Solution Thermodynamic:

Vapor/Liquid Equilibrium (VLE)

PTT 201/4 THERMODYNAMICSSEM 1 (2013/2014)

Nature of Equilibrium– Definition– Measures of composition

VLE : Qualitative behaviorSimple Models for VLE - Raoult’s Law - Dewpoint & Bubblepoint Calculations with Raoult’s Law - Henry’s LawVLE by modified Raoult’s lawVLE from K-value correlations

Chapter Outline (Smith)

THE NATURE OF EQUILIBRIUM

Equilibrium : A static condition in which no changes occur in the macroscopic properties of a system with time.

The T, P, composition reaches final value which will remain fixed: equilibrium

m

m

m

mx iii

V

xC ii

iiiMxM

Measures of composition

VLE: State of coexistence of L & V phases A condition where a liquid phase and vapor phase

are in equilibrium with each other At this condition: rate of evaporation (liquid → vapor) = rate of condensation (vapor → liquid)

VLE: QUALITATIVE BEHAVIOR

Binary mixture: Mixture that contains two constituents e.g: mixture of liquid and vapor at an equilibrium level takes place when liquid and vapor are allowed to contact to each other in a closed location

• Under surface- sat. V states (P-T-y1)

• Upper surface- sat. L states (P-T-x1)

• Liquid at F, reduces pressure at constant T & composition along FG, the first bubble appear at L – bubble point: a point when a liquid forms the first bubble of vapor and begins to evaporate

• As pressure reduces, more & more L vaporizes until completed at W; point where last drop of L (dew) disappear – dew point: a point when a vapor forms the first droplet of liquid and begins to condense

Fig. 10.1 – Shows the P-T-composition surfaces of equilibrium states of

saturated V & saturated L of a binary system

SIMPLE MODELS FOR VLE

Raoult’s Law

• V phase is an ideal gas– Applicable for low to moderate

pressure• L phase is an ideal solution

– Valid only if the species are chemically similar (size, same chemical nature e.g. isomers such as ortho-, meta- & para-xylene)

Assumptions;

NiPxPy satiii ,...,2,1

Where;

pressure Total :

species pure of pressureVapor :

fraction mole phase:

fraction mole phase:

P

iP

Vy

Lx

sati

i

i

1

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

Dewpoint & Bubblepoint Calculations with Raoult’s Law

FIND GIVEN

For binary systems to solve for bubblepoint calculation (T is given);

1i iy

i

satiiPxP 1212 xPPPP satsatsat

PPx

ysat

11

1

2

3

i

satii Py

P1

Raoult’s law equation can be solved for xi to solve for dewpoint calculation (T is given) 1i i

x

satsat PyPyP

2211//

1

satPPy

x1

1

1

4

5

Example 1

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

00.209

64.972,22043.14ln

00.224

47.945,22724.14ln

02

01

CtkPaP

CtkPaP

sat

sat

a)Prepare a graph showing P vs. x1 and P vs. y1 at temperature 750C

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

i

ii

a) BUBL P calculations are required. Since this is a binary system, Eq. 2 may be used.

)(1212 AxPPPP satsatsat

At 750C, the saturated pressure is given by Antoine equation;

98.4121.83 21 satsat PP

Substitute both values in (A) to find P;

kPaP

P

72.66

6.098.4121.8398.41

The corresponding value of y1 is found from Eq. 1, sat

iii PxPy

x1 y1 P/kPa

0.0 0.0000 41.98

0.2 0.3313 50.23

0.4 0.5692 58.47

x1 y1 P/kPa

0.6 0.7483 66.72

0.8 0.8880 74.96

1.0 1.0000 83.21

7483.0

72.66

21.836.0111

P

Pxy

sat

At point c, the vapor composition is y1=0.6, but the composition of liquid at c’ and the pressure must read from graph or calculated. Thus DEW P calculations are required. By using Eq. 3;

satsat PyPyP

2211

1

For y1=0.6 and t=750C

kPaP 74.5998.414.021.836.0

1

And by Eq. 1,

4308.0

21.83

74.596.0

1

11

satP

Pyx

This is the liquid-phase composition at point c’

b) When P is fixed, the T varies along T1sat and

T2sat, with x1 & y1. T1sat & T2sat are calculated

from Antoine equation;

ii

isati C

PA

Bt

ln

For P=70kPa, T1sat=69.840C, T2sat=89.580C. Select T between these two temperatures and calculate P1sat &

P2sat for the two temperatures.

Evaluate x1 by Eq. (A). For example;

satsat

sat

PP

PPx

21

21

5156.0

84.4676.91

84.46701

x

Get y1 from Eq. 1

6759.0

70

76.915156.0111

P

Pxy

sat

e.g; select T= 78˚C

Substituting T= 78˚C into (i) and (ii)

P1sat = 91.76 kPa

P2sat = 46.84 kPa

Summary;

x1 y1 T/˚C

0.0000 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 1.0000 69.84 (t1sat)

1. For pressure low It is so low that it can be assume as ideal gas

2. For species present as a very dilute solution in liquid phase

Assumptions;

Henry’s Law

NiHxPy iii ,...,2,1

Where;

pressure Total :

constant sHenry' :

fraction mole phase:

fraction mole phase:

P

H

Vy

Lx

i

i

i

Henry’s Law

6

Example 2

Assuming that carbonated water contains only CO2(1) and H2O(2), determine the compositions of the V & L phases in a sealed can of ‘soda’ & the P exerted on the can at 100C. Henry’s constant for CO2 in water at 100C is about 990 bar and x1=0.01.

Henry’s law for species 1 & Raoult’s law for species 2 are written;

111 HxPy satPxPy 222

With H1=990 bar & P2sat = 0.01227 bar (from steam tables at 100C)

barP

P

912.9

01227.099.099001.0

satPxHxP 2211

Then by Raoult’s law, Eq. 1 written for species 2;

0012.0

912.9

01227.099.0222

P

Pxy

sat

Whence y1=1-y2=0.9988, and the vapor phase is nearly pure CO2, as expected.

The 2nd assumption of Raoult’s Law is abandoned, taking into account the deviation from solution

ideality in L phase.

Thus, activity coefficient is introduced in Raoult’s Law

NiPxPy satiiii ,...,2,1

VLE BY MODIFIED RAOULT’S LAW

7

Activity coefficients are function of T & liquid phase composition, x

1i iy

i

satiii PxP

i

satiii Py

P

1

For bubble point

For dew point

Since;

1i ix

8

9

AZEOTROPE

A mixture that has a constant composition of liquid and vapor phase

When x1=y1, the dew point and bubble point curves are tangent to the same horizontal line

A boiling L of this composition produce a vapor exactly the same composition; L does not change in composition as it evaporates

VLE FROM K-VALUE CORRELATTIONS

The partition between liquid and vapor phases of a chemical species is equilibrium ratio, Ki.

i

ii x

yK

This quantity is called K-value.

10

satiii PxPy K-value for Raoult’s Law

P

PK

sati

i

K-value for modified Raoult’s Lawsatiiii PxPy

P

PK

satii

i

11

12

Hence,

For binary systems to solve for bubble point calculation;

1i iy

1 ii ixK

For binary systems to solve for dew point calculation;

1i ix

Hence, 1ii

i

K

y

13

14

K-value from DePriester chart-Low T range

K-value from DePriester chart-High T range

When given a mixture of composition at certain T or P;

Bubble point

- System is almost vaporized

- The given mole fraction is yi

- Need to satisfy equation 13

- Composition of dew is xi=yi/Ki

Dew point

- System is almost condensed

- The given mole fraction is xi

- Need to satisfy equation 14

- Composition of bubble is yi=Kixi

The End