THE PHASE RULE
9.6 TWO-COMPONENT PHASE DIAGRAMS
TWO-COMPONENT PHASE DIAGRAMS 135 The mole fraction of a two-component system runs from 0 to 1.0 (Fig. 9.4) where nB is the number of moles of component B and nA+nBis the total number of moles of both components.
9.6.1 Type 1
In a two-component one-phase Type I diagram, pressure is taken as constant. Setting p=const reduces the number of degrees of freedom by 1: f =3−P =2 for 1 phase. (Please do not confuse p and P.) The system can be represented in two dimensions as in Fig. 9.8. If one phase is present, the system can exist at a temperature and mole fraction represented by any point (T,XB) in either area above or below the coexistance curves. The upper and lower curves represent the locus of mole fractions of the vapor (upper) and liquid (lower), respectively, in equilibrium with one another at a temperature specified on the vertical axis. A system at any temperature and mole fraction between the coexistence curves splits into two phases, a vapor (upper curve) and a liquid (lower curve).
When there are two phases in equilibrium (liquid and vapor), the number of degrees of freedom is further reduced ( f =3−P=3−2=1); and specifying XB
of the system, as the mole fraction in the liquid phase or the mole fraction in the vapor, automatically specifies the temperature on one of the two coexistence curves.
These two temperatures will not be the same. The lower curve in Fig. 9.8 is the locus of points at which a trace of vapor is in equilibrium with liquid. The upper curve in Fig. 9.8 is the locus of points at which a trace of liquid is in equilibrium with vapor.
The horizontal lines connecting them represent the l→v or v→l phase change at any specific temperature, T (K).
Fractional distillation is one of our most important laboratory and industrial meth- ods of chemical purification. The separation between the upper and lower curves in a Type I diagram makes fractional distillation possible. At any temperature, the mole fraction of component B in the mixture can be read along the horizontal in Fig. 9.8.
As seen from the figure, the mole fractions are different for coexisting liquid and vapor phases except at the end points of the curves. If we allow a liquid mixture of
XB T (K)
FIGURE 9.8 A Type I phase diagram. Liquid–vapor equilibriums are expressed by each of the three horizontal lines.
A and B to come to equilibrium with its vapor and then separate and condense the vapor, we shall have two solutions, one richer in A than the original solution and one richer in B. Three such equilibrations are represented in Fig. 9.8 as horizontal lines at different temperatures T, between the two T vs. XBcurves. If we start with a solution of A and B, well to the right of the figure, and allow it to come to equilibrium with its vapor, the composition of the vapor is given by the leftmost terminus of the top horizontal. Now separate that vapor from the original solution and condense it. The new liquid is richer in A than the original solution was.
A second equilibration and separation gives a solution still richer in A according to the middle horizontal in Fig. 9.8. A third repetition yields a vapor still richer in A at the lowest temperature. Now the purified concentration is given by the leftmost terminus of the three equilibration steps diagrammed in Fig. 9.8.
Of course, real distillations are not carried out by such a laborious stepwise process of equilibration and condensation. Nevertheless, a process very like this, of continual vaporization equilibration followed by condensation, does take place in the distillation columns we actually use. Real distilling columns may range in height from a hand’s breadth for use in the laboratory to columns several stories tall used in the petroleum industry. By comparing the composition of the input with that of the effluent, the number of theoretical plates of a “still” can be calculated. Within normal practical considerations, the more theoretical plates, the better the still.
9.6.2 Type II
Type II phase diagrams describe liquid–liquid systems in which the components are completely miscible at some temperatures but undergo phase separation at others.
Systems with a composition and temperature above the dome-shaped coexistence curve in Fig. 9.9 are completely miscible. Those with a temperature and composition below the curve split into two phases. The transition can be brought about by cooling a miscible solution so that the temperature drops along a vertical and enters the two- phase zone or it can be brought about at constant temperature by adding one or the
XB T, K
A B
FIGURE 9.9 A Type II phase diagram. Solutions having a composition and temperature on the line split into an A-rich phase and a B-rich phase. The horizontal tie line is unique at each temperature.
COMPOUND PHASE DIAGRAMS 137 other of the components to a miscible solution until the coexistence curve is crossed.
The transition from a clear miscible system to an opaque two-phase emulsion can be quite dramatic in some cases, making it look as though water has turned into milk. Like Type I diagrams, the number of degrees of freedom is 3. We usually hold p=const at 1 atm so that the rest of the phase behavior can be represented in two dimensions. On the dome-shaped coexistence curve, there is 1 degree of freedom.
The horizontal under the dome is called a tie line. The two intersections of the tie line with the coexistence curve give the composition of the two coexisting phases.
The dome need not be symmetrical. Quite a variety of shapes are possible, including one that is closed at the top and at the bottom, forming a closed irregular oval that is essentially an island of immiscibility in a sea of miscibility. The coexistence curve can approach the verticals representing pure A and pure B quite closely, giving rise to the folk saying “oil and water don’t mix.” Actually, they do mix but one phase is overwhelmingly oil-rich and the other is overwhelmingly water-rich. That is why one does not wish to drink water that has come into contact with oil or gasoline.
9.6.3 Type III
Type III solid–liquid phase diagrams are familiar as having a eutectic point. The locus of melting points of mixtures of A and B vs. XBfollows the two curves in Fig. 9.10.
In general, mixtures of the two components have a lower melting point than either component, A or B, alone. Ordinary electrical solder is a eutectic mixture of lead and tin having a melting point that is lower than either pure Pb or pure Sn. We rely on the fact that the melting temperature of a pure compound is higher than that of an impure (mixed) sample of the same compound according to Fig. 9.10. Generations of pre-medical students have been judged partly on the basis of the melting point behavior of the compounds they prepare in the organic chemistry laboratory.