Clapeyron equation
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Every line on a phase diagram — the melting line, the boiling line, the sublimation line — slopes a certain way. Why that slope, and how steep? The Clapeyron equation answers exactly that. It gives the slope of any coexistence line, telling you how much you must change the pressure to shift a transition temperature, for any phase change at all.
The equation says the slope of a phase boundary equals the transition's latent heat (more precisely its entropy change) divided by both the temperature and the change in volume across the transition. In plain terms: a transition with a big heat and a big volume jump gives a particular tilt. Unlike its famous offspring the Clausius–Clapeyron equation, the Clapeyron equation is exact and makes no approximations — it works for melting and solid–solid changes, not just boiling.
Its power is in explaining the shapes of phase diagrams. Because melting involves only a tiny volume change, its line is nearly vertical — pressure barely shifts a melting point. And it nails water's most famous quirk: since ice is less dense than liquid water, the volume change on melting is negative, so the equation predicts a leftward-leaning melting line, which is exactly what we see. Applying gas approximations to it yields the Clausius–Clapeyron equation.
Because ice is less dense than water, melting shrinks the volume, so the Clapeyron equation gives the ice–water line a negative slope — squeeze ice hard enough and it melts. That same negative slope is why the melting line on water's phase diagram tilts to the left.
The equation pins down the slope of every coexistence line.
The Clapeyron equation is exact and applies to any first-order transition; the Clausius–Clapeyron equation is the special, approximate case for liquid (or solid) to vapor, obtained by treating the vapor as an ideal gas and ignoring the condensed phase's volume.