The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter of a single constituent. 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. 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 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 characterizes behavior of a closed system during a phase change 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