Use of the Clausius-Clapeyron Equation
The vapor pressure of liquid 2,2,4-trimethyl pentane at various temperatures is given below. Estimate the heat of vaporization of this compound at 25°C.
| Vapor pressure (kPa) | 0.667 | 1.333 | 2.666 | 5.333 | 8.00 | 13.33 | 26.66 | 53.33 | 101.32 |
| Temperature (°C) | -15.0 | -4.3 | 7.5 | 20.7 | 29.1 | 40.7 | 58.1 | 78.0 | 99.2 |
Over a relatively small range of temperature (say from 20.7 to 29.1°C), \Delta_{ vap } \underline{H} may be taken to be constant. Using Eq. 7.7-6, we obtain
\ln \frac{P^{ vap }\left(T_{2}\right)}{P^{\operatorname{vap}}\left(T_{1}\right)}=-\frac{\Delta_{ vap } \underline{H}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right) (7.7-6)
\frac{\Delta_{ vap } \underline{H}}{R}=\frac{-\ln \left[P^{ vap }\left(T_{2}\right) / P^{ vap }\left(T_{1}\right)\right]}{\frac{1}{T_{2}}-\frac{1}{T_{1}}}=\frac{-\ln (8.000 / 5.333)}{\frac{1}{302.25}-\frac{1}{293.85}}=4287.8 K
so that
\Delta_{ vap } \underline{H}=35.649 kJ / mol
One can obtain an estimate of the temperature variation of the heat of vaporization by noting that the integration of Eq. 7.7-5 can be carried out as an indefinite rather than definite integral. In this case we obtain
\frac{d \ln P^{\text {vap }}}{d T}=\frac{\Delta_{\text {vap }} \underline{H}}{R T^{2}} (7.7-5a)
\ln \frac{P^{\text {vap }}\left(T_{2}\right)}{P^{\text {vap }}\left(T_{1}\right)}=\int_{T_{1}}^{T_{2}} \frac{\Delta_{\text {vap }} \underline{H}}{R T^{2}} d T (7.7-5b)
\ln P^{ vap }(T)=-\frac{\Delta_{ vap } \underline{H}}{R T}+C
where C is a constant. Therefore, if we were to plot \ln P^{\text {vap }} versus 1/T , we should get a straight line with a slope equal to -\Delta_{\text {vap }} \underline{H} / R f the heat of vaporization is independent of temperature, and a curve if \Delta_{ vap } \underline{H} varies with temperature. Figure 7.7-1 is a vapor pressure-temperature plot for the 2,2,4-trimethyl pentane system. As is evident from the linearity of the plot, \Delta_{ vap } \underline{H} is virtually constant over the whole temperature range.
