CHAPTER 2VAPOR LIQUID EQUILIBRIUM

The nature of equilibrium The phase rule: Duhem’s Theorem VLE: Qualitative behavior Simple models for vapor-liquid equilibrium VLE by modified Raoult’s law VLE from K-value correlations

• Equilibrium is a static condition in which no changes occurs in the macroscopic properties of a system with time.

• The temperature, pressure and phase compositions reach final values which thereafter remain fixed.

The Nature of Equilibrium

Measures of Composition

• Most common measures of composition are mass fraction, mole fraction and molar concentration

• Mass or mole fraction is defined as the ratio of mass or number of moles of a particular chemical species in a mixture or solution to the total mass or number of moles of mixture or solution:

m

m

m

mx ii

i

n

n

n

nx ii

i

V

xC i

i

q

nC i

i

Molar concentration is defined as the ratio of mole fraction of a particular chemical species in a mixture or solution to molar volume of the mixture of solution

For flow process, gives the expression as a ratio of rates:

where is the molar flow rate of species i, and q is the volumetric flow rate.n

moles of i per unit volume

The molar mass of a mixture or solution : mole fraction-weighted sum of the molar masses of all species

i

ii MxM

The Phase Rule: Duhem’s Theorem• The no. of variables that may be independently fixed in a system

at equilibrium is the difference between total no. of variables that characterize the intensive state of the system and the no. of independent equations.

• The intensive state of a PVT system containing N chemical species and π phases in equilibrium is characterize by the intensive variables, temperature T, pressure P, and N–1 mole fractions for each phase. No. of phase-rule variables : 2 + (N-1)(π)

- masses of the phases are not phase-rule variable (no influence on intensive state of system)

No. of independent phase equilibrium equations : (π–1)(N)• Degree of freedom of the system, F

F = 2 + (N – 1)(π) – (π–1)(N)• Upon reduction, the phase rule becomes

F = 2–π + N

Duhem’s theorem• It applies to closed systems at equilibrium for which the

extensive state as well as intensive state of the system is fixed.

• Intensive phase rule variables : 2 + (N-1) • Extensive phase rule variable (masses or mole no. of the phases) :

Total number of variables : 2 + (N-1) + = 2 + N

• If the system is closed and formed from specified amounts of each species,No. of independent phase equilibrium equations: (–1)(N)+N = N

• The difference between the no. of variables and the no. of equations,

2 + N–N = 2

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.

The two independent variables subject to specification may in general be either intensive or extensive.

However, the number of independent intensive variables is given by the phase rule.

When F = 1, at least one of the two variables must be extensive, and when F = 0, both must be extensive.

VLE: Qualitative Behavior VLE is the state of coexistence of liquid and vapor phases.

Consider a system comprising of two chemical species (N=2), the phase rule becomes

F = 2- + N = 2- + 2 =4 - Because there must at least one phase ( = 1),

maximum number of phase rule variables = 3 P, T, and one mole (or mass) fraction.

All equilibrium states of the system can be represented in three-dimensional (3D) P-T-composition space.

Within this space, the states of pairs of phases coexisting at equilibrium (F = 4 - 2 = 2) define surfaces.

3D diagram: surfaces for VLE

P-T-composition surfaces contain the equi. states of sat. vapor and sat. liquid for species 1 and 2 of a binary system.

Species 1 is the 'lighter' or more volatile.

Upper surfaces contains sat. liquid states [P-T-x1]

Under surfaces contains sat. vapor states [P-T-y1]

Surfaces intersect along the line RKAC1 & UBHC2 : vapor P vs. T curves for pure species 1 and 2

Rounded surfaces between C1 and C2: critical locus –points at which vapor and liquid phases in equilibrium become identical

Critical point: highest T & P where a pure chemical species is observed to exist in VLE

Subcooled-liquid region – above upper surface

Superheated vapor region – below under surface

Interior space between two surfaces: region of coexistence of both liquid and vapor phase

If starts with liquid at F and reduce the P at constant T and composition along FG line

First bubble of vapor appears at point L: bubblepoint;upper surface: bubblepoint surface

The state of vapor bubble in equilibrium with the liquid at L, is represented by a point on the under surface at the T,P of L : point V. Line LV is tie line.

As the pressure is further reduced, more liquid is vaporized until at W, the process is complete.

W : point where last drop of liquid (dew) disappear: dewpoint; Lower surface: dewpoint surface

Continued reduction of P leads into superheated region.

Complexity of figure, the detailed characteristics of binary VLE are usually depicted by 2D by cutting the 3D diagram.

Vertical plane perpendicular to T axis – AEDBLA

The lines on this plane form P-x1-y1 phase diagram at constant T

• Ta ~ AEDBLA• Horizontal lines ~ tie line connecting

the compositions of phases in equilibrium

• Tb and Td lie between two pure species critical temperatures, Tc identified by C1 and C2.

Horizontal plane perpendicular to P axis – KJIHLK

The lines on this plane form T-x1-y1 phase diagram at constant P

• Pa ~ KJIHLK• Pb ~ lies between the critical pressures,

Pc of two pure species at point C1 and C2

• Pd ~ above critical pressure, Pc of two pure species

Vertical & perpendicular to composition axis – SLMN and Q

UC2 and RC1: vapor-pressures curves for the pure species

At point A & B , sat. liquid and sat. vapor lines intersect – sat. liquid of one composition and sat. vapor of another composition have same T & P and two phases are in equilibrium.

Tie line connecting A & B are perpendicular to P-T plane same as tie line LV.

Critical point of a binary mixture occurs where nose of a loop tangent to the envelope curve: critical locus

Location of the critical point on the nose of the loop varies becomes varies with composition

Under certain conditions, a condensation process occurs as the result of a reduction in pressure.

Fig. 10.4 shows the enlarged nose section of a single P-T loop

Critical point at C Point of maximum pressure ~ Mp

Point of maximum temperature~ MT

Interior dashed curved ~ liquid fraction in a two phase mixture

Pressure reduction along line BD ~ vaporization of liquid from bubblepoint to dewpoint

Retrograde condensation: initial point is at F (sat. vapor) reduction in P causes liquefaction until max at G, after that vaporization take place until dewpoint is reached at H Retrograde condensation: Refer page 344

Fig. 10.5: P-T diagram for typical mixtures of non-polar substances such as hydrocarbon. The P-T diagram for the ethane/n-heptane system.

Fig. 10.6: y1-x1 diagram of ethane/n-heptane for several pressure.

Fig. 10.7: P-T diagram a very different kind of system, methanol(1)/benzene(2). The nature of the curves suggests the difficulties in predicting phase behavior for species so dissimilar as methanol and benzene

Fig. 10.8 and Fig. 10.9: Display common types of P-x-y and t-x-y behavior at condition well removed from the critical region Explanation: Refer page 345-347

Fig. 10.10: The y1-x1 diagrams at constant P four systems

Simple Models for Vapor/Liquid Equilibrium Goal thermodynamics applied in VLE: To find the

temperatures, pressures and compositions of phases in equilibrium.

Therefore, we need models for the behavior of systems in vapor/liquid equilibrium

Two simplest: Raoult’s Law

System at low to moderate pressures and only for systems comprised of chemically similar species.

Henry’s Law For any species present at low concentration and

limited to systems at low to moderate pressures.

Raoult’s Law Assumptions:

The vapor phase is an ideal gas It means can apply only for low to moderate

pressures. The liquid phase is an ideal solution

It have approximate validity when the species that comprise the system are chemically similar

Mathematical expressions to Raoult’s Law:

10.1 Eq. )1,2,...,( NiPxPy satiii

system theof temp.at the species pure of pressure phase vapor

fraction mole phasevapor

fraction mole phase liquid

where;

isat

iP

iy

ix

Dewpoint and Bubblepoint Calculations With Raoult’s Law Because iyi = 1, Eq. (10.1) may be summed over all species to

yield

This equation applied in bubblepoint calculations, where the vapor-phase composition is unknown.

For a binary system with x2 = 1 – x1,

10.2 Eq. i

satii PxP

1212 xPPPP satsatsat

1212

21111

2211

xsat

Psat

Psat

PP

sat)Px(

satPxP

satPx

satPxP

P vs. x1 at constant T straight line connecting

P2sat at x1=0 to P1

sat at x1=1

Equation (10.1) may also be solved for xi and summed over all species. With ixi = 1, this yield

This equation applied in dewpoint calculations, where the liquid-phase composition is unknown.

Pisat, or the vapor pressure of component i, is commonly

represented by Antoine Equation (Appendix B, Table B.2, SVNA 7th ed.):

10.3 Eq.

1sat

ii

i PyP

i

isati CKT

BAkPaP

)()(ln

Dew Point Pressure:Given a vapor composition at a specified temperature, find the composition of the liquid in equilibrium

Given T, y1, y2,... yn find P, x1, x2, ... xn

Dew Point Temperature:Given a vapor composition at a specified pressure, find the composition of the liquid in equilibrium

Given P, y1, y2,... yn find T, x1, x2, ... xn

Bubble Point Pressure:Given a liquid composition at a specified temperature, find the composition of the vapor in equilibrium

Given T, x1, x2, ... xn find P, y1, y2,... yn

Bubble Point Temperature:Given a liquid composition at a specified pressure, find the composition of the vapor in equilibrium

Given P, x1, x2, ... xn find T, y1, y2,... yn

Example 1

Binary system acetonitrile(1)/nitromethane(2) conforms closely to Raoult’s law. Vapor pressures for the pure species are given by the following Antoine equations:

15.64

64.29722043.14ln

15.49

47.29452724.14ln

2

1

TP

TP

sat

sat

a) Prepare a graph showing P vs. x1 and P vs. y1 for a temperature of 75oC (348.15K).

b) Prepare a graph showing t vs. x1 and t vs. y1 for a pressure of 70kPa.

Solution (a)BUBL P calculations

1) Calculate P1sat and P2

sat using Antoine equationsAt 384.15K (75oC), P1

sat = 83.21kPa and P2sat = 41.98kPa

2) Calculate P using equation for binary system:

3) Lets x1 = 0.6,

4) Calculate value of y1 using Raoult’s Law expression:

1212 xPPPP satsatsat

kPaP 72.666.098.4121.8398.41

7483.072.66

)21.83)(6.0(111

P

Pxy

sat

It means that at 75°C (348.15K), a liquid mixture of 60mole% acetonitrile and 40mole% nitromethane is in equilibrium with a vapor containing 74.83mole% acetonitrile at a pressure of 66.72kPa.

DEW P calculations

1) Calculate P using Eq. 10.3:

2) Lets y1 = 0.6 (at point c) and T = 348.15K, calculate P :

3) Calculate value of x1 using Raoult’s Law expression:

satsat PyPyP

2211

1

kPaP 74.5998.414.021.836.0

1

4308.021.83

)74.59)(6.0(

1

11

satP

Pyx Liquid phase composition

at point c’

The results of calculations for 75oC (348.15K) at a number of values of x1:

x1 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

States of sat. liquid

States of sat. vapor

Two phase region: sat.liq & sat. vapor coexist in equilibrium

States in calculation:X1=0.6,P=66.72kPa, y1=0.7483

Bubblepoint

Locus of bubblepoint

Dewpoint

Locus of dewpoint

Solution (b)1) Calculate T1

sat and T2sat at the given pressure using Antoine

equations

For P = 70kPa, T1sat/ t1

sat = 342.99K/69.84°C and T2sat/ t2

sat = 362.73K/89.58°C

2) Select T1sat<T<T2

sat, calculate P1sat and P2

sat for these temperature. For example take T = 78oC (351.15K),

At 351.15K (78oC), P1sat = 91.76kPa and P2

sat = 46.84kPa

3) Lets evaluate x1 using equation for binary system:

4) Calculate value of y1 using Raoult’s Law expression:

ii

isati C

PA

BT

ln

5156.084.4676.91

84.4670

21

21

satsat

sat

PP

PPx

6759.070

)76.91)(5156.0(111

P

Pxy

sat

The results of this and similar calculations for P = 70kPa are as follows:

x1 y1 t (°C)

1.0000 1.0000 69.84

0.8596 0.9247 72

0.7378 0.8484 74

0.6233 0.7656 76

0.5156 0.6759 78

0.4142 0.5789 80

0.3184 0.4742 82

0.2280 0.3614 84

0.1424 0.2401 86

0.0613 0.1098 88

0.0000 0.0000 89.58

States of sat. liquid

States of sat. vapor

Two phase region: sat.liq & sat. vapor coexist in equilibrium

Bubblepoint

Locus of bubblepoint

Dewpoint

Locus of dewpoint

Bubl T calculationsFor x1 = 0.6 and P = 70kPa, T is determine by iteration.Equation (10.2) is written as

Subtract ln P2sat from ln P1

sat as given by Antoine equations yield

Iterate as follow: With the current value of (found for an arbitrary intermediate T) calculate P2

sat by using Eq. (B) Calculate t from Antoine equation for species 2:

Find a new value of by using Eq.(C) Return to initial step and iterate to convergence for a final value of t

The result is t = 76.42oC (349.57K). From Antoine equation, P1sat=87.17 kPa and

by Eq. (10.1), y1 = 0.7472

(B) Eq. 21 where 21

2sat

Psat

Pxx

PP sat

(C) Eq. 00.209

64.2972

00.224

47.29450681.0ln

tt

00.209ln2043.14

64.2972

2

satP

t

Dew T calculationsWith P1

sat/P2sat, Eq. (10.3) is written as

The iteration steps are as before, but are based on P1sat, with

The result is t=79.58oC (352.73K). From Antoine equation, P1sat= 96.53kPa and

from Eq. (10.1), x1 = 0.4351

)( 211 yyPP sat

00.224ln2724.14

47.2945

1

satP

t

EXERCISEAssuming the validity of Raoult’s law, do the following calculation for the benzene (1)/toluene (2) system.

a)Given x1=0.33 and T=100°C(373.15K), find y1 and P b)Given y1=0.33 and T=100°C(373.15K), find x1 and P c)Given x1=0.33 and P=120kPa, find y1 and T d)Given y1=0.33 and P=120kPa, find x1 and T