A liquid is at equilibrium with its vapour. The vapour is assumed to be an ideal gas. The liquid has a molar latent heat of vaporisation \ell_{\ell g} that depends on temperature, with \ell_{\ell g} = A− BT , where A and B are constants. Apply the Clausius–Clapeyron relation (6.50) and consider that the molar volume of the liquid phase is negligible compared to the vapour phase, i.e. ν_\ell \ll ν_g. Use the ideal gas law (5.47) for the vapour phase. Show that at equilibrium at a temperature T, the vapour pressure p depends on temperature according to Dupré’s law,
\frac{dp}{dT} = \frac{\ell _{s\ell }}{T(\nu _\ell -\nu _s)} and \frac{dp}{dT} = \frac{\ell _{\ell g}}{T(\nu _g -\nu _\ell)} . (6.50)
p V = N R T (5.47)
\ln \Bigl(\frac{p}{p_0}\Bigr) = \frac{A}{R} \Bigl(\frac{1}{T_0} - \frac{1}{T}\Bigr) - \frac{B}{R} \ln \Bigl(\frac{T}{T_0}\Bigr) .
where p_0 is the vapour pressure at T_0.