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
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 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,
Using the appropriate Maxwell relation gives
where is the pressure. Since pressure and temperature are constant, 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 (during a phase change) and the definition of specific enthalpy , we obtain
Given constant pressure and temperature (during a phase change), we obtain
Substituting the definition of specific latent heat gives
Substituting this result into the pressure derivative given above (), we obtain
This result (also known as the Clapeyron equation) equates the slope of the coexistence curve to the function of the specific latent heat , the temperature , and the change in specific volume . Instead of the specific, corresponding molar values may also be used.
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
(where is the specific entropy, is the specific volume, and is the molar mass) 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 (liquid or solid), 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. Instead of the specific, corresponding molar values (i.e. in kJ/mol and = 8.31 J/(mol⋅K)) may also be used.