1. VAPOR/LIQUID EQUILIBRIUM- Introduction
ERT 206: Thermodynamics
Miss Anis Atikah Ahmad
anis

2. OUTLINE The Nature of Equilibrium Duhem’s Theorem
Simple Models for VLE
VLE by Modified Raoult’s Law
VLE from K-value Correlations

3. 1. The Nature of Equilibrium
Equilibrium is a static condition in which no changes occur in the macroscopic properties of a system with time.
Eg: An isolated system consisting of liquid & vapor phase reaches a final state wherein no tendency exists for change to occur within the system. The temperature, pressure and phase compositions reach final values which thereafter remain fixed.

4. The Nature of Equilibrium
At microscopic level, conditions are not static.
Molecules with high velocities near the interface overcome surface forces and pass into the other phase.
But the average rate of passage of molecules is the same in both directions & no net interphase transfer of material occurs.
The Nature of Equilibrium

5. Measures of Composition
1. Mass fraction: the ratio of the mass of a particular chemical species in a mixture or solution to the total mass of mixture or solution. 2. Mole fraction: the ratio of the number of moles of a particular chemical species in a mixture or solution to the number of moles of mixture or solution.

6. Measures of Composition
3. Molar concentration: the ratio of the mole fraction of a particular chemical species in a mixture or solution to the molar volume of mixture or solution. 4. Molar mass of mixture/solution: mole-fraction-weighted sum of the molar masses of all species present.
Molar flow rate
Volumetric flow rate

7. Similar to phase rule, but it considers extensive state.
2. Duhem’s Theorem
Duhem’s Theorem: for any closed system formed initially from given masses of prescribed chemical species, the equilibrium state is completely determined when any two independent variables are fixed.
Applies to closed systems at equilibrium
The extensive state and intensive state of system are fixed
No of variables
No of equations

8. 3. SIMPLE MODELS FOR VAPOR/LIQUID EQUILIBRIUM
Vapor/liquid equilibrium (VLE): the state of coexistence of liquid and vapor phase.
VLE Model: to calculate temperatures, pressures and compositions of phases in equilibrium.
The two simplest models are:
Raoult’s law
Henry’s law

9. 3. SIMPLE MODELS FOR VAPOR/LIQUID EQUILIBRIUM
3.1 Raoult’s Law
Assumptions:
The vapor phase is an ideal gas (low to moderate pressure)
The liquid phase is an ideal solution (the system are chemically similar)
*Chemically similar: the molecular species are not too different in size
and are of the same chemical nature.
eg: n-hexane/n-heptane, ethanol/propanol, benzene/toluene
Vapor pressure of pure species i at system temperature
Liquid phase mole fraction
Vapor phase mole fraction

10. Pxy Diagram

12. 3.2 Dewpoint & Bubblepoint Calculations with Raoult’s Law
BUBL P : Calculate {yi} and P, given {xi} and T
DEW P : Calculate {xi} and P, given {yi} and T
BUBL T : Calculate {yi} and T, given {xi} and P
DEW T : Calculate {xi} and T, given {yi} and P
If the vapor-phase composition is unknown, may be assumed; thus
For bubble point calculation

13. 3.2 Dewpoint & Bubblepoint Calculations with Raoult’s Law
If the liquid-phase composition is unknown, may be assumed; thus
For dew point calculation

14. 3.2.1 BUBL P CALCULATION (Calculate {yi} and P, given {xi} and T)
Find P1sat & P2sat using Antoine equation
Find P
Calculate yi

15. Example 1
Binary system acetronitrile (1)/ nitromethane (2) conforms closely to Raoult’s law. Vapor pressure for the pure species are given by the following Antoine equations: Prepare a graph showing P vs. X1 and P vs. Y1 for a temperature of 75°C.

16. 3.2.1 BUBL P CALCULATION (Calculate {yi} and P, given {xi} and T)
Find P1sat & P2sat using Antoine equation
Find P
Calculate yi
At 75°C, by Antoine Equations,

17. 3.2.1 BUBL P CALCULATION (Calculate {yi} and P, given {xi} and T)
Find P1sat & P2sat using Antoine equation
Find P
Calculate yi
Taking at any value of x1, say x1=0.6,

18. 3.2.1 BUBL P CALCULATION (Calculate {yi} and P, given {xi} and T)
Find P1sat & P2sat using Antoine equation
Find P
Calculate yi
At 75°C, a liquid mixture of 60 mol-% acetonitrile and 40 mol-% nitromethane is in equilibrium with a vapor containing mol-% acetonitrile at a pressure of kPa

19. P-x-y Diagram
To draw P-x-y graph, repeat the calculation with different values of x; x1 y1
P/kPa
0.0
0.0000
41.98
0.2
0.3313
50.23
0.4
0.5692
58.47
0.6
0.7483
66.72
0.8
0.8880
74.96
1.0
1.0000
83.21

20. P x y diagram for acetonitrile/nitromethane at 75°C as given by Raoult’s law
Subcooled liquid
Superheated vapor

21. P x y diagram for acetonitrile/nitromethane at 75°C as given by Raoult’s law
Point a is a subcooled liquid mixture of 60 mol-% acetonitrile and 40 mol-% of nitromethane at 75°C.
Point b is saturated liquid.
Points lying between b and c are in two phase region, where saturated liquid and saturated vapor coexist in equilibrium.
Saturated liquid and saturated vapor of the pure species coexist at vapor pressure P1sat and P2sat a
Subcooled liquid b b' c’ c d
Superheated vapor

22. P x y diagram for acetonitrile/nitromethane at 75°C as given by Raoult’s law
Point b: bubblepoint
P-x1 is the locus of bubblepoints
As point c is approached, the liquid phase has almost disappeared, with only droplets (dew) remaining.
Point c: dewpoint
P-y1 is the locus of dewpoints. a
Subcooled liquid b b' c’ c d
Superheated vapor

23. P x y diagram for acetonitrile/nitromethane at 75°C as given by Raoult’s law
Once the dew has evaporated, only saturated vapor at point c remains.
Further pressure reduction leads to superheated vapor at point d
Subcooled liquid b b' c’ c d
Superheated vapor

24. 3.2.2 DEW P CALCULATION (DEW P : Calculate {xi} and P, given {yi} and T)
What is x1 & P
at point c’?
Step 1: Calculate P
Step 2: Calculate x1 a
Subcooled liquid b b' c’ c d
Superheated vapor

25. Find P from Raoult’s Law assuming
3.2.2 DEW P CALCULATION (DEW P : Calculate {xi} and P, given {yi} and T)
Find P from Raoult’s Law assuming
Calculate xi

26. T-x-y Diagram Find T1sat & T2sat using Antoine equation
Find P1sat & P2sat using T btween T1sat & T2sat
Calculate xi
Calculate yi

27. Example 2
Binary system acetronitrile (1)/ nitromethane (2) conforms closely to Raoult’s law. Vapor pressure for the pure species are given by the following Antoine equations: Prepare a graph showing T vs. X1 and T vs. Y1 for a pressure of of 70kPa.

28. Find T1sat & T2sat using Antoine equation
T-x-y Diagram
Find T1sat & T2sat using Antoine equation
Find P1sat & P2sat using T btween T1sat & T2sat
Calculate xi
Calculate yi

29. Find P1sat & P2sat using T btween T1sat & T2sat
T-x-y Diagram
Find T1sat & T2sat using Antoine equation
Find P1sat & P2sat using T btween T1sat & T2sat
Calculate xi
Calculate yi
T1sat = 69.84°C, T2sat = 89.58°C
Let T=78°C,

30. T-x-y Diagram Find T1sat & T2sat using Antoine equation
Find P1sat & P2sat using T btween T1sat & T2sat
Calculate xi
Calculate yi
P1sat = 91.76kPa, P2sat = 46.84kPa

31. T-x-y Diagram Find T1sat & T2sat using Antoine equation
Find P1sat & P2sat using T btween T1sat & T2sat
Calculate xi
Calculate yi
P1sat = 91.76kPa, x =

32. T-x-y Diagram
To draw T-x-y graph, repeat the calculation with different values of T; x1 y1
T/°C
0.0000
89.58 (T2sat)
0.1424
0.2401 86
0.3184
0.4742 82
0.5156
0.6759 78
0.7378
0.8484 74
1.0000
69.84 (T1sat)

33. T x y diagram for acetonitrile/nitromethane at 70 kPa as given by Raoult’s law
T-y1
Superheated vapor
T1sat = 69.84°c
T-x1
T2sat = 89.58°C
Subcooled liquid

34. 3.2.3 BUBL T CALCULATION (Calculate {yi} and T, given {xi} and P)
Superheated vapor
T-y1
What is y1 and T at point b’ (with x1=0.6 and P= 70 kPa)?
T1sat = 69.84°c
T-x1
T2sat = 89.58°C c c’ b' b
Subcooled liquid

35. 3.2.3 BUBL T CALCULATION (Calculate {yi} and T, given {xi} and P)
Start with α=1, find P2sat
Find T using Antoine eq & substitute P2sat obtained in step 1
Find new α by substituting T
Repeat step 1 by using new α until similar value of α is obtained
Find P1sat & find y1 using Raoult’s law
The substraction of ln P1sat & P2sat from Antoine Equation

36. 3.2.3 BUBL T CALCULATION (Calculate {yi} and T, given {xi} and P)
Start with α=1, find P2sat
Find T using Antoine eq & substitute P2sat obtained in step 1
Find new α by substituting T
Repeat step 1 by using new α until similar value of α is obtained
Find P1sat & find y1 using Raoult’s law
Iteration 1
Iteration 3
Iteration 2
Iteration 4

37. 3.2.4 DEW T CALCULATION (Calculate {xi} and T, given {yi} and P)
Superheated vapor
T-y1
What is x1 and T at point c’ (with y1=0.6 and P= 70 kPa)?
T1sat = 69.84°c
T-x1
T2sat = 89.58°C c c’ b' b
Subcooled liquid

38. 3.2.4 DEW T CALCULATION (Calculate {xi} and T, given {yi} and P)
Start with α=1, find P1sat
Find T using Antoine eq & substitute P1sat obtained in step 1
Find new α by substituting T
Repeat step 1 by using new α until similar value of α is obtained
Find x1

39. 3.3 Henry’s Law
Used for a species whose critical temperature is less than the temperature of application, in which Raoult’s Law could not be applied (since Raoult’s Law requires a value of Pisat).
Where Hi is Henry’s constant and obtained from experiment.

40. 4. VLE by Modified Raoult’s Law
Used when the liquid phase is not an ideal solution.
Where ɣi is an activity coefficient
(deviation from solution ideality in liquid phase).

41. 4. VLE by Modified Raoult’s Law
For bubblepoint calculation, (assuming )
For dewpoint calculation, (assuming )

42. 5. VLE from K-value Correlations
Equilibrium ratio, Ki
When Ki > 1, species exhibits a higher concentration of vapor phase
When Ki < 1, species exhibits a higher concentration of liquid phase (is considered as heavy constituent.)

43. 5. VLE from K-value Correlations
K value for Raoult’s Law
K value for modified Raoult’s Law
since
since

44. 5. VLE from K-value Correlations
For bubblepoint calculations,
For dewpoint calculations

45. Example For a mixture of 10 mol-% methane, 20 mol-%
ethane, and 70 mol-% propane at 50°F, determine:
The dewpoint pressure
The bubblepoint pressure

46. Example The dewpoint pressure
For a mixture of 10 mol-% methane, 20 mol-%
ethane, and 70 mol-% propane at 50°F, determine:
The dewpoint pressure
When the system at its dewpoint, only an insignificant amount of liquid is present.
Thus 10 mol-% methane, 20 mol-% ethane, and 70 mol-% propane are the values of yi.
assuming, thus,
By trial, find the value of pressure that satisfy

48. Example The dewpoint pressure Thus, the dewpoint pressure is 126 psia.
For a mixture of 10 mol-% methane, 20 mol-%
ethane, and 70 mol-% propane at 50°F, determine:
The dewpoint pressure
Species yi
P=100psia
P=150psia
P=126psia Ki
yi/Ki
Methane
0.10
20.0
0.005
13.2
0.008
16.0
0.006
Ethane
0.20
3.25
0.062
2.25
0.089
2.65
0.075
Propane
0.70
0.92
0.761
0.65
1.077
0.762
0.919
Thus, the dewpoint pressure is 126 psia.

49. Example (b)The bubblepoint pressure assuming , thus
For a mixture of 10 mol-% methane, 20 mol-%
ethane, and 70 mol-% propane at 50°F, determine:
(b)The bubblepoint pressure assuming , thus
By trial, find the value of pressure that satisfy

51. Example (b) The bubble point pressure
For a mixture of 10 mol-% methane, 20 mol-%
ethane, and 70 mol-% propane at 50°F, determine:
(b) The bubble point pressure
Species xi
P=380psia
P=400psia
P=385psia Ki
Kixi
Methane
0.10
5.60
0.560
5.25
0.525
5.49
0.549
Ethane
0.20
1.11
0.222
1.07
0.214
1.10
0.220
Propane
0.70
0.335
0.235
0.32
0.224
0.33
0.231
Thus, the bubblepoint pressure is 385 psia.