Transcript Solution Thermodynamics: Theory

Chemical Engineering Thermodynamics
Lecturer: Zhenxi Jiang (Ph.D. U.K.)
School of Chemical Engineering
Zhengzhou University
1
Chapter 12
Solution Thermodynamics: Application
2
Chapter 12
Solution Thermodynamics: Application
All of the fundamental equations and
necessary definitions of solution
thermodynamics are given in the
preceding chapter. In this chapter, we
examine what can be learned from
experiment. Considered first are
measurements of vapor/liquid equilibrium
(VLE) data, from which activity coefficient
correlations are derived.
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Chapter 12
Solution Thermodynamics: Application
Second, we treat mixing experiments,
which provide data for property changes of
mixing. In particular, practical applications
of the enthalpy change of mixing, called
the heat of mixing, are presented in detail
in Sec. 12.4.
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12.1 Liquid phase property from VLE data
Figure 12.1 shows a vessel in which a
vapor mixture and a liquid solution coexist
in vapor/liquid equilibrium.
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12.1 Liquid phase property from VLE data
The temperature 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withdrawn for analysis, and this provides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}.
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12.1 Liquid phase property from VLE data
Fugacity
For species I in the vapor mixture, Eq.
(11.52) is written:
v
v
ˆ
ˆ
f i  yii P
The criterion of vapor/liquid equilibrium, as
given by Eq. (11.48), is that fˆ li  fˆ iv .
Therefore,
l
v
ˆ
ˆ
f i  yii P
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12.1 Liquid phase property from VLE data
Although values for vapor-phase fugacity
v
ˆ
coefficient i are easily calculated (Secs.
11.6 and 11.7), VLE measurements are
very often made at pressure low enough
(P ≤ 1 bar) that the vapor phase may be
v
ˆ
assumed an ideal gas. In this case i = 1,
and the two preceding equations reduce
to:
l
v
ˆ
ˆ
f i  f i  yi P
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12.1 Liquid phase property from VLE data
Thus, the fugacity of species i (in both
the liquid and vapor phases) is equal to
the partial pressure of species i in the
vapor phase. Its value increases from
zero at infinite dilution to Pisat for pure
species i. this is illustrated by the data of
Table 12.1 for the methyl ethyl
ketone(1)/toluene(2) system at 50℃.
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12.1 Liquid phase property from VLE data
The first three columns list a set of experimental P-x1-y1 data and columns 4 and 5 show:
fˆ 1  y1P
and
fˆ2  y2 P
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12.1 Liquid phase property from VLE data
The fugacities are plotted in Fig 12.2 as
solid lines. The straight dashed lines
represent Eq. (11.83), the Lewis/Randall
rule, which expresses the composition
dependence of the constituent fugacities
in an ideal solution:
ˆf id  x f
i
1 i
(11.83)
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12.1 Liquid phase property from VLE data
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12.1 Liquid phase property from VLE data
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12.1 Liquid phase property from VLE data
Although derived from a particular set of
data, Fig. 12.2 illustrates the general
nature of the fugacities of components 1
and 2 vs. x1 relationships for a binary
liquid solution at constant T. The
equilibrium pressure P varies with
composition, but its influence on the
liquid phase values of fˆ1 and fˆ2 is
negligible.
Thus a plot at constant T and P would look the same, as indicated in Fig.
12.3 foe species I (I = 1, 2) in a binary solution at constant T and P.
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12.1 Liquid phase property from VLE data
Activity Coefficient
The lower dashed line in Fig. 12.3,
representing the Lewis/Randall rule, is
characteristic of ideal-solution behavior. It
provides the simplest possible model for
the composition dependence of fˆ1 ,
representing a standard to which actual
behavior may be compared.
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12.1 Liquid phase property from VLE data
Activity Coefficient
The activity coefficient as defined by Eq.
(11.90) formalizes this comparison:
fˆi
fˆi
i 
 id
xi f i
fˆi
Thus the activity coefficient of a species in
solution is the ratio of its actual fugacity to
the value given by the Lewis/Randall rule
at the same T, P, and composition.
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12.1 Liquid phase property from VLE data
Activity Coefficient
For the calculation of experimental values,
id
ˆ
ˆ
both f i and f i are eliminated in favor of
measurable quantities:
yi P
yi P
i 

xi fi xi Pi sat
(i  1, 2,
, N)
(12.1)
This is a restatement of Eq. (10.5), modified Raoult’s law, and is
adequate for present purposes, allowing easy calculation of
activity coefficients from experimental low pressure VLE data.
Values from this equation appears in the last two columns of
Table 12.1.
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12.1 Liquid phase property from VLE data
Activity Coefficient
The solid lines in both Figs 12.2 and 12.3,
representing experimental values of fˆi ,
become tangent to the Lewis/Randall rule
lines at xi = 1. This is a consequence of
the Gibbs/Duhem equation. Thus, the ratio
fˆi xi is indeterminate in this limit, and
application of 1’Hopital’s rule yields:
fˆi  dfˆi 
lim

Ή i

 dx 
xi 0 x
i
 i  xi 0
(12.2)
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12.1 Liquid phase property from VLE data
Activity Coefficient
Equation (12.2) defines Henry’s constant
Hi as the limiting slope of the fˆi  xi
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.
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12.1 Liquid phase property from VLE data
Activity Coefficient
The equation of this tangent line
expresses Henry’s law:
fˆi  xiΉ i
(12.3)
Applicable in the limit as xi -> 0, it is also of
approximate validity for small values of xi. Henry’s law
as given by Eq. (10.4) follows immediately from this
equation when fˆ i  yi P , i.e., when fˆi has its ideal
gas value.
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12.1 Liquid phase property from VLE data
Activity Coefficient
Henry’s law is related to the Lewis/Randall
rule through the Gibbs/Duhem equation.
Writing Eq. (11.14) for a binary solution
ˆ  μ gives:
and replacing Mˆ i by G
i
i
x1 dμ1 + x2 dμ2 = 0
(const T, P)
Differentiation of Eq. (11.46) at constant T
and P yields: dμi = RT dln fˆ; whence,
i
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12.1 Liquid phase property from VLE data
Activity Coefficient
x1dln fˆ 1 + x2dln fˆ 2 = 0
(const T, P)
Upon division by dx1 this becomes:
d ln fˆ1
d ln fˆ2
x1
 x2
 0 (const T , P) (12.4)
dx1
dx2
This is a special form of the Gibbs/Duhem
equation.
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12.1 Liquid phase property from VLE data
Activity Coefficient
Through more operations the Eq. (12.5) is
obtained.
 dfˆ1 
 f1


 dx 
1  x 1

1
(12.5)
This equation is the exact expression of
the Lewis/Randall rule as applied to real
solutions.
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12.1 Liquid phase property from VLE data
Activity Coefficient
Henry’s law applies to a species as it
approaches infinite dilute in a binary
solution, and the Gibbs/Duhem equation
insures validity of the Lewis/Randall rule
for the other species as it approaches
purity.
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12.1 Liquid phase property from VLE data
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12.1 Liquid phase property from VLE data
The fugacity shown by Fig. 12.3 is for a species with
positive deviations from ideality in the sense of the
Lewis/Randall rule. Negative deviations are less
common, but are also observed; the fˆi  xi curve then
lies below the Lewis/Randall line. In Fig. 12.4 the
fugacity of acetone is shown as a function of
composition for two different binary liquid solutions at
50 ℃. When the second species is methanol, acetone
exhibits positive deviations from ideality. When the
second species is chloroform, the deviations are
negative. The fugacity of pure acetone is of course the
same regardless of the identity of the second species.
However, Henry’s constants, represented by slopes of
the two dotted lines, are very different for the two
cases.
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12.1 Liquid phase property from VLE data
Excess Gibbs Energy
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12.1 Liquid phase property from VLE data
Excess Gibbs Energy
In Table 12.2 the first three columns
repeat the P-x1-y1 data of Table 12.1 for
system methyl ethyl ketone(1)/ toluene(2).
These data points are also shown as circles
on Fig. 12.5(a). Values of lnγ1 and lnγ2 are
listed in columns 4 and 5, and are shown
by the open squares and triangles of Fig.
12.5(b).
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12.1 Liquid phase property from VLE data
Excess Gibbs Energy
They are combined for a binary system in accord
with Eq. (11.99):
GE
 x1 ln  1  x2 ln  2
(12.6)
RT
The values of G E/ RT so calculated are then
divided by x1 x2 to provide values of
G E/x1x2RT; the two sets of numbers are listed
in columns 6 and 7 of Table 12.2, and appear as
solid circles on Fig. 12.5(b).
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12.1 Liquid phase property from VLE data
Excess Gibbs Energy
The four thermodynamic functions, lnγ1,
lnγ2 , G E/ RT , and G E/x1x2RT, are
properties of the liquid phase. Figure
12.5(b) shows how their experimental
values very with composition for a
particular binary system at a specified
temperature. This figure is characteristic of
systems for which:
i 1
and
ln i  0
(i  1,2)
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12.1 Liquid phase property from VLE data
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12.1 Liquid phase property from VLE data
Excess Gibbs Energy
In such cases the liquid phase shows positive deviations from Raoult’s law behavior.
Because the 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. At the other limit, where
xi → 0 and species i becomes infinitely
dilute, ln i approaches a finite limit,
namely, ln i .
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12.1 Liquid phase property from VLE data
Excess Gibbs Energy
The Gibbs/Duhem equation, written for a
binary system, is finally divided to give:
d ln  1
d ln  2
x1
 x2
0
dx1
dx1
(const T , P) (12.7)
And
1
d (G / RT )
 ln
dx1
2
E
(12.8)
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12.1 Liquid phase property from VLE data
Data Reduction
Of the sets of points shown in Fig. 12.5(b),
those for G E/x1x2RT most closely
confirm to a simple mathematical relation.
Thus a straight line provides a reasonable
approximation to this set of points, and
mathematical expression is given to this
linear relation by the equation:
E
G
 A 21 x1  A12 x2
x1 x2 RT
(12.9a)
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12.1 Liquid phase property from VLE data
Data Reduction
where A21 and A12 are constants in any
particular application. Alternatively,
E
G
(12.9b)
 (A 21 x1  A12 x2 ) x1 x2
RT
Expressions for ln 1 and ln 2 are derived
from Eq. (12.9b) by application of Eq.
(11.96).
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12.1 Liquid phase property from VLE data
Data Reduction
Further reduction leads to:
ln  1  x [A12  2(A 21  A12 ) x1 ] (12.10a)
2
2
ln  2  x [A21  2(A12  A21 ) x2 ] (12.10b)
2
1
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12.1 Liquid phase property from VLE data
Data Reduction
These are the Margules equations, and
they represent a commonly used empirical
model of solution behavior. For the limiting
conditions of infinite dilution, they become

ln 1  A12  x1  0 and ln  2  A21  x2  0
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12.1 Liquid phase property from VLE data
Data Reduction
For the methyl ethyl ketone/toluene
system considered here, the curves of Fig.
12.5(b) for G E/ RT, ln  1 and ln  2
represent Eqs. (12.9b) and (12.10) with:
A12 = 0.372 and A21 = 0.198
These are values of the intercepts at x1= 0
and x1 = 1 of the straight line drawn to
represent the G E / x1 x2 RT data points.
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12.1 Liquid phase property from VLE data
Data Reduction
A set of VLE data has here been reduced
to a simple mathematical equation for the
dimensionless excess Gibbs energy:
E
nG
  0.198 x1  0.372 x2  x1 x2
RT
This equation concisely stores the
information of the data set.
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12.1 Liquid phase property from VLE data
Data Reduction
For binary system:
P  x P x  P
sat
1 1 1
sat
2 2 2
x P
y1 
x P  x  P
sat
1 1 1
sat
sat
1 1 1
2 2 2
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12.1 Liquid phase property from VLE data
Data Reduction
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12.1 Liquid phase property from VLE data
Data Reduction
A second set of data, for chloroform(1)/
1,4-dioxane(2) at 50°C, is given in Table
12.3, along with values of pertinent
thermodynamic functions. Figures 12.6(a)
and 12.6(b) display as points all of the
experimental values. This system shows
negative deviations from Raoult's-law
behavior.
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12.1 Liquid phase property from VLE data
Data Reduction
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12.1 Liquid phase property from VLE data
Data Reduction
Although the correlations provided by the
Margules equations for the two sets of VLE
data presented here are satisfactory, they
are not perfect.
Finding the correlation that best represents
the data is a trial procedure.
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12.1 Liquid phase property from VLE data
Thermodynamic Consistency
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12.1 Liquid phase property from VLE data
Thermodynamic Consistency
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12.1 Liquid phase property from VLE data
Thermodynamic Consistency
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12.1 Liquid phase property from VLE data
This end of Section 1 of Chapter 12
Questions?
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12.1 Liquid phase property from VLE data
Assignment
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12.1 Liquid phase property from VLE data
THANKS
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