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.