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. 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, where d P / d T {displaystyle mathrm {d} P/mathrm {d} T} is the slope of the tangent to the coexistence curve at any point, L {displaystyle L} is the specific latent heat, T {displaystyle T} is the temperature, Δ v {displaystyle Delta v} is the specific volume change of the phase transition, and Δ s {displaystyle Delta s} is the specific entropy change of the phase transition. Using the state postulate, take the specific entropy s {displaystyle s} for a homogeneous substance to be a function of specific volume v {displaystyle v} and temperature T {displaystyle T} .:508 The Clausius–Clapeyron relation characterizes behavior of a closed system during a phase change, during which temperature and pressure are constant by definition. Therefore,:508 Using the appropriate Maxwell relation gives:508 where P {displaystyle P} is the pressure. Since pressure and temperature are constant, by definition the derivative of pressure with respect to temperature does not change.:57, 62 & 671 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 α {displaystyle alpha } to a final phase β {displaystyle eta } ,:508 to obtain where Δ s ≡ s β − s α {displaystyle Delta sequiv s_{eta }-s_{alpha }} and Δ v ≡ v β − v α {displaystyle Delta vequiv v_{eta }-v_{alpha }} 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