1. Chapters 8-9 Phase Diagrams

2. One component Phase Diagram Analysis using The Phase Rule

3. Example#1
The vapor pressure of HNO3 is 14 mmHg at 0 0C and 47.9 mm Hg at 20 0C. Estimate: a) the normal boiling point of HNO3 and b) its molar enthalpy of vaporization.

4. F = 2 + c − p The Gibbs Phase Rule liquid solid p = 1 F = 2
gas
solid + gas
p = 2, F = 1
solid + liquid
liquid + gas
solid +liquid+ gas
p = 3, F = 0 T
variance = number of variables − number of equations ≥ 0

5. One component Phase Diagram Analysis using The Phase Rule
In the one−phase regions one can vary either the temperature, or the pressure, or both (within limits) without crossing a phase line. We say that in these regions there is a variance of 2. We have indicated in the solid, liquid, and gas regions that there is one phase and the variance is two.
Along a phase line we have two phases in equilibrium with each other, so on a phase line the number of phases is 2. However, if we want to stay on a phase line, we can't change the temperature and pressure arbitrarily. If we change the temperature − and keep two phases − then the pressure must change also to keep us on the phase line. Otherwise we go off the line and we no longer have two phases in equilibrium. So on a phase line the number of phases is 2, but the variance is 1.
At the triple point there are three phases in equilibrium, but there is only one point on the diagram where we can have three phases in equilibrium with each other. The variance is 0, this point is completely determined.

6. Vapor Pressure Diagrams for Ideal Mixtures
Figure 1. On this diagram the dotted line that runs from "Tol VP" to lower right Bz corner is the Raoult's law vapor pressure of toluene. The dotted line that runs from the lower left Tol corner to "Bz VP" is the Raoult's law vapor pressure of benzene. The solid line from "Tol VP" to "Bz VP" is the total vapor pressure of the solution which is just the sum of the two Raoult's law vapor pressures (the sum of two straight lines is a straight line).

7. Vapor Pressure Diagrams for no Ideal Mixtures with Positive Deviations
dT=0
Raoult's law
Xi → 1, Solvent
Henry's law
Xi → 0, Solute
Ki Solubility constant
Figure 3. This vapor pressure diagram shows qualitatively how the vapor pressures vary with composition for positive deviations from Raoult's law. We have not tried to draw the vapor composition curve on the diagram. We have tried to show that each component obeys Raoult's law in the limit as its mole fraction goes to one. Also, the vapor pressures of the individual components becomes linear in mole fraction as that mole fraction approaches zero. This latter phenomenon is known as "Henry's law" and will discussed further below.

8. Vapor Pressure Diagrams for no Ideal Mixtures with Negative Deviation
dT=0
Raoult's law
Xi → 1, Solvent
Henry's law
Xi → 0, Solute
Ki Solubility constant
Figure 4. Once again we have tried to show that each component obeys Raoult's law as its mole fraction approaches one and becomes linear in its mole fraction as that mole fraction approaches zero. We have not shown the total vapor curve or the vapor composition curve. The total vapor pressure curve would just be the sum of the two individual vapor pressures. Since the solution does not obey Raoult's law it would require experimental data or a more sophisticated theory to construct the vapor composition curve. 

9. Example#1: Henry’s Law

10. Two component Boiling diagrams
Figure 5. This diagram is called a "boiling diagram." Notice that at the edges we have points representing the boiling points of the pure components. The lower curve gives the boiling point of the liquid mixture as a function of composition.

11. Heating of a Two component Mixture
Figure 6. This point gives the boiling point of a mixture which is 0.33 mole percent benzene. We can ask, what is the composition of the vapor which is in equilibrium with the liquid at this temperature? Since we are asking a question about something at this temperature, the temperature of the point, clearly the vapor composition must lie on a horizontal (constant temperature) line going through this point. We know that at any given temperature and pressure the composition of the vapor must be richer in the more volatile component and benzene is the more volatile component. Thus we must draw our constant temperature line from the point proceeding to the right. 

12. Heating of a Two component Mixture
Figure 7. We call this line a tie line and it intersects the upper curve at the composition of the vapor in equilibrium with the liquid at the same temperature. The region between the two curves is a two-phase region. In this region two phases, liquid and gas, are in equilibrium with each other and we must be able to keep track of the compositions of both phases.

13. Heating of a Two component Mixture
Figure 8. The first vapor to come off will have the composition shown at point "b." As we continue to heat the system the more volatile component will evaporate preferentially and the liquid phase will become richer in the less volatile component (toluene, here). As we continue to heat the mixture the toluene will gradually catch up to the benzene in the vapor phase. The very last bit of liquid to evaporate will have the composition shown at point "c," and the vapor will have the original composition that we started with, "d" (or "a"). Notice that as we heat the system the tie line "tracks" with the lower and upper curves to give the composition of liquid and vapor in equilibrium.

14. Fractional distillation
Figure 9. As we heat the liquid it will begin to boil when the temperature reaches the temperature of point "a." The first vapor to come off has the composition shown at point "b." Capture the vapor, condense it, and heat it up. The new liquid will boil at point "c" giving a vapor with composition at point "d." Capture this vapor, cool it, and boil it at point, "e," and so on. by continuing this process the vapor can be made as pure in benzene as desired.

15. Fractional distillation of Two component Mixture
Fig. 6.36  A binary temperature– composition diagram can be used to discuss zone refining, as explained in the text.

16. Petrol Distillation Process

17. Boiling diagrams for nonideal solutions
Figure 10. Point "a" denotes a constant boiling mixture called an "azeotrope." In this case it is a ”minimum boiling azeotrope." We have drawn in some "zig-zag" lines to show why fractional distillation will not give complete separation of the mixture into its components.

18. The point "a" indicates the high boiling azeotrope.
Figure 11. It turns out that the composition of the azeotrope is a function of the pressure. One can obtain a solution with very accurately known composition by letting it boil until the high boiling azeotrope is reached. For example, there are tables in the literature that give the composition of the high boiling HCl-H2O azeotrope as a function of pressure. By measuring the local atmospheric pressure and going to the table you can obtain the composition of the high boiling azeotrope to an accuracy sufficient for acid-base analytical work. 

19. Liquid-Vapor Phases Compositions
Fig The lever rule. The distances lα and lβ are used to find the proportions of the amounts of phases α (such as vapour) and β (for example,liquid) present at equilibrium. The lever rule is so called because a similar rule relates the masses at two ends of a lever to their distances from a pivot ( mαlα = mβlβ for balance).

20. Distillation of Mixtures, theoretical plates
Fig The number of theoretical plates is the number of steps needed to bring about a specified degree of separation of two components in a mixture. The two systems shown correspond to (a) 3, (b) 5 theoretical plates.

21. Example 6.2 Interpreting a liquid–liquid phase diagram
Answer We denote hexane by H and nitrobenzene by N; refer to Fig. 6.20, which is a simplified version of Fig The point xN = 0.41,T = 290 K occurs in the two-phase region of the phase diagram. The horizontal tie line cuts the phase boundary at xN = 0.35 and xN = 0.83,so those are the compositions of the two phases. According to the lever rule, the ratio of amounts of each phase is equal to the ratio of the distances l and lβ:
That is, there is about 7 times more hexane-rich phase than nitrobenzene-rich phase. Heating the sample to 292 K takes it into the single-phase region.

22. d(pressure)=0 vap p=2 F=2  + vap  T p=1 F=3
In the one−phase regions (liquid or vapor) we have a variance of three because
we can change the temperature, the composition, X, and the pressure (by moving in and out of the plane of the screen). In the two phase region the variance is only two because if we change the temperature the compositions of both the liquid and vapor phases must track as indicated by the tie−lines connecting the liquid and vapor composition lines.

23. Exam #3 Study Guide 1. You should know from memory:
The temperature and pressure dependence of G and G G = H  T S for a constant temperature process The Gibbs-Helmholtz equation The formula for entropy of mixing for ideal solutions The expressions for chemical potential for ideal gases, nonideal gases, and solutions The definition of the chemical potential, i The definitions of activity and activity coefficient The equation for rG when reaction components are not in their standard states The Gibbs phase rule The Clapeyron and Clausius-Clapeyron equations The definition of an ideal solution *Raoult's law and Henry's law The definition of partial molar volume and other partial molar quantities The formula for ionic strength *Debye-Hückel’s limit law

24. Exam #3 Study Guide
2. You should know how to: Write expressions for H, A, and G from the integrated form for U Calculate rG o at 25oC and other temperatures. Sketch and/or analyze a one-component phase diagram (p as a function of T) Use the Clapeyron and Clausius-Clapeyron equations to calculate various properties of materials
Understand the meaning of dG (maximum no-expansion work)
You should know the expression of dG in terms of P and T
Calculate the vapor pressure of a substance under an applied external pressure Calculate the partial molar volume given appropriate volume and composition data Derive expressions for Smix, Hmix and etc. for ideal solutions Find vapor pressures of, and vapor compositions above, ideal liquid solutions *Derive the expression for vapor pressure depression (from Raoult's law) Sketch and/or analyze (both p vs X and T vs X versions of) a two-component liquid-liquid phase diagram

25. Exam #3 Study Guide
*Sketch and/or analyze a two-component, liquid-solid phase diagram Identify solid solutions, compound formation, and incongruent melting points in phase diagrams Construct cooling curves from a phase diagram and vice versa *Define a suitable ± for an ionic compound in water solution *Find the expression for asalt for ionic solutions in terms of molality and ± Calculate the ionic strength of an ionic solution 3. You should understand:
The third law of thermodynamics and why it is important The meaning and utility of chemical potential *The meaning and utility of activity and activity coefficient *What is meant by components, phases and degrees of freedom (variance) Vapor pressure The origin of the Gibbs phase rule The triple point, critical point, vapor pressure curve, sublimation curve, and melting curve Vapor pressure depression, melting point depression, and boiling point elevation

26. Exam #3 Study Guide
- The temperature–composition diagram corresponding to an ideal mixture with the component A more volatile than component B
- temperature–composition diagram showing only the compositions and temperatures at which the liquids are partially miscible
-Temperature–composition diagram with azeotropes
- Eutectic With Compound Formation
- temperature–composition diagram for a binary system in which boiling occurs before the two liquids are fully miscible