## Multiple phase systems

We proceed with the case of water in both its liquid and vapor forms in equilibrium in a container. This is a one-component (only one chemical species is present) system with two phases (liquid and gas) in equilibrium. Experience tells us that the two phases can coexist in equilibrium in this configuration. In fact, we have seen that for a given mass of the substance there is a range of values of volume for which the equilibrium exists with transfers of mass from one phase to the other as the volume is changed (at constant pressure and temperature). This is the horizontal line CB in Figure 5.2. Let the temperature and pressures be T0 andp0 along CB. In the T-p plane this line is a single point (To,po), see Figure 5.5. If we were to make an infinitesimal change in the temperature reservoir to T0 + AT (see Figure 5.6), then we would move to a higher horizontal line in Figure 5.2, thereby operating at

1 Callen gives an expression for the entropy of a van der Waals gas: S(u, v) = vR* ln ((v - b) (u + a/v)c) + vs0

where c is the molar heat capacity at constant pressure. For water vapor, a = 0.544 Pam6, b = 30.5 x 10-6 m3, and c = 3.1.

Figure 5.5 A point in the T-p plane in which liquid and vapor are in equilibrium.

P Po

Figure 5.6 As the temperature is increased from T0 to T0 + AT, the saturation vapor pressure will increase fromp0 to p0 + Ap.

(po + Ap, T0 + AT). As we change from one flat line in the V-p plane, we trace out a new curve in the T-p plane. Let us call it pequii (T).

Along this curve, p = pequu (T), the two phases can exist in equilibrium. In fact, if T and p lie on the curve (i.e., p = pequu(T)) then the volume can be varied isothermally and isobarically causing mass to transfer from one phase to the other until one of the phases is exhausted. The point in Figure 5.5 lies between points B and b in Figure 5.7. The variation can be thought of as into or out of the T-p plane along the V (volume) axis.

The upshot of all this is that when the two phases are together in equilibrium there will be a unique curve in the T-p plane. This line is of great interest to us. For example its slope tells us how much the saturation vapor pressure will increase for a small change in the temperature.

Water can form ice as its solid phase. It turns out that a single-component system such as pure water can coexist in all three phases simultaneously only at a single point in the phase diagram called the triple point. The triple point for water is 273.16 K at a pressure of 6.11 hPa. At pressures below 6.11 hPa ice and vapor can

Figure 5.7 A schematic phase diagram in the T-p plane of the phases of water. Below the line ABC the phase is vapor. To the left of ABD the phase is solid. Above DBC the phase is liquid. These lines are called phase boundaries since along them two phases can coexist. The point B is the so-called triple point since all three phases can coexist at this point. The dashed line ab indicates atmospheric pressure. The boiling point is at b.

Figure 5.7 A schematic phase diagram in the T-p plane of the phases of water. Below the line ABC the phase is vapor. To the left of ABD the phase is solid. Above DBC the phase is liquid. These lines are called phase boundaries since along them two phases can coexist. The point B is the so-called triple point since all three phases can coexist at this point. The dashed line ab indicates atmospheric pressure. The boiling point is at b.

coexist, but no liquid water will exist in equilibrium. There are many other phases of water in its different crystalline forms, and these are important in high pressure situations well outside the range encountered in atmospheric applications.

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