Clausius–Clapeyron equation
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You already know the pattern from the kitchen: a little warmer, and a liquid evaporates a lot more eagerly. The Clausius–Clapeyron equation puts numbers on that gut feeling. It tells you, quantitatively, how a liquid's vapor pressure climbs as you raise the temperature — and the climb is steep, rising exponentially rather than gently.
The equation is a streamlined version of the Clapeyron equation, specialized for a liquid (or solid) turning into a gas. To get its simple form, it makes two reasonable approximations: the vapor behaves as an ideal gas, and the liquid's volume is tiny next to the vapor's. The result links just two things — the heat needed to vaporize the substance (its enthalpy of vaporization) and the temperature — to predict the vapor pressure at any temperature once you know it at one.
This makes it genuinely handy. Measure vapor pressure at two temperatures and you can extract the enthalpy of vaporization without a calorimeter; know that enthalpy and one boiling point, and you can predict the boiling point at a different pressure — say, at altitude. It is one of the most-used working equations in physical chemistry, though its approximations break down near the critical point, where vapor is no longer ideal.
Knowing water boils at 100 °C at sea level and its enthalpy of vaporization (about 40.7 kJ/mol), the equation predicts it should boil near 93 °C in Denver, where the air pressure is lower — matching what cooks there actually find.
From one boiling point plus a heat of vaporization, predict another.
The Clausius–Clapeyron equation is the approximate, vapor-specific cousin of the exact Clapeyron equation. It assumes ideal-gas vapor and a negligible liquid volume, so it works well far from the critical point but loses accuracy as you approach it.