Thermodynamics

808 | ThermodynamicsFIGURE 16–17Wet clothes hung in an open areaeventually dry as a result of masstransfer from the liquid phase to thevapor phase.© Vol. OS36/PhotoDiscT, PVAPORm gm fLIQUIDFIGURE 16–18A liquid–vapor mixture in equilibriumat a constant temperature and pressure.16–6 ■ PHASE EQUILIBRIUMWe showed at the beginning of this chapter that the equilibrium state of asystem at a specified temperature and pressure is the state of the minimumGibbs function, and the equilibrium criterion for a reacting or nonreactingsystem was expressed as (Eq. 16–4)1dG2 T,P 0In the preceding sections we applied the equilibrium criterion to reactingsystems. In this section, we apply it to nonreacting multiphase systems.We know from experience that a wet T-shirt hanging in an open areaeventually dries, a small amount of water left in a glass evaporates, and theaftershave in an open bottle quickly disappears (Fig. 16–17). Theseexamples suggest that there is a driving force between the two phases of asubstance that forces the mass to transform from one phase to another. Themagnitude of this force depends, among other things, on the relativeconcentrations of the two phases. A wet T-shirt dries much quicker in dryair than it does in humid air. In fact, it does not dry at all if the relativehumidity of the environment is 100 percent. In this case, there is no transformationfrom the liquid phase to the vapor phase, and the two phases arein phase equilibrium. The conditions of phase equilibrium change, however,if the temperature or the pressure is changed. Therefore, we examinephase equilibrium at a specified temperature and pressure.Phase Equilibrium for a Single-Component SystemThe equilibrium criterion for two phases of a pure substance such as wateris easily developed by considering a mixture of saturated liquid and saturatedvapor in equilibrium at a specified temperature and pressure, such asthat shown in Fig. 16–18. The total Gibbs function of this mixture isG m f g f m g g gwhere g f and g g are the Gibbs functions of the liquid and vapor phases perunit mass, respectively. Now imagine a disturbance during which a differentialamount of liquid dm f evaporates at constant temperature and pressure.The change in the total Gibbs function during this disturbance is1dG2 T,P g f dm f g g dm gsince g f and g g remain constant at constant temperature and pressure. Atequilibrium, (dG) T,P 0. Also from the conservation of mass, dm g dm f .Substituting, we obtain1dG2 T,P 1g f g g 2 dm fwhich must be equal to zero at equilibrium. It yieldsg f g g(16–19)Therefore, the two phases of a pure substance are in equilibrium when eachphase has the same value of specific Gibbs function. Also, at the triple point(the state at which all three phases coexist in equilibrium), the specificGibbs functions of all three phases are equal to each other.