5.2 Two-phase System

  • During phase change of a pure substance, \(T\) and \(P\) remain constant. Therefore, from the fundamental energy relation \(dG = VdP - SdT\), we can say that \(dG = 0\)

  • Clapeyron Equation \[\frac{dP^{\text{sat}}}{dT} = \frac{\Delta H^{\alpha\beta}}{T \Delta V^{\alpha\beta}}\] This equation gives the gradient of the phase boundary in the \(P\)-\(T\) plane. It applies to all changes of phase from \(\alpha\) to \(\beta\) (i.e., solid-liquid, liquid-vapor, solid-vapor).

    The Clapeyron equation enables us to determine the enthalpy change associated with a phase change, from knowledge of \(P, V\), and \(T\) data alone.

  • Clausius-Clapeyron Equation The following assumptions are made in deriving Clausius-Clapeyron equation from Clapeyron equation.

    1. \(V^v-V^l \approx V^v\) (molar volume of liquid is negligible in comparison with that of vapor).

    2. \(V^v = RT/P\) (vapor can be considered as an ideal gas).

    3. \(\lambda = \Delta H^{\alpha\beta}\) = constant. Latent heat of vaporization (\(\lambda\)) is assumed to remain constant for small changes in \(T\).

    \[ \ln \frac{P_2^{\text{sat}}}{P_1^{\text{sat}}} = \frac{\Delta H^{\alpha\beta}}{R} \left(\frac{1}{T_1} - \frac{1}{T_2}\right) \tag*{Clausius-Clapeyron equation}\] The fact that all known substances in the two-phase region fulfill the Clausius-Clapeyron equation, provides the general validity of the first and second laws of thermodynamics (as derivation of this equation is from the combination of first and second laws).

  • According to the Clausius-Clapeyron equation \(\Delta H^{\text{lv}}\) is almost constant, virtually independent of \(T\). This is not true; \(\Delta H^{\text{lv}}\) decreases monotonically with increasing temperature from the triple point temperature to the critical point, where it becomes zero. The assumptions on which the Clausius-Clapeyron equation are based, have approximate validity only at low pressures.

  • Antoine Equation

    Clausius-Clapeyron equation can be written in empirical form as \[\ln P^{\text{sat}} = A - \frac{B}{T}\] A satisfactory relation given by Antoine is of the form \[\ln P^{\text{sat}} = A - \frac{B}{T+C} \tag*{(Antoine equation)}\] The values of the constants \(A,B\) and \(C\) are readily available for many species.

  • Trouton’s Rule: Trouton’s rule states that the entropy change of vaporization is almost the same value, about 85–88 J/(mol.K), for most of the liquids at their normal boiling points. Since \(\Delta S=\Delta H/T=\lambda/T\), we can also say, that, according to this rule, molar latent heat of vaporization (\(\lambda\)) at normal boiling point is related to the boiling temperature as \[\lambda \approx 10 RT_b \tag{Trouton's rule}\] where \(T_b\) is normal boiling point in Kelvin. This empirical relationship holds good for many systems. Because of its convenience, the rule is used to estimate the enthalpy of vaporization of liquids whose boiling points are known.

  • Engineering Rule for Vapor Pressure of Water: The following simple formula, which is easy to remember, gives surprisingly good estimates of the vapor pressure for water for temperatures from 100\(^\circ\)C (the normal boiling point) and up to 374\(^\circ\)C (the critical point): \[P_{\ce{H2O}}^{\text{sat}} [\text{bar}] = \left(\frac{T[^\circ\text{C}]}{100} \right)^4\] This formula is very convenient for engineers dealing with steam at various pressure levels. For example, from the formula we estimate \(P^{\text{sat}}= 2.074\) bar at 120\(^\circ\)C (the correct value is = 1.987 bar) and \(P^{\text{sat}}= 81\) bar at 300\(^\circ\)C (the correct value is = 85.88 bar).

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