Another Application of the Gibbs Phase Rule
Determine the number of degrees of freedom for each of the following mixtures.
a. A one-component vapor-liquid mixture
b. A nonreacting two-component vapor-liquid mixture
c. Avapor-liquid mixture of ortho-, meta-, and para-xylenes and ethylbenzene at temperatures high enough that the xylenes can undergo isomerization
a. This system has one component (C = 1) and two phases (P = 2), and there are no chemical reactions (M= 0). Therefore,
F = C – M – P +2=1-0-2+2=1Consequently, this system has one degree of freedom. If the temperature is set, the pressure is fixed; or if the pressure is set, the temperature is fixed. We see this when boiling water in a pot open to the atmosphere. At 101.3 kPa (1 atm), the temperature of the boiling water will be 100◦C and will remain at this temperature no matter how much of the water boils away, provided the water is pure. In order to change the temperature of this vapor-liquid mixture, the pressure must change.
b. This system has two components (C = 2) and two phases (P = 2), and there are no chemical reactions occurring (M= 0). The number of degrees of freedom in this system is
F = C – M – P +2=2-0-2+2=2Therefore, the values of two state parameters—for example, temperature and pressure, temperature and the mole fraction of one of the species, or pressure and the mole fraction of one of the species—must be set to fix the state of this mixture. This suggests that at a fixed pressure the boiling temperature of the mixture will be a function of its composition. To see the implication of this, consider the experiment of preparing a mixture of two species (composition initially known) at one atmosphere and heating this mixture to its boiling point, and removing the vapor as the boiling continues. For most mixtures (an azeotropic mixture, to be discussed in Sec. 10.2, is the exception) as boiling occurs the composition of the vapor will be different from that of the prepared mixture, so that (by a mass balance) the composition of the remaining liquid will change during the boiling process (unless the vapor is continually condensed and replaced). As a result, at fixed pressure, the boiling temperature of this mixture will continually change as the process of boiling continues. This behavior is different from the boiling of a one-component mixture considered above, in which the temperature remains constant as the boiling process continues at fixed pressure. Also, by changing the composition of this mixture, a range of equilibrium temperatures can be obtained at the same pressure, or a range of equilibrium pressures can occur at a fixed temperature.
c. There are three independent reactions for this system. One set of such independent reactions is
\begin{array}{l}m \text {-xylene } \leftrightarrow o \text {-xylene } \\m \text {-xylene } \leftrightarrow p \text {-xylene } \\m \text {-xylene } \leftrightarrow \text { ethylbenzene }\end{array}This system has four components (C = 4) and two phases (P = 2), and there are three independent chemical reactions occurring (M= 3). The number of degrees of freedom in this system is
F = C – M – P +2=4-3-2+2=1Therefore, specification of the value of only one state variable—temperature, pressure, or the mole fraction of one of the species in one of the phases—completely fixes the twophase state of this mixture. For example, consider the experiment of preparing a mixture of these species, and heating the mixture under pressure to a temperature that is high enough that vaporization and chemical reaction occur. In this mixture, once such a temperature is fixed, the pressure, liquid composition, and vapor composition are all fixed. If an additional amount of one of the components is added to this mixture, the total number of moles of vapor and liquid will change, but the pressure, the vapor mole fractions, and the liquid mole fractions will not change. Also for this mixture, as with a pure component, the vaporliquid equilibrium temperature will change when the pressure is changed, but will remain constant as boiling occurs and the vapor is removed.