#### Clapeyron, Clausius-Clapeyron, and Antoine Equations: Screencast

Uses the first law for a reversible process and definitions of H and G to derive the Clapeyron equation. After making some assumptions, the Clausius-Clapeyron equation is derived from the Clapeyron equation.

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##### Important Equations:

Clapeyron equation for vapor-liquid equilibrium

$\frac{dP^{sat}}{dT} = \frac{\Delta H^{vap}}{T(V^V – V^L)}$

where $$P^{sat}$$ is the saturation pressure, $$T$$ is the absolute temperature, $$\Delta H^{vap}$$ is the heat of vaporization, $$V^V$$ is the molar vapor volume, and $$V^L$$ is the molar liquid volume.

Clapeyron equation for vapor-solid equilibrium

$\frac{dP^{sat}}{dT} = \frac{\Delta H^{sub}}{T(V^V – V^S)}$

where $$\Delta H^{sub}$$ is the heat of sublimation and $$V^S$$ is the molar solid volume.

Clapeyron equation for liquid-solid equilibrium

$\frac{dP}{dT} = \frac{\Delta H^{fus}}{T(V^L – V^S)}$

where $$P$$ is the equilibrium pressure and $$\Delta H^{fus}$$ is the heat of fusion.

Clausius-Clapeyron equation (vapor-liquid equilibrium and vapor is an ideal gas)

$dln(P^{sat}) = -\frac{\Delta H^{vap}}{R} d \left( \frac{1}{T} \right)$

Clausius-Clapeyron equation for constant heat of vaporization

$ln\left( \frac{P^{sat} _2}{P^{sat} _1} \right) = \frac{\Delta H^{vap}}{R} \left( \frac{1}{T_2} – \frac{1}{T_1} \right)$

where $$P^{sat} _1$$ is the saturation pressure at temperature $$T_1$$.

Antoine equation

$log_{10} P^{sat} = A – \frac{B}{T+C}$

where $$T$$ is temperature, usually in °C, and $$A, B,$$ and $$C$$ are constants for a given compound.