### Clausius-Clapeyron Equation

Starting from the Clapeyron Equation:

$\frac{dP}{dT} = \frac{h_l-h_v}{T(v_l-v_v)}$

We can note that, in general:

Taking $v_v$ from the ideal gas law $(v_v = \frac{RT}{P}$), and
combining

$\frac{dP}{dT} = \frac{h_l-h_v}{T(-v_v)} =
\frac{-P(h_l-h_v)}{RT^2} = \frac{P(h_v-h_l)}{RT^2} =
\frac{P\Delta h_v}{RT^2}$

rearranging and integrating

$\int \frac{dP}{P} = \int \frac{\Delta
h_v}{R}\frac{dT}{T^2}$

$\ln P = \frac{-\Delta h_v}{R}\frac{1}{T} + Const$

##### Note:

This suggests that a plot of saturation pressure, $P_{sat}$
versus $\frac{1}{T}$ should yield a straight line. This is close
to correct for most materials.

Perhaps a more useful way to use this equation is to know the
saturation pressure at a particular temperature (say, the normal
boiling point ... i.e., $P_{sat}=1 bar$):

$\ln \frac{P_2}{P_1} = \frac{-\Delta h_v}{R}\left
(\frac{1}{T_2} -\frac{1}{T_1} \right )$

##### Definition:

The **Clausius-Clapeyron equation** is a special
case of the Clapeyron equation that assumes ideal gas behavior for
the vapor and exploits the fact that the specific volume of a vapor
is substantially larger than that of a liquid.

##### Outcome:

Use the Clapeyron equation and/or the Clausius-Clapeyron
equation to relate T and P for pure species phase equilibrium