11 Properties of

tions
Learning Objectives
I
Thischapter will enable readers to

1hs ideal
solution:
an
define
the Lewis-Randall rule;
derstand

law from the Lewis-Randall rule;

u n d e r

Raoult's
ce
oredict the phase equilibrium data in ideal solutions;
bble pressure, dew pressure,
perform bubt
ble
calculations;
temperature, dew
temperature.
factor: and flash
know
about the K
define
excess properties, activity and activity coefficient
express the
-Duhem equation in several
the activity coefficient of aa component in
calculate a
binary solution from
activity
fficient of the
other.component; a
knowledge of the
several excess
Gibbs free energy models;
know coefficients from
calculate the activity Margules, van Laar, Wilson and NRTL
calculate activity coefficients from group contribution methods, equations,
calculate activity coefficients using UNIQUAC and UNIFAC methods; and
know about
Henry's law.

The term solution means a hOmogeneous mixture ot two or more components whether in the
liauid or solid phase. When two diferent gases or liquids are mixed, it is sometimes posible
properties or the solution from the properties and the auantities
gas,
to oredict the approximateThus the volume ot a solution is approximately equal to the sum of
of the pure components. the solution. However, this is not true when
the volumes of
the pure components constituting
form a solution. Similarly, the internal energy of a mixture of
water and alcohol are mixed to the sum of the internal energies of sulfuric acid and
water is not equal to
sulfuric acid and Since a chemical engineer deas with a
is liberated in the form of heat.
water because energy the properties of solutions to carry
it is essential for him to estimate
solutions, of
variety of liquid This chapter deals with the thermodynamics
out a thermodynamic analysis of the process.
solutions.

11.1 Ideal solution

independent of
each other. Thesealomyu
Therefore, it
which are between
them.
atoms/molecules forces
consists of attractive
attractive torces by the
presence

dcal gas and there are no

not
influenced
of the
cules do not occupy any volume
sum
ideal gas
are the
equal to
of o n e mixture ia in whieh
ideal are mixed, the proper of the for mixtures torves
wO gases
the internal
energy
c a n be
expected intermolecular

e second ideal gas. Hence behavior

and the
mixtures

Similar size of the

the sun
gases.
nlernal energies of the constituent or rilate properticsas as the
properties sum

are approximate imated

uolecules of the constituent components The can

be. behavior
ofgases
entical.
series or a
P-v-l'

etween the like and unlike molecules al h o m o l o g o u s

the
of an
i d e a l gas

of adjacent members
a
discussing
discussing
that

isotop isomers or
While
approaches

of the proper the constituent compol gas

it was mentioned 0 , the

behavior of a
that as
 376 Chemical engineering thermodynamics

Pv = RT
gas which obeys the relation Pu =RT. In reality, no gas obeys the relation and
IS only an idealization to express the P-V-T behavior of gases. Similar idealization can it
can he be
to express the behavior of solutions also.
made
An ideal solution is defincd as a solution that obeys the Lewis and Randall rule which es.

(the fugacity of a component i in a solution) is equal

to the product of the
that J:
fraction r; and f' (the fugacity of the pure component i in the same mole
phase and at the soe
In other words, in an ideal solution
lution
Temperature and pressure).

(11.1)
Ifthe composition of the mixture is known, the partial fugacity of the component iin
an
ideal solution can be estimated from a knowledge of the pure component fugacity f which can
be evaluated by the methods discussed in Chapter 9. Equation 11.1 can be used to estimate the
partial fugacities in gaseous and vapor mixtures at low to moderate pressures since they obey
the Lewis and Randall rule. From Eqn.11.1, we can obtain

8nf
OP=(
8lnj

see Eqns.9.137-9.142] (11.2)

( (
or h, =
i; (11.3)
Since h = u+Pu, from Eqns.11.2 and 11.3, we obtain

, = u
(11.4)
where the superscript * indicates the properties of the pure componenti at the solution

temperature and pressure. Thus we find that in an ideal solution, the partial molar volume
enthalpy and internal energy of a component i are identical to the pure component properties
at the same temperature and pressure.
The changes in molar volume (Avn), enthalpy (Ahm) and internal energy (Aun) upon mixing
the components to form an ideal solution are given by

At= z , (7, -v;) =0 (11.5)

Ahn= z , (h, - h;) =0 (11.6)

Aun =(7,-u) =0 (11.7)

 l'roperties of solutions 377

i n molar volume, enthalpy and nternal energy upon mixing the purc
change in mola
I8 no solution.
ideal
an
of idcal solution. The volume
Thus therTe

components to to
torm
an
ideal gas xture as an cxample an
Cnts Isider
L e t u s n o w

mixture
and P is given by
N mo
by
occupied

NRT N)RT (11.8)

P
of component i is given by
volume
The partial molar

(8)
R v
P
F3/PM
(11.9)

mixture temperature and pressure.

component i at the
ma mloa
larr volume
vo of pure Hence
s the to pressure.
where v Is cqual
fugacity
ideal gas,
For an (11.10)
or P;
= 7;P
Amagat'e
and 11.10 show that
. Equations 11.9
of component shown in
pressure and Randall rule. It was
1s the partial of the LeWIs
non-identical
where D: follow as c o n s e q u e n c e s due to of
mixing
Dalton's law (per mole oft
the mixture)
law and the entropy change
Section 5.8 that
gases
is given by (5.47)
ideal

As=-R(71 ln «1 + T2 ln z2)
mixture as
multicomponent
for a
This can be generalized
(11.11)
=-R,lnz
C
(11.12)
Ahn-TAsn= RT2r; Ina;

solutions
in ideal
11.2 Phase equilibrium co-existing in
a
ot
statetwo
components, between the
containing c
phases, each
iilibrium
and vapor criterion for equilib
sider liquid T and pressure
P. The
cquilibrium at perature

phases is given by (9.148)

i =
1, 2, .., c Lewis and
Kandall
rut
application of the
solutions,
If the vapor phases are ideal
guid
(11.13)
gives
 378 Chemical emgineering tihermodynams

and f a P

where , and denote the mole fraction of ormporent in the liquid and (11.14
respectively, The fugacity of the pure mponent i at temperature T and re Po h
estimated by the method preented in Sectn 9.7, and is given by ressute Pn

From Eqns.9.148, 11.13, 11.14 and 9.150, we get

(9154
P= a,P or i= 1, 2,..., c)
which is known as Rault's Law. That is, Raoult's law states that the
(11.15
partial nres..
componcnt in the vaporphasc is cyual uo the saturation pressure times the mole fractin
componcnt in the liquid phasc. fraction of h
For a system containing c awmponcnts, Eqn.11.15 gives a et ofc
equations relsi.
variables 7 Pz and n (i = 1, 2, -., )since P is a function of T only. The tng t
variables have to satisfy the following additional constraints also:

2=1 and =1
(11.16
Applying Gibbs phasc rule to this system, we get

=c+2- P=c+2-2 =c

Hence cvariablcs arc requircd for the complete specification of the system.
Tor Pand cither r, or n (i
Usually one measures
1, 2,..., C-1). Then the remaining c variables can be
=

by solving Eqns.11.15. The folkowing cxample illustrates the method of calculations

estimate

Example 11. Benzene (1) and toluenc (2) form an ideal solution. The
vapor pressures of
benzene and toluene arc adequately represented by the Antoine cquation

B
log P=A-4C
+C
where P is in Torr and & is in "C.
(a) Prepare a P-a-y diagram at 95°C.
(b) Prepare a T-a-y diagram at 101.325 kPa (760 Torr) pressure.

Solution The Antoine constants |sce Table A.7 under

Appendix] are given by
A C
Benzene (1) 6.87987 1196.760 219.161
Toluene (2) 6.95087 1342.310 219.187

(a) At 95°C, the saturation pressures arc

1196.760
log1 Pi= 6.87987 or P =1176.21 Torr
954219.161
 Properties of solutions 379

1312.310
6.95087 - or
P 477.03 Torr
log10 P
=
95+219.187

where subs.
rints 1 and 2 refer to benzene and toluenc, respectively. For idcal solutions, thc
relation is given by
phase cquilcquilibrium
1, 2,..., c) (see Eqn.11.15)
P= riPi (
n P=riP (A)

y2 P= #2P2

N + 2 ) P = P = z1Pi + 2 P i = *1Pi +(1 -æi)P

+ (P-Pi)z (B)
or P P
from P to P. dependence The
shows that the total pressure P changes linearly
Eauation (B) in the liquid phase is shown in Fig.11.1.
o n the mole fraction ri of benzene
aftotal pressure

P S (1176.2)
1200

1000 * 2 h

PXP
P +

800 P
X

YP-
600
477)
400
yP X2P
200

0.6 0.8 1.0

0.2 0.4
in
mole fraction a1 of benzene
the vapor phase
versus

and (y2P) n
Fig.11.1 (y1 P)
Plot
of partial pressures

the liquid. with a liquid of

which is in
equilibrium
in the vapor phase
Tho Iraction ( ) of benzene
l e
mposition ri can be calculated from Eqn.nA) a

P
when = 0.1.
 380 Chemical engineering themodynamics

477.03)0.1 546.92 Torr

477.03 + (1176.21
=
P P+(P Pi)r1
-
-
=

0 . 1 x 1176.21 - = 0.2151
546.92
Similarly Pand yi can be calculated for several values of z1 in the range 0z1<1. Theo
of thesc calculations arc presented below.
results
P (Torr) T1 1 P (Torr)

0 0 477.0 0.6 0.7872 896.52

0.1 0.2151 546.92 0.8 0.9079 1036.36
0.2 0.3814 616.84 1.0 1.0 1176.2
0.4 0.6218 756.68

Figure 11.2 shows the P-a-y diagram for benzene-toluene system at 95°C

1200 (1176.2)

1000-Subcooled liquid ()
G

800
+V
P
600 p-Y

47 7

a400 Superheated vapour (V)

200

n2
0.2 0.4 0.6 0.8 1.0

Fig.11.2 Pe-y diagram for benzene-toluene at 95°C.

(b) The total pressure is fixed at P = 101.325 kPa (760 Torr). When the pressure is fixed, ine
saturation temperature changes with ti. To determine the saturation temperature the Ano
cquation can be rearranged as

B
t A-logoP

The saturation temperatures of benzene and toluene at 760 Torr are given by

t =80.1°C and t= 110.61"C

 8
Properties of solutions

a tcmperature t such that 80.1°C < < 110.61'C and at that temperature calculate Pi
Choose

hence 1 and yi. Assume t = 85°C. Then

a n d

881.54 Torr; Pi = 345.22 Torr

be rearranged as
Equation (B) can

P-
P- P
' 1
T60-345.22 = 0.7734
881.54 -345.22

-
.7734 x 881..54 = 0.8971
yi=
Then 760

at other temperatures are presented below.

Results of similar
calculations performed

t (C) Pi (Torr) Pi (Torr)

1.0 1.0
80.1 760.0
345.22 0.7734 0.8971
881.54
85
406.87 0.5753 0.7727
1020.65
90 0.4047 0.6263
1176.21 477.03
95 0.2566 0.4557
556.50
100 1349.47
0.1271 0.2578
646.14
1541.70
105
760.0 0.0 0.0
110.61

benzene-toluene system at 760 Torr is shown in Fig.11.3.

for
The T-t-y diagram
115
A

(T-Y Dew
Superheated vapor (V)

105
curve C
Tie line

C) 95
(T-x)Bub le curve G
85 02
Subcooled liquld ()

1.0
0.8
0.6

at 760 Torn
b e n z e n e - t o l u e n e
system
Pig.11.3 - diagram for