Solution Thermo Part 4

CHE553 Chemical Engineering Thermodynamics 24/04/2015

1

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

OVERVIEW

4

LIQUID PHASE PROPERTIES FROM VLE DATA

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 Vapor and liquid samples may be withdrawn for analysis and this provides experimental values for mole fractions in the vapor {yi} and mole fractions in the liquid {xi}.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

Fugacity

l vi i if y P

l vi i if f y P

v vi i if y P

l vi if f

1vi

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 2 andf 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.

idi i if x f

7

idi i if x f

1 2 1 and 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 2 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. For the calculation of experimental values, both are eliminated in favor of measurable quantities:

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

i i

i idi i i

f f

x f f

Activity coefficient

1 2i ii sati i i i

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

x f x P

and idi 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 lHopital rule yields

Eq. (12.2) defines Henrys 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 Henrys law:

Henrys law as given by eq. (10.4) follows immediately from this equation when i.e. when has its ideal gas value.

10

if

0,i ix fi if x

00

limi

i

i ii

xi i x

f df

x dx

(12.2)

vs i if x

i i if x (12.3)

,i if y P if

10.4i i iy P x H

Henrys 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 21 1

ln ln0 const T,P

d f d fx x

dx dx (12.4)

byi i iM G

0i ii

x d M

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

lni i iT RT f (11.46)

lni id RTd f

1 1 2 2 ln 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

ln lnor

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

lim lim

x x

d f dx d f dx

f x f x

1 1f f

2

12

2 201

1 1 2 210

1

lim

x

xx

d f dxdf

f dx f x

00

limi

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

Henrys 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

x

dff

dx

(12.5)

if

idi 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.

Henrys constants are represented by

the slopes of the two dotted lines.

For a binary system,

15

Excess Gibbs Energy

1 1 2 2ln lnEG

x xRT

(12.6)

Properties

of liquid

phase

ii sat

i i

y P

x P

Experiment

al 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 (Raoults

law) positive deviation.

Raoults 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

17

18

Model Equations

Margules

Van Laar

Wilson

NRTL

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

22 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 21 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 11 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 2EG

A x A x x xRT

20

Margules Equation

21 1 12 2

1 2

EGA x A x

x x RT (12.9a)

(12.9b)

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.

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

22 1 21 12 21 2ln 2x A A A x

(12.10a)

(12.10b)


6

Linear relation for GE/x1x2RT

where A21 and A12 are constant.

Alternatively,

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

(at xi = 0)

Values of A12 and A21 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/A12 and

at x1=1 is 1/A21.

Van Laar Equation

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

Wilson Equation

12 211 1 2 12 21 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 11 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

NRTL Equation

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.

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 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

Thermodynamic Consistency

1 21 21 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 21 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 sati 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 (G

E/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

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

22 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

Barkers 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

REFERENCE

PREPARED BY: NORASMAH MOHAMMED MANSHOR FACULTY OF CHEMICAL ENGINEERING, UiTM SHAH ALAM. [email protected] 03-55436333/019-2368303

Documents

Solution Thermo Part 4