Application of the Clapyron equation to vapor-liquid equilibria is identical to that for vapor-solid equilibrium, so only the former is presented. When one of the phases is a vapor, its molar volume is so much larger than that of the condensed phase that the latter can be neglected in Avvap. In addition, assuming the vapor to behave ideally is generally adequate. With these two approximations the volume change on vaporization is:
Avtr = Avvap = vg - vl S vg S RT/psat where psat is the vapor pressure, or saturation pressure, at temperature T.
Substituting this equation into Eq (5.5) and identifying Ahtt with the enthalpy of vaporization Ahvap yields:
dlnPsat _ Ahvap dT RT2
This relation is the Clausius-Clapyron equation. Its integration requires knowledge of the temperature dependence of the enthalpy of vaporization. Assuming that the heat capacities of the liquid and vapor are independent of temperature, the enthalpy of vaporization is given by a formula analogous to the enthalpy of melting (Eq. (3.2)):
where ACP = Cpg - CPL and Tref is an arbitrary reference temperature.
To assess the importance of the second term in Eq (5.7), consider water between 0 and 100oC. The reference state is chosen as the normal boiling point (i.e., where the vapor pressure is 1 atm, or TR = 373 K). At this temperature, Ahvap(373) = 41.1 kJ/mole. The heat capacities of liquid water and water vapor are 75 and 33 J/mole-K, respectively. Substituting these values into Eq (5.7) for T = 273 K shows that Ahvap changes by about 10% over this 100 degree temperature range. For most other higher-melting inorganic materials (metals and ceramics), the heat of vaporization is much larger than that of water and the second term on the right in Eq (5.7) is only a few percent of the first term. Thus, over a substantial temperature interval, the temperature dependence of Ahvap can be neglected and Eq (5.6) can be integrated from the reference state at Tref to temperature T, resulting in:
Problems 5.3, 5.6 and 5.7 utilize this form of the Clausius-Clapyron equation in typical applications. Equation (5.8) is often written the more compact form that avoids the need to specify a reference state:
psat,w is expressed in atm and T in Kelvins.
A plot of the logarithm of the vapor pressure against the reciprocal of the absolute temperature should be a straight line with a slope of -Ahvap/R. Figure 5.1 shows such a plot for water. The line is nearly straight, but the ~ 10% change in slope over the 100oC temperature range is discernible. Ignoring this slope change, the coefficient values for water are:
Fig. 5.1 The vapor pressure of water as a function of temperature
For most other materials, the approximate version of Eq (5.8) or (5.9) is adequate. These vapor pressure -temperature relations are specified by either of two pairs of constants:
- for Eq (5.8): Ahvap and psat,ref , the vapor pressure at reference temperature Tref
- for Eq (5.9) The constants A and B, with psat usually expressed atm
Equations identical in form to (5.8) and (5.9) apply to sublimation, with Ahsub in place of Ahvap. Problem 5.3 provides an exercise in simultaneous application of the sublimation and vaporization equilibrium pressure equations.
The very large range of vapor pressures of the elements is illustrated in Fig. 5.2. For clarity of presentation, the abcissa is temperature rather than 1/T (as suggested in
Eq (5.9)). However, if these curves were plotted in the form of Eq (5.9), they would be converted to straight lines.
When the ACP term in Eq (5.7) is too large to be neglected, integration of Eq (5.6) leads to an additional term in Eq (5.9) (see Problem 5.1). Other applications of the more accurate vapor-pressure formula are given in Problems 5.2, 5.8 and 5.9 and in Chap. 11. The temperature dependence of the enthalpy of vaporization caused by the ACP term in Eq (5.7) also complicates analysis of processes in which a condensed phase (i.e., a liquid or solid) undergoes both a phase transition and a temperature change. Problem 5.8 illustrates this situation.
Was this article helpful?
Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.