THE PHASE RULE
9.8 TERNARY PHASE DIAGRAMS
Any mixture of three components can be represented within an equilateral triangle.
All points within the triangle represent one and only one of the infinitely many possi- ble mixtures. Because there are now three composition variables f =C−P+2= 3−P+2, both T and p are held constant for the three-component triangular rep- resentation on a two-dimensional surface. If only one phase is present, all solutions are permitted and the composition can take any point in the triangular 2-space. When two phases are present, the composition is restricted to the locus of points on the coexistence curve. Like the Type II phase diagram, many three-component phase diagrams are known having a dome-shaped coexistence curve. They are widely used for characterizing and selecting solvents used in industrial processes. Intersections of tie lines with the coexistence curve indicate the composition of phases in two-phase systems formed when the total composition point is below the coexistence curve.
In Fig. 9.12, solvents B and C are nearly completely immiscible, but addition of sol- vent A brings the composition of the two coexisting phases, the termini of the tie lines, closer together until finally there are ternary solutions of A, B, and C that are homoge- neous. These solutions correspond to ternary mixtures above the coexistence dome.
PROBLEMS AND EXAMPLES 139 A
B C
FIGURE 9.12 A ternary phase diagram with a tie line. At constant p and T, f =3− 1=2 within the two-dimensional triangular surface, but f =3−2=1 on the coexistence curve. In general, the area under the curve is smaller at higher temperatures. Numerous, more complicated forms are known.
Addition of A may cause a milky suspension of immiscible phases to suddenly clear. Conversely, addition of C to a clear stirred solution of A and B may cause it to go milky at some concentration. The transition is sudden and sharp and can be used as the end point of a phase titration.
PROBLEMS AND EXAMPLES
Example 9.1 The Enthalpy of Vaporization of H2O
Handbook values for the vapor pressures of water between 273 and 373 K are available. Plot pvapvs.−1/T for several temperatures in the vicinity of 298 K and use a commercial curve fitting routine to determine the enthalpy change for the vaporization of water at that temperature.
Solution 9.1 The handbook gives pvap at temperatures from 273 to 373 K. We have selected six values symmetrically grouped around 298 K and presented them as 1/T K−1in Table 9.1.
TABLE 9.1 Negative Inverse Temperatures and Vapor Pressures for Water.
−1/T pvap
−3.6600e−3 0.6110
−3.5300e−3 1.2300
−3.4100e−3 2.3400
−3.3000e−3 4.2500
−3.1900e−3 7.3800
−3.1000e−3 12.3400
The curve of pvapvs.−1/T for water is shown in Fig. 9.13.
-1/T (K-1)
-0.0037 -0.0036 -0.0035 -0.0034 -0.0033 -0.0032 -0.0031 -0.0030
Vapor Pressure (kPa)
0 2 4 6 8 10 12 14
FIGURE 9.13 The liquid–vapor coexistence curve of water leading to vapH (H2O)= 44.90 kJ mol−1.
The vapor pressures over pure water were selected at 10 K intervals from 273 to 323 K, plotted, and submitted to a standard curve-fitting rou- tine (SigmaPlot 11.0). A two-parameter exponential curve fit of pvap vs.
−1/T (Statistics → Nonlinear → Regression Wizard → Ex- ponential Growth → Single, 2-Parameter) gave the empirical con- stant b (Outputb 5400.6916) K, where the unit of the empirical constant is kelvins in order to make the exponent−b/T unitless. The curve fit is good, having a mean residual (difference between calculated and experimental values) of 1 part per thou- sand or 0.1% over a range of about 12 kPa. We have the relationship between the parameter b andvapH :
b= vapH R which gives us
vapH= R(b)=8.314
5.401×103 =44.90×103=44.90 kJ mol−1 This value is “averaged out” by the curve fitting technique over the temperatures symmetrically distributed around 298 K. The handbook value ofvapH of water at 298 K is 43.99 kJ mol−1.
Example 9.2 Ternary Phase Diagrams
The ternary phase diagram ABC in which A is completely miscible with B and C but B and C are only partially miscible in each other looks like a combination of Figs.
9.9 and 9.12 except that the tie lines are not horizontal.
PROBLEMS AND EXAMPLES 141 A
B C
FIGURE 9.14 A ternary phase diagram in which B and C are partially miscible. Figure 9.13 might approach this form at higher temperatures.
Starting with a solution of 0.5 mol of A and 0.5 mol of B at a constant temperature corresponding to that of phase diagram Fig. 9.14, we add component C in small portions. What happens?
Solution 9.2 The phase behavior is somewhat complicated. At the first few small increments of component C, a clear homogeneous solution of A, B, and C results, corresponding to points on a straight line from the midpoint of axis AB in the direction of apex C. Soon the coexistence curve is crossed and the solution splits into two phases. The overall composition continues along the straight line toward C as increments are added, but the system now consists of two phases corresponding to the end points of the tie line. At first, a minute amount of the second phase appears; but further along in the addition process, substantial amounts of both phases are present.
Their compositions always correspond to the end points of the tie line as it intersects the coexistence curve.
There comes a time when the short end of the tie line approaches the overall composition line. The amount of each phase is in the inverse ratio of the length of the tie line cut off by the overall composition line such that the AC-rich phase predominates over the AB rich phase. Ultimately the coexistence curve is crossed again and the solution clears, its composition corresponding to the points on the lower right of the overall composition curve approaching the C apex.
Problem 9.1
(a) How many components are there in a dilute solution of sodium acetate NaAc in water?
(b) A drop of HCl is added to the solution in part a. Now the anions of the weak acid Ac− and the strong acid Cl− are competing for the protons H+. How many components are there in this system of Na+, Ac−, H+, Cl−, H3O+, and a minimal concentration of OH−?
Problem 9.2
(a) What is the effect of a decrease in atmospheric pressure on the freezing and boiling points of water?
(b) What is the effect of a decrease in atmospheric pressure on the freezing and boiling points of benzene?
(c) Describe the behavior of a system that is carried along a horizontal line above the critical point of the phase diagram for water (Fig. 9.3).
(d) Describe the behavior of a system that is carried along a horizontal line below the critical point, but above the triple point of the phase diagram for water (Fig. 9.3).
(e) Describe the behavior of a system that is carried along a horizontal line below the triple point of the phase diagram for benzene (solid lines Fig. 9.3).
Problem 9.3
Describe the behavior of the system with a mole fraction 0.75 well above the dome- shaped coexistence curve in Fig. 9.9 as the temperature is slowly decreased.
Problem 9.4
Describe the behavior of the system with a mole fraction 0.25 well above the coex- istence curve in Fig. 9.11 as the temperature is slowly decreased.
Problem 9.5
The vapor pressure of liquid benzene is given by ln Pvap= −4110
T +18.33
in the approximate location of the triple point (McQuarrie and Simon, 1997). The sublimation pressure of solid benzene is given by
ln Psub= −5319
T +22.67 What is the triple point for benzene?
Problem 9.6
Sketch a binary phase diagram resembling Fig. 9.10.
(a) Locate the points corresponding to pure components. Call them components X and Y.
(b) Locate a point corresponding to an equimolar mixture of X and Y.
(c) Locate a point corresponding to a mixture of 20% X and 80% Y.
(d) Locate the point corresponding to the compound X2Y.
PROBLEMS AND EXAMPLES 143 Problem 9.7
Find an area with two degrees of freedom, a coexistence curve with one degree of freedom, and a point with no degrees of freedom on the phase diagram (Fig. 9.10).
Problem 9.8
Sketch a triangular phase diagram resembling Fig. 9.12.
(a) Locate the point corresponding to a binary solution containing components in equal amounts.
(b) Locate the point corresponding (approximately) to a solution with three com- ponents in the ratio 45%, 45%, 10%.
(c) Locate the point corresponding to 33.3% for each of the three components.
Problem 9.9
Sketch the three-component phase diagram for a system in which all three of the components are miscible (soluble) in all proportions X in Y, X in Z, and Y in Z, but phase separation takes place for some of the ternary mixtures XYZ.