Question

Derive Clapeyron equation and Clausius modification for liquid-vapor equilibrium conditions.

Answer 1

According to Clapeyron equation :

dp/dT = Htransition/TVtransition

whenthis equation is applied to the liquid-vapour phase coexistance, several simplyfying assumption can be made

First, since the volume of a gas is much greater than the volume of a condensed phase,

V = Vgas-Vliq ~ Vgas

Second, the gas is assumed to obey the ideal gas law

V = nRT/P

Applying the above assumption the Clapeyron equation becomes:

dP/dT = H*P/nRT²

dP/P = H*dT/nRT²

Third, H is assumed to be independent of temperature and pressure, allowing indefinite integration over p and T

Choosing the constant of integration to equal ln p₀, where p₀ equals one pressure unit, e.g., 1 Torr if pressure is measured in Torr, allows the pressure units to cancel and yields

This equation is useful for determining DH from a plot of ln (p/p₀) vs. 1/T.

If definite integration from p₁ to p₂ and from T₁ to T₂ is performed, the result is