1. SOLUTION THERMODYNAMICS
Chapter 2
SOLUTION THERMODYNAMICS
(Introduction)
(Part 1)

2. THE GIBBS PHASE RULE
The system can be described with a number of thermodynamic properties including T, P, V, S, H, and G.
However not all of these properties are independent.
In fact, for the glass of water, only two intensive thermodynamic properties are independent.
Thus, if we specify T and P, all the other properties, in their intensive (i.e. their value per mole of substance) form, can be determined.
H2O (

3. Temperature and pressure are often taken as independent intensive variables (see P-T diagram)
This is because it is usually easy to experimentally vary and measure these two properties directly.
However, any intensive property can be chosen to be one of the independent variables.
If the intensive enthalpy (J/mol) and intensive entropy (J/mol K) of the water in the glass were specified, its temperature and pressure could then be found using the steam tables or a Mollier diagram.
The Gibbs phase rule tells how many independent intensive properties, F, can be chosen.
This will depend on the number of chemical species, N, and number of phases, , present. (

5. T and P are common independent intensive variables
Consider a in equilibrium system comprising of  phases with N components
T and P are common independent intensive variables 1
N components
No. of intensive variables = N – 1  2
N components
No. of intensive variables = N – 1  3 
N components
No. of intensive variables = N – 1 
Total number of variables =  (N – 1) + 2

6. As will become clear later in this chapter, the chemical potential of each component must be equal in all phases.
Comp. 1
Comp. 2
Comp. N
Total number of equations/constraints = N ( – 1)

7. Subtract the number of constraints from the number of variables to obtain the number of degrees of freedom as
F =  (N – 1) + 2 – N ( – 1)
Upon reduction, this becomes the phase rule:
(2.1)

8. Single Component System
 = 1 ; F = 2
Solid + gas
 = 2 ; F = 1
Liquid + gas
Solid + liquid
Solid + liquid + gas
 = 3 ; F = 0
Vapor

9. Two Component System
The possible phases are the vapor, two immiscible (or partially miscible) liquid phases, and two solid phases.
Of course, they don’t have to all exist.
The liquids might turn out to be miscible for all compositions.

10. Two-Component Vapor-Liquid Equilibrium
Vapor-liquid equilibrium (VLE) is the state of coexistence of liquid and vapor phases.
Since there must be at least one phase ( = 1), the maximum number of phase-rule variables which must be specified to fix the intensive state of the system is three: namely, P, T, and one mole (or mass) fraction.
All equilibrium states of the system can therefore be represented in three-dimensional P-T- composition space.
Within this space, the states of pairs of phases coexisting at equilibrium define surfaces.

11. Let us place two gases (A and B) in an isothermal cell.
If we move the piston down, we would compress the gases, causing a decrease in volume and, of course, an increase in pressure.
The process starts at point A, as shown in the figure in the next page,  an all-vapor condition.

12. P-v Diagram for a Binary Mixture(

13. As pressure increases, volume decreases accordingly.
After some compression, the first droplet of liquid will appear  we have found the dew point of the mixture (point B).
As we further compress the system, more liquid will appear and the volume will continue to decrease.
During the phase transition, pressure does not remain constant.
As compression progresses and more liquid is formed, pressure keeps rising — although not as sharply as in the single-phase vapor region.
When the entire system has become liquid, with only an infinitesimal bubble of vapor left, we are at point C — the bubble point of the mixture.

14. Please note that, for binary mixtures (as is the case for multi- component mixtures,) the dew point and bubble point do not occur at the same pressure for isothermal compression.
If you recall, for the single-component system, the dew point and the bubble point coincide.
This is not true for binary and multi-component systems.
Compare two figures in the following page.

15. P-V Diagram for a Binary Mixture
P-V Diagram For A Pure Component

16. Why is pressure increasing during the phase transition?
In a single-component system, both liquid and vapor in the two-phase region have the same composition (there is only one chemical substance within the system).
Now, when a mixture exists in a two-phase condition, different molecules of different species are present and they can be either in a liquid or vapor state (two-phase condition).

17. Some of them would “prefer” to be in the gas phase while the others would “prefer” to be in the liquid phase.
This “preference” is controlled by the volatility of the given component.
When we reach point B and the first droplet of liquid appears, the heaviest molecules are the ones that preferentially go to that first tiny droplet of liquid phase.
For ‘heavy’ molecules, given the choice, it is more desirable to be in the condensed state

18. As we keep on compressing, mainly light molecules remain in the vapor phase.
At the end point of the transition (point C) we have forced all of them to go to the liquid state.
If we compare a sample of liquid at dew point conditions (point B) to one taken in the middle of the transition, it is clear that the former would be richer in heavy components than the latter.
The properties of the heaviest component would be most influential at the dew point (when the liquid first appears); while the properties of the lighter component would be most influential at the bubble point (when the last bubble is about to disappear.)

19. In the two-phase region, pressure increases as the system passes from the dew point to the bubble point.
The composition of liquid and vapor is changing; but the overall composition is always the same!
At the dew point, the composition of the vapor is equal to the overall composition of the system; however, the infinitesimal amount of liquid that is condensed is richer in the less volatile component.
At the bubble point, the composition of the liquid is equal to that of the system, but the infinitesimal amount of vapor remaining at the bubble point is richer in the more volatile component than the system as a whole.

20. In general, when two different species are mixed, some of the behaviors of the individual species and their properties will change.
Their usual behavior (as pure components) will be altered as a consequence of the new field of molecular interactions that has been created.
While kept in a pure condition, molecules only interact with like molecules.
On the other hand, in a mixture new interactions between dissimilar molecules occur.

21. Family of Isotherms on a P-V diagram Binary Mixture 
Family of Isotherms on a P-V diagram Binary Mixture (

22. The line connecting all of the bubble and dew points will generate the bubble and dew point curve, both of which meet at the critical point.
Notice that the critical point does not represent a maximum in the P-V diagram of a mixture.
Also note that bubble point pressures and dew point pressures are no longer the same.

23. The isotherms through the two-phase region are not horizontal but have a definite slope.
This must have an implication. In fact, it does. What if we now want to plot the P-T diagram for this mixture?
The bubble point curve will shift to the upper left (higher pressures) and dew point curve will shift to the lower right (lower pressures) — both of them meeting at the critical point.

24. P-T Phase Envelope For A Binary System 
P-T Phase Envelope For A Binary System

25. Vapor pressure curve (PT diagram) for pure component

26. The critical point is no longer at the apex or peak of the two-phase region; hence, vapor and liquid can coexist in equilibrium at T > Tc and P > Pc.
In fact, we can identify two new maxima: condition Pcc is the maximum pressure and condition Tcc is the maximum temperature at which L+V will be found in equilibrium.
We assign special names to these points. They are the cricondenbar and cricondentherm, respectively.

27. Cricondentherm (Tcc):
The highest temperature in the two-phase envelope.
For T > Tcc, liquid and vapor cannot co-exist at equilibrium, no matter what the pressure is.
Cricondenbar (Pcc):
The highest pressure in the two-phase envelope.
For P > Pcc, liquid and vapor cannot co-exist at equilibrium, no matter what the temperature is.
For pure substances only:
Cricondentherm = Cricondenbar = Critical Point.

28. Retrograde Phenomenon
Although the critical point for binary systems is the common point between the dew and bubble point curves (the point for which liquid and vapor phases are indistinguishable), in general, this point neither represents the maximum pressure nor the maximum temperature for vapor-liquid coexistence.
In fact, we gave new names to these maxima: cricondenbar (for the maximum pressure) and cricondentherm (for the maximum temperature).
Let’s look at this again in following Figure, where the critical point (Pc, Tc), cricondentherm (Tcc), and cricondenbar (Pcc) are highlighted.

29. Cricondenbar, Cricondentherm and Critical Point(

30. Please recall that the bubble point curve represents the line of saturated liquid (100 % liquid with an infinitesimal amount of vapor) and the dew point curve represents the line of saturated vapor (100 % vapor with an infinitesimal amount of liquid).
These conditions are all shown in previous Figure.
Let us now consider the isothermal processes taking place at T = T1 and T = T2, represented in the following Figure.
Notice that these temperatures are such that T1 < Tc and Tc < T2 < Tcc.

31. Isothermal Compression At T1 and T2(

32. Isothermal compression at T = T1.
At point A, we are in an ALL VAPOR condition (0 % liquid) and we are starting to cross over into the two- phase region.
As we compress from point A to B, more and more liquid is formed until the entire system has been condensed (point B).
We went all the way from 0 % liquid to 100 % liquid, as we expected, by compressing the vapor.

33. In the second case (Tc < T2 < Tcc), we have a different behavior.
At point C, we are starting in an ALL VAPOR condition (0 % liquid); by increasing pressure, we force the system to enter the two-phase region.
Thus, some liquid has to drop out; we expect that as the pressure keeps increasing, we will produce more and more liquid.
That is true to some extent, but at the final point (point D), where we have 0 % liquid in the system again.

34. Portion of a PT diagram in the critical region
How so??
Portion of a PT diagram in the critical region

35. Vapor pressure curve (pure component)
Effect of Composition on Phase Behavior
Vapor pressure curve (pure component) (

36. P-T Graphs For The Pure Components A and B(

37. Since we are considering A to be the more volatile, it is expected to have higher vapor pressures at lower temperatures, thus, its curve is located towards the left.
For B, the less volatile component, we have a boiling point curve with lower vapor pressures at higher temperatures. Hence, the boiling point curve of B is found towards the right at lower pressures.
Now, if we mix A and B, the new phase envelope can be anywhere within curves A and B.
This is shown in the following Figure, where the effect of composition on phase behavior of the binary mixture Methane/Ethane is illustrated.

38. PT diagram for methane – ethane(

39. In the previous figure, each phase envelope represents a different composition or a particular composition between A and B (pure conditions).
The phase envelopes are bounded by the pure-component vapor pressure curve for component A (Methane) on the left, that for component B (Ethane) on the right, and the critical locus (i.e., the curve connecting the critical points for the individual phase envelopes) on the top.
Note that when one of the components is dominant, the curves are characteristic of relatively narrow-boiling systems, whereas the curves for which the components are present in comparable amounts constitute relatively wide-boiling systems

40. Notice that the range of temperature of the critical point locus is bounded by the critical temperature of the pure components for binary mixtures.
Therefore, no binary mixture has a critical temperature either below the lightest component’s critical temperature or above the heaviest component’s critical temperature.
However, this is true only for critical temperatures; but not for critical pressures.

41. A mixture’s critical pressure can be found to be higher than the critical pressures of both pure components — hence, we see a concave shape for the critical locus.
In general, the more dissimilar the two substances, the farther the upward reach of the critical locus.
When the substances making up the mixture are similar in molecular complexity, the shape of the critical locus flattens down.

42. Px Diagram (

43. Tx Diagram (

44. PTxy diagram for vapor-liquid equilibrium

45. yx Diagram
yx diagram for ethane-n-heptane

46. According to convention, y1 and x1 represent the mole fractions of the more volatile species in the mixture.
The maximum and minimum concentrations of the more volatile species obtainable by distillation at a given pressure are indicated by the points of intersection of the appropriate y-x curve with the diagonal, for at these points the vapor and liquid have the same composition.
They are in fact mixture critical points, unless y1 = x1 = 0 or y1 = x1 = 1.
Point A in previous figure represents the composition of the vapor and liquid phases at the maximum pressure at which the phases can coexist in the ethane-n-heptane system.
The composition is about 77 mol -% ethane and the pressure is about 87.1 bar.

47. Px relation for ideal liquid solutions (Raoult's law)
Pxy diagrams at constant T of tetrahydrofuran (1 )/carbon tetrachloride (2) at K

48. Px relation for ideal liquid solutions (Raoult's law)
Pxy diagrams at constant T of chloroform(1)/tetrahydrofuran(2) at K

49. Px relation for ideal liquid solutions (Raoult's law)
Pxy diagrams at constant T of furan(1)/carbon tetrachloride(2) at K

50. Pxy diagrams at constant T of ethanol (1)/toluene (2) at
Px relation for ideal liquid solutions (Raoult's law)
Pxy diagrams at constant T of ethanol (1)/toluene (2) at K

51. Appreciable negative departures from P-x1 linearity reflect stronger liquid-phase intermolecular attractions between unlike than between like pairs of molecules.
Conversely, appreciable positive departures result for solutions for which liquid-phase intermolecular forces between like molecules are stronger than between unlike.
In this latter case the forces between like molecules may be so strong as to prevent complete miscibility, and the system then forms two separate liquid phases over a range of compositions.
Since distillation processes are carried out more nearly at constant pressure than at constant temperature, T-x1-y1, diagrams of data at constant P are of practical interest.

52. Pxy diagrams at constant T of tetrahydrofuran (1 )/carbon tetrachloride (2) at 1 atm

53. Pxy diagrams at constant T of tetrahydrofuran (1 )/carbon tetrachloride (2) at 1 atm

54. Pxy diagrams at constant T of furan(1)/carbon tetrachloride(2) at 1 atm

55. Pxy diagrams at constant T of ethanol (1)/toluene (2) at
1 atm

56. yx diagrams at 1 atm: (a) tetrahydrofuran (1)/carbon tetrachloride (2); (b) chloroform (1)/ tetrahydrofuran (2); (c) furan (1)/carbon tetrachloride (2); (d) ethanol (1)/toluene (2)

57. The Lever Rule
P-x and T-x diagrams are quite useful, in that information about the compositions and relative amounts of the two phases can be easily extracted.
In fact, besides giving a qualitative picture of the phase behavior of fluid mixtures, phase diagrams can also give quantitative information pertaining to the amounts of each phase present, as well as the composition of each phase.

58. At a given temperature or pressure in a T-x or P-x diagram (respectively), a horizontal line may be drawn through the two-phase region that will connect the composition of the liquid (xA) and vapor (yA) in equilibrium at such condition — that is, the bubble and dew points at the given temperature or pressure, respectively.

59. If, at the given pressure and temperature, the overall composition of the system (zA) is found within these values (xA < zA < yA in the T-x diagram or yA < zA < xA in the P-x diagram), the system will be in a two-phase condition and the vapor fraction (αG) and liquid fraction (αL) can be determined by the lever rule:
Note that αL and αG are not independent of each other, since αL + αG = 1.

61. VLE by Raoult’s Law
The two major assumptions required to reduce VLE calculations to Raoult's law:
The vapor phase is an ideal gas.
The liquid phase is an ideal solution (Chapter 3).
The first assumption means that Raoult's law can apply only for low to moderate pressures.
The second implies that it can have approximate validity only when the species that comprise the system are chemically similar.

62. Just as the ideal gas serves as a standard to which real gas behavior may be compared, the ideal solution represents a standard to which real solution behavior may be compared.
Ideal solution behavior is often approximated by liquid phases wherein the molecular species are not too different in size and are of the same chemical nature.
Thus, a mixture of isomers, such as ortho-, meta-, and para-xylene, conforms very closely to ideal solution behavior.
So do mixtures of adjacent members of a homologous series, as for example, n-hexane /n-heptane, ethanol/propanol, and benzene/toluene.

63. The mathematical expression which reflects the two listed assumptions and which therefore gives quantitative expression to Raoult's law is
(i = 1, 2, …, N) (2.2)
where xi is a liquid-phase mole fraction, yi is a vapor-phase mole fraction, and P is the vapor pressure of pure species i at the temperature of the system.
The product yi P on the left side of Eq. (2.2) is known as the partial pressure of species i.

64. Dewpoint and Bubblepoint Calculations with Raoult's Law
Because i yi = 1, Eq. (2.2) may be summed over all species to yield:
(2.3)
Equation (2.2) may also be solved for xi and summed over all species.
With i xi = 1, this yields:
(2.4)

66. Example 2.1
Binary system acrilonitrile (1)/nitromethane (2) conforms closely to Raoult’s law. Vapor pressures for the pure species are given by the following Antoine equations:
Prepare a graph showing P vs. x1 and P vs. y1 to a temperature of K.
Prepare a graph showing T vs. x1 and T vs. y1 to a pressure of 70 kPa.

67. Solution 2.1
a) Bubble pressure calculation is required.
(A)
At K by Antoine equations:
P1sat = kPa P2sat = kPa
For x1 = 0.6
Using eq. (2.2):

68. The result s of calculation for entire values of x1 are:y1
P (kPa)
0.0
0.0000
41.98
0.1
0.1805
46.10
0.2
0.3313
50.23
0.3
0.4593
54.35
0.4
0.5692
58.47
0.5
0.6647
62.60
0.6
0.7483
66.72
0.7
0.8222
70.84
0.8
0.8880
74.96
0.9
0.9469
79.09
1.0
1.0000
83.21

69. P1sat = 83.21
Subcooled liquid
Superheated vapor
P2sat = 41.98

70. b) When P is fixed, the temperature varies along with x1 and y1.
Generalized Antoine equation:
For P = 70 kPa:

71. The simplest way to prepare T-x1-y1 diagram is to
select values of T between these two temperatures,
calculate P1sat and P2sat and
evaluate x1 by Eq. (A):
Calculate y1 by Eq. (2.2)

72. T
P1sat
P2sat x1 y1
362.73
131.55
69.99
0.0001
0.0003
359.15
118.02
62.03
0.1424
0.2401
355.15
104.24
54.00
0.3184
0.4742
351.15
91.76
46.84
0.5156
0.6759
347.15
80.50
40.46
0.7378
0.8484
342.99
34.59
1.0003
1.0002

74. Henry’s law
Application of Raoult's law to species i requires a value for PiSat at the temperature of application, and thus is not appropriate for a species whose critical temperature is less than the temperature of application.
If a system of air in contact with liquid water is presumed at equilibrium, then the air is saturated with water.
The mole fraction of water vapor in the air is usually found from Raoult's law applied to the water with the assumption that no air dissolves in the liquid phase.

75. Thus, the liquid water is regarded as pure and Raoult's law for the water (species 2) becomes y2 P = P2sat.
At K and atmospheric pressure, this equation yields:
where the pressures are in kPa, and P2sat comes from the steam tables.

76. If one wishes to calculate the mole fraction of air dissolved in the water, then Raoult's law cannot be applied, because the Tc of air is much lower than K.
This problem can be solved by Henry's law, applied here for pressures low enough that the vapor phase may be assumed an ideal gas.
For a species present as a very dilute solute in the liquid phase, Henry's law then states that the partial pressure of the species in the vapor phase is directly proportional to its liquid-phase mole fraction. Thus,
yi P = xi Hi (2.5)
Where Hi is Henry’s constant