Question 6.7: The Clapeyron equation for vaporization at low pressure is o...
The Clapeyron equation for vaporization at low pressure is often simplified through reasonable approximations, namely, that the vapor phase is an ideal gas and that the molar volume of the liquid is negligible compared with the molar volume of the vapor.
How do these assumptions alter the Clapeyron equation?
For the assumptions made,
\Delta Z^{l v}=Z^v-Z^l=\frac{P^{s a t} V^v}{R T}-\frac{P^{s a t} V^l}{R T}=1-0=1
Then by Eq. (6.87),
\frac{d \ln P^{\text {sat }}}{d T}=\frac{\Delta H^{l v}}{R T^2 \Delta Z^{l v}} (6.87)
\Delta H^{l v}=-R \frac{d \ln P^{\mathrm{sat}}}{d(1 / T)}
Known as the Clausius/Clapeyron equation, this approximate expression relates the latent heat of vaporization directly to the vapor-pressure curve. Specifically, it indicates a direct proportionality of ΔH^{lv} to the slope of a plot of ln P^{sat} vs. 1/T. Such plots of experimental data produce lines for many substances that are very nearly straight. The Clausius/ Clapeyron equation implies, in such cases, that ΔH^{lv} is constant, virtually independent of T. This is not in accord with experiment; indeed, ΔH^{lv} decreases monotonically with increasing temperature from the triple point to the critical point, where it becomes zero. The assumptions upon which the Clausius/ Clapeyron equation are based have approximate validity only at low pressures.