1

3.9 Azeotropic binary mixtures

If total pressure data are available for the entire concentration range of a binary solution at constant temperature, and the total pressure is a maximum at the composition x1 = a, it can be shown that

at the composition x1 = a, this solution has an azeotrope, i.e., that the relative volatility, , at this

composition is unity.

and = 1. This means that the conditions for azetrope formation

are:1. dP/dx = 0 at a given x1 = a = y1

or

2. dP/dx = 0 and = = 1

If the vapour phase is ideal this is easy to prove (see problem VII).

3.10. Principles of Phase Equilibrium

As we now wish to apply the principles of thermodynamics to phase equilibrium, let us consider the meaning of an equilibrium state as applied to a multicomponent, multiphase system comprised of nonreacting substances. As an example, suppose we introduce liquid water and liquid ethanol into a vessel equipped with means of measuring pressure and temperature and of withdrawing samples of liquid and vapor for analyses. We now place the vessel in a bath maintained at a specified temperature. Within the vessel water and ethanol will begin to distribute themselves between the liquid and vapor phases. This distribution process involves vaporization and diffusion; it is usually slow but can be accelerated by mechanical agitation. When diffusion has ceased, the system will be in an equilibrium state which can be verified when successive readings of temperature, pressure, liquid mol fractions, and vapor mol fractions are identical. This equilibrium state is reproducible, and according to the phase rule, this is possible for our two-component, two-phase system when any two of its intensive properties are duplicated. Here we are saying that all the intensive properties of the system are duplicated, although the quantities of the phases, and thus the extensive properties, may differ. In applying thermodynamics to phase equilibrium, we are mainly interested in intensive properties and the relationships between them; therefore, the phase rule specifies the conditions necessary to determine the equilibrium state adequately for our purposes.

As we have already seen, the successful application of the methods of thermodynamics requires an empirical understanding of the system under study. To assist in cultivating this understanding, we first consider the various ways in which vapor-liquid equilibrium data may be presented and thereby seek to develop a physical appreciation of the variables of phase equilibrium and how they are interrelated. Often, however, the concept of phase equilibrium and the variables expressing it do not acquire physical meaning until one considers the experimental determination. The chapter therefore begins with a descriptive account of the presentation and experimental determination of phase equilibrium data. Tools for the thermodynamic treatment of phase equilibrium are then developed. Finally, the thermodynamic approach to phase equilibrium is delineated.

Gas-Liquid systems

2

In studying vapor-liquid equilibrium in a binary system, there are four intensive variables which will concern us: temperature, pressure, a single liquid mol fraction,1 and a single vapor mol fraction. Application of the phase rule to a binary system shows that the maximum number of intensive variables which can be specified is three-this occurs when only a single phase is present.

The phase rule for the system:

ph + f = k´ - z - r +2 = 4 and fmax = 3 when 1 phase is present (z = r = 0) k = 2

When a liquid and a gas phase are present f=1, and we can vary x, y, T or P.Therefore, to graphically represent the phase behavior of a binary system, a three-dimensional plot, such as shown in Fig. 3.7, is necessary .This figure is worthy of considerable study2: It shows that each phase is represented by a surface on PTxy coordinates. The upper surface represents the liquid phase, and the lower surface represents the vapor phase. When there is equilibrium between the two phases, we know that the temperatures and pressures must be equal. Therefore, the compositions of the equilibrium phases are determined by the intersection of a constant-temperature plane with a constant-pressure plane. In Fig. 3.7 a constant-temperature plane is shown by the lens-shaped envelope AB which lies in the vertical plane DAVaLaBE. This envelope is shown to intersect a constant- pressure envelope KV aLaH which lies in the horizontal plane. The intersection of the envelopes occurs in two places: on the liquid surface at the point La and on the vapor surface at the point Va. These points represent values of xI and y1 that are in equilibrium at the temperature and pressure we have selected. Equilibrium liquid and vapor compositions are also shown as Lb and Vb where the isothermal plane CIVVbLbNP corresponding to Tb intersects the isobaric plane MLVbLbCI corresponding to Pb.

1 Only a single mol fraction is required to describe the composition of a phase in a binary system because of the condition that mol fraction in a phase must sum to unity .

2 Visualization and understanding of these diagrams can be greatly enhanced through the use of interactive com- puter graphics. See G. N. Charos. P. Clancy. and K. E. Gubbins. Chem. Eng. Educ., 20. 80 (1986) and K. R. Jolls, Chem. Eng. Educ., 32, 113 (1998).

Fig. 3.7 PTxy diagram

3

Because three variables are required to describe the state of a single phase, the representation of phase behavior on a two-dimensional plot can be accomplished by holding one variable constant; three such plots are possible. These three types of plots corresponding to the three- dimensional diagram of Fig. 3.7 are shown on Figs. 3.8, 3.9, and 3.10. Unless we are concerned with equilibrium near the critical region, we will have little use for the PT plot and we will find that the envelopes on the Pxy and Txy diagrams will span the entire range of mol fractions. Usually we are interested in systems at low to moderate pressure where the constant-temperature envelope will resemble Ta in Fig. 3.8 and the constant-pressure envelope will resemble Pa in Fig. 3.9.

Consider equilibrium at a constant temperature of Ta as represented on Fig. 3.8. At equilibrium the pressure must be equal for each phase and a horizontal line ties together the saturated liquid and vapor curves. Such lines determine the equilibrium liquid and vapor composition at a given pressure and are called tie lines. The phase rule requires specification of two variables to determine a two-component, two-phase system. As we have already specified Ta only one other variable may be chosen. If we choose P, then x1 and yI are found at the ends of the tie line. If we choose either x1 or y1, we have fixed a point on either saturated curve and a tie line determines p and the mol fraction of the other phase. In Fig. 3.9, a constant-pressure diagram, tie lines are characterized by temperature. Figure 3.10 shows the curves formed by the intersection of the two constant-composition planes, TSRVaCIII and WVULaCII , with the liquid and vapor surfaces of Fig. 3.7. Note that it is possible for mixtures to have critical temperatures, CII and CIII, above those of either pure component. The intersection of the projections of the vapor portion of one curve with the liquid portion of another locates an equilibrium point, La, Va.

Fig. 3.8 Pxy diagram.

As the temperature increases there is a narrowing of the range of composition over which a liquid phase can exist. For example, from Fig. 3.7 it is seen that for the isothermal plane CIVVbLbNP corresponding to Tb component 2 is above its critical temperature and therefore cannot exist as a pure liquid-in fact, there is a maximum mol fraction of component 2 above which no liquid phase can form at this temperature. The composition range also narrows as the pressure is increased. Fig. 3.7 shows that the lens-shaped envelope formed by the intersection of the constant-pressure plane, MLVbLbCI at Pb with the liquid and vapor surfaces does not extend to x1= YI = 1. This limited composition range can also be discerned from Fig. 3.10 where it is observed that the pressure Pb lies above the vapor pressure curve for component 1. As the temperature or pressure continues to increase, the composition range steadily diminishes. Often, a point is reached where neither component can exist as pure conjugate liquid and vapor phases.

4

The preponderance of experimentally determined vapor-liquid equilibrium data is re- ported at either constant temperature or pressure as either of these constraints can be easily imposed on an experimental determination. Unless we are dealing with systems following ideal solution behavior where the phase behavior may be calculated from pure component properties, experimental data are needed to construct the phase diagrams.

Fig. 3.10 PT diagram

Let us consider some simple Pxy and Txy diagrams that represent different type of binary interactions. Some typical diagrams are shown in Figs. 3.11-3.12

Fig. 3.9 Txy diagram

P

Fig 3.11 Fig. 3.12

5

The azeotropic diagrams with max P or T and min P or T. How to calculate the azeotropic

composition and the equilibrium conditions have been described previously.

Solid-Liquid systems

The phase diagrams, Gibbs free energy, and thermodynamic activity for liquid-solid systems.

Complete mutual solid solubility of the components A and B requires that A and B have the same

crystal structures, be of comparable atomic size, and have similar electronegativities and valencies.

If anyone of these conditions is not met, then a miscibility gap will occur in the solid state. Consider

the system A-B, the phase diagram of which is shown in Fig. 3.13 a, in which A and B have

differing crystal structures. Two terminal solid solutions, and , occur. The molar Gibbs free

energy of mixing curves, at the temperature T1 are shown in Fig. 3.13b. In this figure, a and c,

located at GM = 0, represent, respectively, the molar Gibbs free energies of pure solid A and pure

liquid B, and b and d represent, respectively, the molar Gibbs free energies of pure liquid A and

pure solid B. The curve aeg (curve I) is the Gibbs free energy of mixing of solid A and solid B to

form homogeneous a solid solutions which have the same crystal structure as has A. This curve

intersects the XB = 1 axis at the molar Gibbs free energy which solid B would have if it had the

same crystal structure as has A. Similarly, the curve dh (curve II) represents the Gibbs free energy-

of mixing of solid B and solid A to form homogeneous solid solutions which have the same

crystal structure as has B. This curve intersects the XA = 1 axis at the molar Gibbs free energy which

A would have if it had the same crystal structure as B. The curve bfc ( curve III) represents the

molar Gibbs free energy of mixing of liquid A and liquid B to form a homogeneous liquid solution.

As curve II lies everywhere above curve III, solid solutions are not stable at the temperature T1.

The double tangent to the curves I and III identifies the solidus composition at the temperature T1

as and the liquidus composition as f. Fig. 3.13 c shows the activity-composition relationships of

the components at the temperature T1 drawn with respect to solid as the standard state for A and

liquid as the standard state for B. These relationships are drawn in accordance with the assumption

that the liquid solutions exhibit Raoultian ideality and the solid solutions show positive deviations

from Raoult's law.

As the temperature decreases below T1 the length of ab increases and the length of cd

decreases until, at T = T m(B), the points c and d coincide at GM = 0. At T2 < Tm(B) the point c (liquid

B) lies above d in Fig. 3.14b, and, as curve II lies partially below curve III, two double tangents can

be drawn: one to the curves I and III, which defines the compositions of the solidus and its

conjugate liquidus, and one to the curves II and III, which defines the compositions of the solidus and its conjugate liquidus. The activity-composition curves at T 2 are shown in Fig. 3.13 c, in which

the solid is the standard state for both components.

6

With further decrease in temperature the two liquidus compositions, m and n in Fig. 3.14b,

approach one another and, at the unique temperature, TE the eutectic temperature, they coincide,

which means that the two double tangents merge to form the triple tangent to the three curves

shown in Fig. 3.15b. At compositions between o and p in Fig. 3.15b a doubly saturated eutectic

liquid coexists in equilibrium with and solid solutions. From the Gibbs phase rule discussed

earlier, this three-phase equilibrium has one degree of freedom, which is used to specify the

pressure of the system. Thus, at the specified pressure, the three-phase equilibrium is invariant.

Fig.3.15c shows the activities of A and B at T E. At T3 < TE curve III lies above the double tangent to

curves l and II, and thus the liquid phase is not stable. This behavior and the corresponding activity-

composition relationships are shown, respectively, in Figs. 3.16b and c.

If the ranges of solid solubility in the and phases are immeasurably small, then, as a

reasonable approximation, it can be said that A and B are insoluble in one another in the solid state.

The phase diagram for such a system is shown in Fig.3.17a. As all Gibbs free energy of mixing

curves have vertical tangents at their extremities, any pure substance presents an infinite chemical

sink to any other substance, or conversely, it is impossible to obtain an absolutely pure substance.

As the range of solid solubility in Fig.3.17a is so small that it may be neglected on the scale of Fig.

3.17a, then also the Gibbs free energy curves for formation of and (curves I and II in Figs.

3.13-3.16) are so compressed toward the XA = 1 and XB = 1 axes, respectively, that on the scale of

Figs. 3.13-3.16, they coincide with the vertical axes. The sequence in Fig. 10.16 shows how, as the

solu- bility of B in a decreases, the Gibbs free energy curve for a is compressed against the X A = 1

axis. The Gibbs free energy of formation of the liquid solutions in the system A-B at the temperature

T is shown in Fig.3.17b. The "double tangent" to the solid solution and liquid solution curves is

reduced to a tangent drawn from the point on the XA = 1 axis which represents pure solid A to the

liquid solutions curve. The corresponding activity-composition relations are shown in Fig.3.17c.

Again these are drawn in accordance with the assumption of ideal liquid solutions. In Fig. 3.17c pqr

is the activity of A with respect to pure solid A at p, s is the activity of pure liquid A with respect to

solid A at p, str is the activity of A with respect to liquid A having unit activity at s and Auvw is the

activity of B with respect to liquid B having unit activity at w.

In a binary system which exhibits complete miscibility in the liquid state and virtually

complete immiscibility in the solid state, e.g., Fig.3.17a, the variations of the activities of the

components of the liquid solutions can be obtained from consideration of the liquidus curves. At

any temperature T (Fig.3.17a), the system with a composition between pure A and the liquidus

composition exists as virtually pure solid A in equilibrium with a liquid solution of the liquidus

composition. Thus, at T

+ RT ln aA

in which aA is with respect to liquid A as the standard state. Thus

= - RT ln aA (3.64)

or, if the liquid solutions are Raoultian,

= - RT ln XA (3.65)

7

Consider the application of Eq. (3.65) to calculation of the liquidus lines in a binary eutectic system.

In the system Cd-Bi, the phase diagram for which is shown in Fig. 10.17 , cadmium is virtually

insoluble in solid bismuth, and the maximum solubility of bismuth in solid cadmium is 2.75 mole

percent at the eutectic temperature of 419 K. If the liquidus solutions are ideal, the Bi liquidus is

obtained from Eq. (3.65) as

= - RT ln XBi(liquidus)

= 10,900 J at Tm(Bi) = 554 K, and thus

= 20.0 J/K at 544 K

Figures 3.13-3.16 The effect of temperature on the molar Gibbs free energies of mixing and the activities of the components of the system A-B

The molar constant pressure heat capacities of solid and liquid bismuth vary with temperature as

Cp,Bi(s) = 18.8 + 22.6 x 10-3 TJ/K

Cp,Bi(l) = 20 + 6.15 x 10-3 T + 21.1 x 105T-2 J/K

Thus

Cp,Bi(l) - Cp,Bi(s) = Cp,Bi = 1.2- 16.45 X 10-3 T+ 21.1 x 105 T -2J/K

3.13 3.14 3.15 3.16 3.15 3.16

8

and

= 16,560 -23.79T -1.2T ln T + 8.225x 10-3 T2 -10.55 x 105T -I

= -RT In X Bi(liquidus) (3.66) or

ln XBi(liquidus) = -:1992/T + 2.861 + 0.1441n T- 9.892 x 10-4 T + 1.269 x 105/T2

Figure 3.17 The molar Gibbs free energy of mixing and the activities in a binary eutectic system that exhibits complete liquid miscibility and virtual complete solid immiscibility

Figure 3.18 The effect of decreasing solid solubility on the molar Gibbs free energy of mixing curve

This equation is drawn as the broken line (i) in Fig. 3.19.

Similarly, if the small solid solubility of Bi in Cd is ignored,

= - RT ln XCd(liquidus)

= 6,400 J at Tm(Cd) = 594 K, and thus

= 6400/594 = 10.77 J/K at 594 K.

The constant pressure molar heat capacities are

Cp.Cd(s) = 22.2 + 12.3 x 10-3 TJ/K

Figure 3.19 The phase diagram for the system Bi-Cd. The full lines are the measured liquidus lines, and the broken lines are calculated assuming no solid solution and ideal mixing in the liquid solutions.

and

Cp.Cd(/) = 29.7 J/K

Thus

Cp.Cd(l) -Cp,Cd(s) = Cp.Cd = 7.5 -12.3 x 10-3 T J/K

= 4155 + 37.32T -7.5T ln T + 6.15 x 10-3T2 J (3.67) = -RT ln XCd(liquidus) (3.67)

or ln XCd(liquidus)= -495/T - 4.489 + 0.90 ln T- 7.397 x 10-4T

which is drawn as the broken line (ii) in Fig. 3.19. Lines (i) and (ii) intersect at the composition of the Raoultian liquid which is simultaneously saturated with Cd and Bi and at 406 K, which would

be the eutectic temperature if the liquids were ideal. The actual liquidus lines lie above those calculated, and the actual eutectic temperature is 419 K. From Eq. (3.66), = 2482 J,

and from Eq. (3.67), = 1898 J. Thus, from Eq. (3.64), in the actual eutectic melt,

aBi = exp = 0.49

and

aCd = exp = 0.58

The actual eutectic composition is XCd = 0.55, XBi = 0.45, and thus the activity coefficients are

= 1 09

and

= 1 05 .

Thus, positive deviations from Raoultian ideality cause an increase in the liquidus temperatures. It is now of interest to examine what happens to the liquidus line as the magnitude of the

positive deviation from Raoultian behavior in the liquids increases, i.e.,

as GXS becomes increasingly positive. Assuming regular solution behavior, Eq. (3.64), written in the form

- = RTln XA + RTln A becomes

- = RTln XA + RT(l -XA)2 (3.68)

Consider a hypothetical system A-B in which = 10 kJ at Tm,A = 2000 K. Thus, for this system is -10,000 + 5T = RT ln XA + (1 -XA)2 since So

m(A) = -10.000/2000= -5.

Here XA is the composition of the A liquidus at the temperature T. The A liquidus lines, drawn for = 0, 10,20,25.3,30,40, and 50 kJ, are shown in Fig. 3.20. As exceeds some critical value (which is 25.3 kJ in this case), the form of the liquidus line changes from a monotonic decrease in liquidus temperature with decreasing XA to a form which contains a maximum and a minimum. At the critical value of the maximum and minimum coincide at XA = 0.5 to produce a horizontal inflexion in the liquidus curve. It is apparent that, when exceeds the critical value, isothermal tie-

lines cannot be drawn between pure solid A and all points on the liquidus lines, which, necessarily, means that the calculated liquidus lines are impossible. From Eq. (3.64) it can easily be shown that

ln aA = =

Thus

d ln aA = (3.69)

or

(3.70)

and also

(3.71)

cr = 25.4 kJ and the horizontal inflexion in the critical liquidus curve occurs at XA = 0.5, T = 1510 K. This is in agreement with the limit for immiscibility observed for regular solutions:A/RT = /RT= 2 since 25.4 / (8.314*1510) = 2.

Inflection points are observed when

This is obtained for = 2RT as we found earlier

when we used the thermodynamic criterion

for incipient instability

. = 0 and =0

Figure 3.20 Calculated liquidus lines assuming regular solution behavior in the liquid solutions and no solid solubility.