Clausius–Clapeyron relation

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. 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. 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. 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.