solution thermo(4)

CPE553 CHEMICAL ENGINEERING THERMODYNAMICS 5/9/2013

1

SOLUTION THERMODYNAMICS:

APPLICATION

Fundamental equations of solution thermodynamics are given in the

preceding chapter.

In this chapter, experimental vapor/liquid equilibrium (VLE) data are

considered, from which the activity coefficient correlations are derived.

2

• LIQUID PHASE PROPERTIES FROM VLE DATA

• MODELS FOR THE EXCESS GIBBS ENERGY

3

4

The figure shows coexistence of a vapor mixture and liquid solution in vapor/liquid equilibrium. T and P are uniform throughout the vessel and can be measured with appropriate instruments. Vapor and liquid samples may be withdrawn for analysis and this provides experimental values for mole fractions in the vapor {yi} and mole fractions in the liquid {xi}.


2

For species i in the vapor mixture, eq. (11.52) is written as

The criterion of vapor/liquid equilibrium, as given by eq. (11.48), is that

. Therefore,

VLE measurements are made at low pressure (P 1 bar) that the vapor

phase may be assumed an ideal gas. In this case . Therefore,

The fugacity of species i (in both the liquid and vapor phase) is equal to

the partial pressure of species i in the vapor phase.

5

ˆ ˆl v

i i if y P

ˆ ˆl v

i i if f y P

ˆ ˆv v

i i if y P

ˆ ˆl v

i if f

ˆ 1v

i

ˆˆ ii

i

f

y P (11.52)

Fugacity increases from zero at infinite dilution (xi = yi 0) to Pisat for

pure species i. This is illustrated by the data of Table 12.1 for methyl ethyl

ketone(1)/toluene(2) at 50oC (323.15K). The first three columns list a set

of experimental P-x1-y1 data and columns 4 and 5 show

6

1 1 2 2ˆ ândf y P f y P

ii sat

i i

y P

x P

Eqn. (12.1)

The fugacities are plotted in Fig. 12.2

as solid lines. The straight dashed

lines represent the Lewis/Randall

rule expressing the composition

dependence of the constituent

fugacities in an ideal solution,

This figure illustrates the general

nature of relationships

for a binary liquid solution at

constant T.

îd

i i if x f

7

ˆ id

i i if x f

1 2 1ˆ ând vs. f f x

7

The equilibrium pressure P varies

with compositions but has negligible

influence on the liquid phase values

of

Therefore, a plot at constant T and P

is as shown in Fig. 12.3 for species i

(i = 1, 2) in a binary solution at

constant T and P.

8

1 2ˆ ˆ and .f f

8


3

The lower dashed line in Fig. 12.3 represent the Lewis/Randall rule, which characterize ideal solution behavior.

Activity coefficient as defined by eq. (11.90) provide the actual behavior (non-ideal) from the idealize one:

The activity coefficient of a species in solution is the ratio of its actual fugacity to the value given by Lewis/Randall rule at the same T, P and composition. To obtain experimental values, both are defined as follow:

This is restatement of eq. (10.5), modified Raoult’s law, and allow calculation of activity coefficient from VLE data as shown in the last two columns of Table 12.1.

ˆ ˆ

î i

i idi i i

f f

x f f

9

1 2i ii sat

i i i i

y P y P (i , , ....N)

x f x P

ˆ ˆ and id

i if f

(12.1)

9

The solid lines in Fig. 12.2 and 12.3, representing experimental values of ,

become tangent to the Lewis/Randall rule lines at xi = 1.

In the other limit, becomes zero. The ratio is indeterminate

in this limit, and application of l’Hopital rule yields

Eq. (12.2) defines Henry’s constant, Hi as the limiting slope of the

curve at xi = 0.

As shown by Fig. 12.3, this is the slope of a line drawn tangent to the curve

at xi = 0. The equation of this tangent line expresses Henry’s law:

Henry’s law as given by eq. (10.4) follows immediately from this equation

when i.e. when has its ideal gas value.

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îf

ˆ0,i ix f î if x

00

ˆ ˆlimi

i

i ii

xi i x

f df

x dx

(12.2)

ˆ vs i if x

î i if x (12.3)

ˆ ,i if y P îf

10.4i i iy P x H

Henry’s law is related to the Lewis/Randal rule through the Gibbs/Duhem

equation.

Writing eq. (11.14) for a binary solution and replacing

Differentiate eq. (11.46) at constant T and P yields:

Therefore,

Divide by dx1 becomes

This is the special form of the Gibbs/Duhem equation.

11

1 21 2

1 1

ˆ ˆln ln0 const T,P

d f d fx x

dx dx (12.4)

11

byi i iM G

0i i

i

x d M

(11.14) 1 1 2 2 0 const , x d x d T P

ˆlni i iT RT f (11.46)

ˆlni id RTd f

1 1 2 2ˆ ˆln ln 0 const , x d f x d f T P

Substitute dx1 by –dx2 in the second term produces

In the limit as x1 1 and x2 0,

Because when x1 = 1, this may be rewritten as

According to eq. (12.2), the numerator and denominator on the right side

of this equation are equal.

12

1 1 2 21 21 2

1 2 1 1 2 2

ˆ ˆˆ ˆln lnor

ˆ ˆ

d f dx d f dxd f d fx x

dx dx f x f x

1 2

1 1 2 2

1 01 1 2 2

ˆ ˆlim lim

ˆ ˆx x

d f dx d f dx

f x f x

1 1f̂ f

2

12

2 201

1 1 2 210

ˆˆ1

ˆlim

x

xx

d f dxdf

f dx f x

00

ˆ ˆlimi

i

i i

xi i x

f df

x dx

(12.2)


4

Therefore the exact expression of Lewis/Randall rule applied to real

solutions is:

It also implies that eq. (11.83) provides approximately correct values of

when xi 1:

13

Henry’s law applies to a species as it approaches infinite

dilution in a binary solution, and the Gibbs/Duhem equation

insures validity of the Lewis/Randall rule for the other species

as it approaches purity.

1

11

1 1

ˆ

x

dff

dx

(12.5)

ˆif

ˆ ˆ id

i i i if f x f

14

When the second species is

methanol, acetone exhibits positive

deviations from ideality and with

chloroform it exhibits negative

deviations.

The fugacity of pure acetone, facetone

is the same regardless of the identity

of the second species.

Henry’s constants are represented

by the slopes of the two dotted

lines.

For a binary system,

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1 1 2 2ln lnEG

x xRT

(12.6)

Properties

of liquid

phase

ii sat

i i

y P

x P

Experimen

tal data

16

Activity coefficient of a species in solution becomes unity as the species becomes pure,

each ln i (i=1,2) tends to zero as xi 1.

As xi 0, species i becomes infinitely dilute, ln i approaches a finite limit, namely ln i.

For i 1 and ln i 0

P-x1 data points all lie

above dash line (Raoult’s

law) – positive deviation.

Raoult’s law


5

The most important step in determining composition of species in

solution (that is in equilibrium with vapor) is to calculate the species

activity coefficients. There are a number of models that can be used to

determine the activity coefficients depending on the type of species.

◦ Excess Gibbs free energy model e.g. Margules and Van Laar

◦ Local composition model e.g. Wilson, NRTL and UNIQUAC

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18

Model Equations

Margules

Van Laar

Wilson

NRTL

2

1 2 12 21 12 1ln 2x A A A x

2

2 1 21 12 21 2ln 2x A A A x

(12.10a)

(12.10b)

21 1 12 2

1 2

EGA x A x

x x RT (12.9a)

' '

12 21

' '

1 2 12 1 21 2

EG A A

x x RT A x A x

(12.16)

2'

' 12 11 12 '

21 2

ln 1A x

AA x

2'

' 21 22 21 '

12 1

ln 1A x

AA x

(12.17a)

(12.17b)

12 211 1 2 12 2

1 2 12 2 1 21

ln ln x x xx x x x

1 1 2 12 2 2 1 21ln ln

EGx x x x x x

RT

12 212 2 1 21 1

1 2 12 2 1 21

ln ln x x xx x x x

21 21 12 12

1 2 1 2 21 2 1 12

EG G G

x x RT x x G x x G

2

2 21 12 121 2 21 2

1 2 21 2 1 12

lnG G

xx x G x x G

2

2 12 21 212 1 12 2

2 1 12 1 2 21

lnG G

xx x G x x G

exp (i j)j ij

ij

i

V a

V RT

12 12 21 21

12 2112 21

exp( ) exp( )

G G

b b

RT RT

(12.18)

(12.24)

(12.19a)

(12.19b)

(12.20) (12.21a)

(12.21b)

19

Set of points in Fig. 12.5(b) provide linear

relation for GE/x1x2RT

where A21 and A12 are constant.

Alternatively,

From eqn. (12.10a) & (12.10b), the limiting

conditions of infinite dilution (at xi = 0)

21 1 12 2 1 2

EGA x A x x x

RT

20

21 1 12 2

1 2

EGA x A x

x x RT (12.9a)

(12.9b)

20

1 12 1 2 21 2ln at 0 and ln at 0A x A x

Fig. 12.5(b)

A12 = 0.372

A21 = 0.198

From Fig. 12.5(b), the intercepts at

x1 = 0 and x1 =1 of the straight line

GE/x1x2RT gives the parameters A12

and A21.

2

1 2 12 21 12 1ln 2x A A A x

2

2 1 21 12 21 2ln 2x A A A x

(12.10a)

(12.10b)


6

Linear relation for GE/x1x2RT

where A’21 and A’12 are constant.

Alternatively,

From eqn. (12.17a) & (12.17b), the limiting conditions of infinite dilution

(at xi = 0)

Values of A’12 and A’21 are obtained from the intercept of plot GE/x1x2RT vs. x1 at

x1 = 0 and x1=1. Or from plot of x1x2RT/GE vs. x1, intercept at x1=0 is 1/A’12

and at x1=1 is 1/A’21.

21

' '

12 21

' '

1 2 12 1 21 2

EG A A

x x RT A x A x

(12.16)

' '

1 2 12 1 21 2 1 2

' ' ' '

12 21 21 12

E

x x RT A x A x x x

G A A A A

2'

' 12 11 12 '

21 2

ln 1A x

AA x

2'

' 21 22 21 '

12 1

ln 1A x

AA x

(12.17a) (12.17b)

1 12 1 2 21 2ln ' at 0 and ln ' at 0A x A x

22

12 211 1 2 12 2

1 2 12 2 1 21

ln ln x x xx x x x

1 1 2 12 2 2 1 21ln lnEG

x x x x x xRT

12 212 2 1 21 1

1 2 12 2 1 21

ln ln x x xx x x x

exp (i j)j ij

ij

i

V a

V RT

(12.18)

(12.24)

(12.19a)

(12.19b)

The limiting conditions of infinite dilution (at xi = 0),

1 12 21 1 2 21 12 2ln ln 1 at 0 and ln ln 1 at 0x x

23

21 21 12 12

1 2 1 2 21 2 1 12

EG G G

x x RT x x G x x G

2

2 21 12 121 2 21 2

1 2 21 2 1 12

lnG G

xx x G x x G

2

2 12 21 212 1 12 2

2 1 12 1 2 21

lnG G

xx x G x x G

12 12 21 21

12 2112 21

exp( ) exp( )

G G

b b

RT RT

(12.20)

(12.21a)

(12.21b)

1 21 12 12 2 12 21 21ln exp( ) and ln exp( )

The limiting conditions of infinite dilution (at xi = 0),

All models provide eqns. for ln 1 and ln 2. This allow construction of a correlation of the original P-x1-y1 data set (experimental values). Eq. (12.1) is rearranged to give

Addition gives

From eqn. (12.1), therefore

Values of 1 and 2 from all models with their parameters are combined with experimental values of P1

sat and P2sat to calculate P and y1 by

eqs. (12.11) and (12.12) at various x1.

Then, P-x1-y1 diagram can be plotted to compare the experimental data and calculated values.

24

24

1 1 1 1 2 2 2 2andsat saty P x P y P x P

1 1 1 2 2 2

sat satP x P x P (12.11)

1 1 1 1 1 11

1 1 1 2 2 2

sat sat

sat sat

x P x Py

P x P x P

(12.12)

i ii sat

i i i i

y P y P

x f x P

(12.1)

24


7

25

Comparison of

experimental and

calculated data by

Margules eqn. These

clearly provide an

adequate

correlation of the

experimental data

points.

experimental

calculated

Fig. 12.5(a)

The Gibbs Duhem eqn imposes a constraint on activity coefficients that may

not be satisfied by a set of experimental values derived from P-x1-y1 data.

The Gibbs Duhem eqn is implicit in eq. (11.96), and activity coefficients

derived from this equation necessarily obey the Gibbs Duhem eqn.

These derived activity coefficients cannot possibly be consistent with the

experimental values unless the experimental values also satisfy the Gibbs

Duhem eqn.

26

1 21 2

1 1

ln ln0 const T,P

d dx x

dx dx

(12.7)

, ,

ln

j

E

i

iP T n

nG RT

n

(11.96)

If the experimental data are inconsistent with the Gibbs Duhem eqn, they are

necessarily incorrect as the result of systematic error in the data.

Therefore simple test is develop for the consistency with respect to the

Gibbs Duhem eqn of a P-x1-y1 data set.

Application of the test for consistency is represented by Eq. (12.13) which

requires calculation of the residuals

The right side of this equation is exactly the quantity that eq. (12.7), the

Gibbs/Duhem equation, requires to be zero for consistent data.

The residual on the left therefore provides a direct measure of deviation

from the Gibbs/Duhem equation.

The extent to which a data set departs from consistency is measured by the

degree to which these residuals fail to scatter about zero.

27

* *

1 1 21 2

2 1 1

ln lnln

d dx x

dx dx

(12.13)

Asterisk * denote the experimental values

1 21 2

1 1

ln ln0 const T,P

d dx x

dx dx

(12.7)

28

Experimental

values

calculated from

eqn. (12.1) &

(12.6)

1 2i ii sat

i i i i

y P y P (i , , ....N)

x f x P (12.1)

1 1 2 2ln lnEG

x xRT

(12.6)


8

29

P-x1-y1 data and experimental values,

ln 1*, ln 2

* and (GE/x1x2RT )* are

shown as points on Figs. 12.7(a) and

12.7(b) .

The data points of Fig. 12.7(b) for

(GE/x1x2RT )* show scatter. The

straight line drawn is represented by

This is eq. (12.9a) with A21 = 0.70

and A12 = 1.35.

Values of ln 1, ln 2 at the given

values of x1, derived from this eqn,

are calculated by eqs. (12.10) and

derived values of P and y1 at the

same values of x1 come from eqs.

(12.11) and (12.12).

This results are plotted as the solid

lines of Fig. 12.7(a) and 12.7(b). They

clearly do not represent a good

correlation of the data.

1 2

1 2

0.70 1.35EG

x xx x RT

2

1 2 12 21 12 1ln 2x A A A x (12.10a)

2

2 1 21 12 21 2ln 2x A A A x (12.10b)

1 1 1 2 2 2

sat satP x P x P (12.11) 1 1 11

1 1 1 2 2 2

sat

sat sat

x Py

x P x P

(12.12)

experimental

calculated

Barker’s method

30

Application of test for consistency

represented by Eq. 12.13 requires

calculation of the residuals

which plotted vs. x1 in Fig. 12.8.

The residuals distribute

themselves about zero, as required by

the test, but the residual ,

do not, which show the data fail to

satisfy the Gibbs/Duhem eqn.

1 2 and lnEG RT

EG RT

1 2ln

Smith, J.M., Van Ness, H.C., and Abbott, M.M. 2005. Introduction to

Chemical Engineering Thermodynamics. Seventh Edition. Mc

Graw-Hill.

31

PREPARED BY:

MDM. NORASMAH MOHAMMED MANSHOR

FACULTY OF CHEMICAL ENGINEERING,

UiTM SHAH ALAM.

[email protected]

03-55436333/019-2368303

solution thermo(4)

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