Clapeyron’s equation and application to vaporization

4.6 Clapeyron’s equation and application to vaporization

For the vaporization (liquid \(\rightarrow\) gas) \(dp/dT\) is always positive, since \(\Delta V\) is positive in any case. For the integration of Eq. (4.5) we will use three approximations:

Approximation 1: Neglecting \(V(liquid)\) we get \(\Delta V = V(vapor)\).
Approximation 2: The vapor is a perfect gas.
Approximation 3: The \(T\)-dependence of \(C_p\) will be neglected.
(Approximation 4): Instead of Approximation 3 sometimes Trouton’s rule (cf. Eq. (3.19)) is used: \(\Delta S_{vap,m} = \Delta H_{vap,m}/T = 85\) J/(mol K) is used.

\begin{equation*} \begin{split} \frac{dp}{dT} = &\;\;\frac{\Delta H_{vap,m}}{T\,\Delta V_{vap,m}} \quad \mbox{with} \quad \Delta V_{vap,m} \approx V_{vap,m} \approx \frac{RT}{p}\\ \Rightarrow \frac{d \ln p}{dT} = &\;\;\frac{\Delta H_{vap,m}}{R T^2} \quad \Rightarrow \quad p_2 = p_1 \, \exp\left[ - \frac{\Delta H_{vap,m}}{R} \left(\frac{1}{T_2} - \frac{1}{T_1} \right) \right] \end{split} \label{eq:dpdT_Clapeyron_vap_1} \end{equation*}

(4.8)

The last line is the Clausius-Clapeyron equation and holds for \(\Delta H_{vap,m}\) to be constant.
If we instead assume Kirchoff’s law, i.e. \(\Delta H_{vap,m} = \Delta H_{vap,m}^0 + \Delta C_p\left(T-T^0 \right)\) we find

\begin{equation*} \begin{split} \frac{d \ln p}{dT} = &\;\; \frac{\Delta H_{vap,m}^0 + \Delta C_p\left(T-T^0 \right)}{R T^2} \\ \Rightarrow \ln p = &\;\; \left(\frac{- \Delta H_{vap,m}^0 + \Delta C_p \,T^0}{R}\right) \frac{1}{T} +\frac{\Delta C_p}{R} \ln T + const. = \frac{A}{T} + B \ln T + C\\ \end{split} \label{eq:dpdT_Clapeyron_vap_2} \end{equation*}

(4.9)

For many materials extended tables for the parameters \(A\), \(B\), and \(C\) are available.