Calculation Manual for
PhaseEquilibrium Engineers and Investigators
Preface
This
manual includes four chapters 14 which were presented in MTMSf24 held at
Koriyama, Japan, Aug. 27 – 29, 2024. Chapter 5 is its application which is
still added. This manual, pdf version including figures, is sold at the price
of 3000 JPY from TC Lines JP. A Perl program including kparameter
tables for predicting the Á_{i}^{}power functions of critical
ratios is sold (90,000 JPY). I hope that this manual contributes to phaseequilibrium
engineers and investigators.
Satoru Kato
TC Lines JP
Professor, Emeritus,
Tokyo Metropolitan University
kato@tclines.com
(order in this email address)
30
Aug. 2024, Chino, Japan
Contents.
Chapter 1. How to Classify
Liquid Mixtures
Chapter
2. How to Use Critical Ratios (the critical ratio is defined in the World
Patent application, PCT/ JP2012/
074514)
Chapter 3. How to Predict Infinite Dilution Activity
Coefficients
Chapter 4. How to Define
Hypothetical Liquids
Chapter 5. Applications
5.1 How
to Calculate Constant Pressure VLE
5.2 How
to Estimate Molecular Disorder and Strength in Liquid Mixtures
5.3 How
to Apply Entropy and Enthalpy Calculations to War Game Performance and
Intermediatestep Chess Game Performance
Chapter 1. How to Classify Liquid
Mixtures
Introduction
The
Margules equation can predict the activities of nonazeotropic binary liquid
mixtures, while the van Laar equation covers azeotropic and twoliquid phase
forming mixtures. The remaining problem is the application criteria of the
Wilson equation applying to alcoholhydrocarbon mixtures. The present manual
uses the Wohl equation for establishing the criteria.
Models and Calculation Manual for
Chapter 1
The Margules
equation is used for the activity calculations of binary nonazeotropic
mixtures:
lnÁ_{1} = x_{2}^{2} [A
+ 2(B – A) x_{1}] (11)
lnÁ_{2}
= x_{1}^{2} [B + 2(A – B) x_{2}] (12)
A
= lnÁ_{1}^{} (13)
B
= lnÁ_{2}^{} (14)
Azeotropic and twoliquid
phase forming mixtures.
The
van Laar equation is used for the activity calculations of binary azeotropic
and twoliquid phase forming mixtures:
lnÁ_{1}
= z_{2}^{2} [A + 2(BCA) z_{1}] (15)
lnÁ_{2} = z_{1}^{2}
[B + 2(A/CB) z_{2}] (16)
z_{1} = Cx_{1}/(Cx_{1}
+ x_{2}) (17)
z_{2} = 1 – z_{1} (18)
C
= A/B (19)
Alcoholhydrocarbon
azeotropic mixtures.
The
Wilson equation is used for the activity calculations of binary
alcoholhydrocarbon mixtures:
lnÁ_{1 }= ln(x_{1} + Š_{12}x_{2})
+ x_{2}[Š_{12}/(x_{1} + Š_{12}x_{2})
 Š_{21}/(Š_{21}x_{1}
+ x_{2})] (110)
lnÁ_{2
}= ln(x_{2} + Š_{21}x_{1})
– x_{1}[Š_{12}/(x_{1} + Š_{12}x_{2})
 Š_{21}/(Š_{21}x_{1} + x_{2})] (111)
lnÁ_{1}^{}
= ln(Š_{12}) + (1  Š_{21}) (112)
lnÁ_{2}^{}
= ln(Š_{21}) – (Š_{12} – 1) (113)
Distinction
criteria for the use of the Wilson equation.
Alcoholhydrocarbon
mixtures are classified into azeotropic mixtures. However, if the van Laar
equation is used, correlations result in twoliquid phase forming mixtures
(Prausnitz, ref. 1 page 232). Therefore, a criterion is needed for the use of the
Wilson equation. The Wilson equation should be used for alcoholhydrocarbon
mixtures, if the following relationship is satisfied:
˘_{ave} = (1/100)°y_{k,vanLaar}
– y_{k,Wohl} > 0.01 (114)
The
quantity, y_{k,vanLaar}, is calculated using the van Laar equation
at kth liquid phase mole fraction increasing with 0.01 division from 0
to 1. Furthermore, the quantity, y_{k,Wohl}, is calculated using
the Wohl Equation at the kth liquid phase mole fraction as follows (K. Wohl, Trans. AIChE 42 (1946) 215 
249) :
lnÁ_{1} = z_{2}^{2}
[A + 2(BCA)z_{1}] (115)
lnÁ_{2} = z_{1}^{2}
[B + 2(A/CB)z_{2}] (116)
z_{1} = Cx_{1}/(Cx_{1}
+ x_{2}) (117)
z_{2} = 1 – z_{1 }(118)
C = r_{1}/r_{2} (119)
Parameters
A and B in Equations (115) and (116) are determined at the
minimized ˘_{ave} value. Meanwhile, the A and B values of
the van Laar equation must represent the experimental VLE data. The Wohl
equation is used in the criterion equation, because it can express the
azeotropic and twoliquid phase forming behavior appearing in xy
relationships, while the Px relation shows nonazeotropic behavior, as
shown in Chapter 4.
Multicomponent activity
coefficients.
The present manual recommends the
multicomponent NRTL equation. The binary NRTL equation satisfactorily predicts
the activity coefficients of nonazeotropic, azeotropic and twoliquid phase forming
mixtures, except for alcoholhydrocarbon mixtures. Determine the binary
parameters of the NRTL equation, in advance, using binary Px or xy
data.
Data
Sources
Vapor
pressure calculations
Vapor pressures are calculated
using the Clapeyron equation. Details are shown in Chapter 3.
Constant
temperature binary VLE and LLE data
Constant temperature VLE and LLE
data are cited from ref. 2. Furthermore, constant temperature LLE data are cited
from ref. 3. The vaporliquid equilibria are expressed as follows;
Py_{i}_{ }= Á_{i}x_{i}p_{is}
(i
= 1, 2) (120)
Á_{i}
= Á_{i}^{(Pa)} (Ó_{is}/Ó_{i}^{V}) exp[V_{i}^{L}(PP_{a})/RT]
exp[V_{i}^{0}(Pp_{is})/RT] (121)
Activity coefficient
equations are used in Equation (120), assuming that pressure effects cancel
out in Equation (121). Parameters A, B and Š_{ij} are
optimized using fittings to constant temperature Px data (ref. 2).
Results and Discussion for Chapter 1
The
xy calculations of nonazeotropic mixtures.
In
Figure 11, y_{1} is plotted versus x_{1}
for the methanol (1) + water (2) binary at 323.15 [K]. The Margules equation is
used for the calculation of nonazeotropic mixture, in which Px data are
correlated, in advance, for determining optimum A and B
parameters. In Figure 11, the determined parameters are used for calculating xy
relationships. Therefore, data scatterings have been removed in Figure 11.
Figure 11 shows that experimental data are in good agreement with pure
prediction data described in Chapter 3. The mixture properties of the
predicting system are not used in the pure prediction. In Figure 12, y_{1}
is plotted versus x_{1} for the water (1) + n, ndimethylformamide (2) binary at
373.15 [K]. The Margules equation is used for the calculations of
nonazeotropic mixtures. Other activity coefficient equations, including the
Wilson, NRTL and UNIQUAC equations, satisfactorily correlate nonazeotropic
mixture data. However, a twoliquid phase forming behavior appears, if the van
Laar equation is used in this case (ref. 2, part 1 page XXXVI).
The
xy calculations of azeotropic mixtures.
In Figure 13,
y_{1} is plotted versus x_{1} for the ethanol (1)
+ water (2) binary at 323.15 [K]. The van Laar equation is used for the
calculations of minimum azeotropic mixtures. In Figure 14, y_{1}
is plotted versus x_{1} for the acetone (1) + chloroform (2)
binary at 323.15 [K]. The van Laar equation is used for the calculations of
maximum azeotropic mixtures. Other activity coefficient equations, including
the Wilson, NRTL and UNIQUAC equations, can calculate the VLE of azeotropic
mixtures.
The calculations of
twoliquid phase forming mixtures.
In Figure 15, y_{1}
is plotted versus x_{1} for the 2butanone (1) + water (2)
binary at 323.15 [K]. As shown in Figure 15, correlations using the van Laar
equation are satisfactory at x_{1} < x_{1low}
= 0.04. However, correlations are insufficient at x_{1high} ( =
0.726) < x_{1}. The problem occurs, if other activity
coefficient equations, including the NRTL and UNIQUAC equations, are used. The
problem is not solved yet. In Figure 16, y_{1} is
plotted versus x_{1} for the water (1) + furfural (2)
system at 293.15 [K]. In Figure 17, y_{1} is plotted
versus x_{1} for the cyclohexane (1) + aniline (2) system
at 298.15 [K]. Figures 16 and 17 demonstrate that the van Laar equation
satisfactory calculate the xy data of asymmetric binary systems.
Distinction criteria
applying to the Wohl equation.
In Figure 18, y_{1}
is plotted versus x_{1} for the ethanol (1) + 2, 2,
4trimethylpentane (2) binary at 273.15 [K]. The van Laar equation is used for
the correlation of experimental Px data. However, the correlation
provides a twoliquid phase forming behavior. The best fitting obtained from
the Wohl equation to the xy data calculated using the van Laar equation
provides ˘_{ave} = 0.058, satisfying the criteria, Equation (114).
Therefore, the Wilson equation is fitted to the xy data calculated using
the van Laar equation. As shown in Figure 18, the agreements between
experimental xy and calculated xy using the Wilson equation are
satisfactory. Other activity coefficient equations, including the Margules, NRTL
and UNIQUAC equations, provide similar twoliquid phase forming behaviors for
alcoholhydrocarbon binary systems. Equation (114) should be used to
alcoholhydrocarbon mixtures, although the Equation satisfies other mixtures,
such as amin + water and acetate + water binaries.
Figure Captions for Chapter
1
Figure 11. Nonazeotropic y_{1}
vs. x_{1} for the methanol (1) + water (2) system at 323.15 [K]
calculated using the Margules equation, xaxis:
x_{1}, yaxis: y_{1}, () Dul
Itskaya et al. cited from ref. 2, part 1 page 45, ()
Mc Glashan et al. cited from ref. 2, part 1 page 56, (~) Kurihara et al. cited from ref. 2, part 1c
page 77, (yellow solid line) pure prediction shown in Chapter 3.
Figure 12. Nonazeotropic y_{1} vs. x_{1}
for the water (1) + n, ndimethylformamide (2)
system at 373.15 [K] calculated using the Margules equation, xaxis: x_{1},
yaxis: y_{1}, () Doering cited from ref. 2, part 1 page
389, (yellow solid line) pure prediction shown in Chapter 3.
Figure 13. Minimum azeotropic
y_{1} vs. x_{1} for the ethanol (1) + water (2)
system at 323.15 [K] calculated using the van Laar equation, xaxis: x_{1}, yaxis: y_{1},
() Dul Itskaya et al. cited from ref. 2, part 1 page 161, () Udovenko and Fatkulina cited from ref. 2, part 1
page 191, (~) Chaudhry et al. cited
from ref. 2, part 1a pages 117 and 118, Pomberton and Mash cited from ref. 2,
part 1a page 143, Wilson et al. from ref. 2, part 1a page 155 and Kurihara et
al. from ref. 2, part 1c page 198, (yellow solid line) pure prediction shown in
Chapter 3.
Figure 14. Maximum azeotropic y_{1} vs. x_{1}
for the acetone (1) + chloroform (2) system at 323.15 [K] calculated using the
van Laar equation, xaxis: x_{1}, yaxis: y_{1},
() Abbott et al. cited from ref. 2, part 3b page 9, ()
Abbott and van Ness from ref. 2, part 3b page 10, (~)
Abbott and van Ness from ref. 2, part 3b page 1, (~) Goral et al. cited
from ref. 2, part 3b pages 17 and 18, Abbott and van Ness from ref. 2, part 3c
page 108, Gorbunov from ref. 2, part 3c page 110, Mueller and Kearns from ref.
2, part 3+4 page 101, Ravincvich and Nikolaev from ref. 2, part 3+4 page 104,
Roeck and Schrceder from ref. 2, part 3+4 page 113 and Schmidt from ref. 2,
part 3+4 page 120, (yellow solid line) pure prediction shown in Chapter 3.
Figure 15. twoliquid phase forming y_{1}
vs. x_{1} for the 2butanone (1) + water (2) system at 323.15
[K] calculated using the van Laar equation, xaxis:
x_{1}, yaxis: y_{1}, ()
experimental data by Sokolova and Morachevsky cited from ref. 2, part 1b page
208, () correlation of the experimental data using the van Laar equation with A = 3.44 and B =
1.82, (yellow solid line) pure
prediction providing mutual solubilities x_{1low} = 0.04001 and x_{1high}
= 0.7267.
Figure 16. Asymmetric twoliquid phase forming y_{1}
vs. x_{1} for the water (1) + furfural
(2) system at 293.15 [K], xaxis: x_{1}, yaxis: y_{1},
() Briggs and Comings cited from ref. 3, part 1 page 272, (yellow solid line)
pure prediction
Figure 17. Asymmetric twoliquid phase forming y_{1}
vs. x_{1} for the cyclohexane (1) +
aniline (2) system at 298.15 [K], xaxis: x_{1},
yaxis: y_{1}, () Buchner and Kleyn cited from ref. 3,
part 1 page 365, (yellow solid line) pure prediction.
Figure 18. y_{1} vs. x_{1}
for the ethanol (1) + 2,2,4trimethylpentane at 273.15 [K], (Ł)
experimental data by Kretschmer et al. cited from ref. 2, part 2a page 501 ()
correlation fitted to the experimental data using the van Laar equation, (red
solid line) correlation calculated using the Wilson equation, minimizing ˘_{ave}
= 0.058
Conclusion for Chapter 1
Nonazeotropic VLE data are calculated
using the Margules equation, while the van Laar equation should be used for the
VLE calculations of azeotropic and twoliquid phase forming mixtures. Use the
Wilson equation, if Equation (114) is satisfied for the alcoholhydrocarbon
mixtures.
Nomenclature for Chapter 1
A Margules
parameter defined in Equation (13)
B Margules
parameter defined in Equation (14)
C A/B
defined in Equation (19)
P system
pressure
P_{a} reference
pressure
P_{Ci} critical
pressure of component i
p_{is} vapor
pressure of component i
R gas
constant
r_{1} molar
volume of component 1
r_{2} molar
volume of component 2
T system
temperature
V_{i}^{L} partial molar volume of component i^{ }^{}
V_{i}^{0} molar
volume of pure liquid i at T
x_{1} liquid
phase mole fraction of component 1
x_{2} liquid
phase mole fraction of component 2
x_{1low}, x_{1high} mutual
solubility
y_{i} vapor
phase mole fraction of component i
y_{k} kth
vapor phase mole fraction
z_{1} modified
mole fraction of component 1 defined in Equation (17)
z_{2} modified
mole fraction of component 2 defined in Equation (18)
˘_{ave} average
deviation determined using Equation (114)
Á_{i} activity
coefficient of component i
Á_{i}^{} infinite
dilution activity coefficient of component i
Ó_{is} fugacity
coefficient of the saturated vapor i at p_{is}
Ó_{i}^{V} vapor
phase fugacity coefficient of component i in the mixture at P
Š_{ij} Wilson
parameter defined in Equations (110) and (111)
Superscript
(P) reference
pressure at P
infinite
dilution
Chapter 2. How to Use Critical Ratios
Introduction
Using binary constant temperature VLE and LLE data, the
present manual shows that infinite dilution activity coefficients are expressed
using power functions of critical ratios.
Models
and Calculation Manual for Chapter 2
Using binary constant temperature VLE and LLE data, Kato showed that
infinite dilution activity coefficients are expressed using critical ratios as
follows (S. Kato, MTMSf24):
lnÁ_{i}^{}
= lna_{Xi} + b_{Xi}lnX (i=1,
2) (21)
The parameter, Á_{i}^{}, denotes
the infinite dilution activity coefficient of component i. The critical
ratio, X, is defined as follows:
X
= (p_{1s} + p_{2s}) / (P_{C1} + p_{2s}) (22)
The quantities, p_{1s} and p_{2s},
denote the vapor pressures of pure component 1 and 2, respectively, while P_{C1}
denotes the critical pressure of component 1. It should be noted that the value
of X is determined, if temperature, T, and pure substances 1 and
2 are fixed. In VLE analyses, lower boiling point substances are specified as
component 1. Therefore, at critical point of component 1, X = 1 holds.
In many cases, lowboiling point VLE data cover X < 0.2. The utmost
advantage of using Equation (21) is that it enables us to purely predict
prepower constant, a_{Xi}, and consonant, b_{Xi}.
The details of the pure prediction of a_{Xi} and b_{Xi}
are shown in Chapter 3.
Data
Sources
Constant temperature binary VLE data are cited from ref. 2, in which Px
data are correlated using the Margules and van Laar equations. Furthermore,
mutual solubility data in ref. 3 are used to determine van Laar parameters.
Results
and Discussion for Chapter 2
In Figure
21, Á_{1}^{}
and Á_{2}^{}
are plotted versus X for the benzene (1) + heptane (2) binary at
20 [K] < T – 273.15 < 155 [K].
The Margules equation is used for the nonazeotropic mixtures. Figure 21 shows
that Á_{1}^{}
and Á_{2}^{}
data are represented using power functions. The power functions are almost
passing through the origin, lnÁ_{i}^{} = 0 at lnX = 0,
which characterizes binary hydrocarbon mixtures. In Figure 22, Á_{1}^{}
is plotted versus 1/T for the benzene (1) + heptane (2) binary at
20 [K] < T – 273.15 < 155 [K]. Figure 22 demonstrates that data
scatterings are the same between Figures 21 and 22. The utmost disadvantage
of Á_{i}^{} = ae^{b/T}
type function is that partial molar excess enthalpy and entropy loses
temperature effects. In Figure 23. Á_{1}^{} and Á_{2}^{}
are plotted versus X for the benzene (1) + cyclohexane (2) binary
at 8 [K] < T – 273.15 < 150 [K]. Figure 23 shows that correlation
lines are passing through the origin.
In Figures 24 and 25,
Á_{1}^{} and Á_{2}^{}
are plotted versus X for the methanol (1) + water (2) and ethanol
(1) + water (2) systems, respectively. Figures 24 and 25 demonstrate that
data scatterings are not small for alcohol + water mixtures. In Figures 26
and 27, Á_{1}^{}
and Á_{2}^{} are plotted versus X
for the acrylonitrile (1) + water (2) and acetaldehyde (1) + water (2)
systems, respectively. Figures 26 and 27 show that straight lines are
obtained at Á_{i}^{}
>> 1. In Figures 28,
29 and 210, Á_{1}^{} and Á_{2}^{}
are plotted versus X for the water (1) + n, ndimethylformamide
(2), water (1) + n, ndimethylacetamide (2) and acetone (1) + chloroform (2)
systems, respectively. Figures 28, 29 and 210 show that straight lines are
obtained at Á_{i}^{}
1. Figures
211 and 212 demonstrate that twoliquid phase forming mixtures
and minimum azeotropic mixtures provide straight lines, if lnÁ_{i}^{}
is plotted versus lnX.
Figure Captions for Chapter 2
Figure 21. Á_{i}^{}
vs. X for the benzene (1) +
heptane (2) binary at 20 [K] < T – 273.15 < 155 [K] calculated using
the Margules equation,
xaxis: X, yaxis: Á_{1}^{}
and Á_{2}^{}, () Á_{1}^{}, () Á_{2}^{}, data are cited from ref.
2, part 6b pages 120 – 157, 6c 464 – 472 and 6e 574 – 587.
Figure
22. Á_{1}^{} vs. 1/T for the benzene (1) + heptane (2)
binary at 20 [K] < T – 273.15 < 155 [K] calculated using the
Margules equation, xaxis: X,
yaxis: Á_{1}^{}, () Á_{1}^{}, data
are cited from ref. 2, part 6b pages 120 – 157, 6c 464 – 472 and 6e 574 – 587.
Figure
23. Á_{i}^{} vs. X for the benzene (1) + cyclohexane
(2) binary at 8 [K] < T – 273.15 < 150 [K] calculated using the
van Laar equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 6b pages 204 – 239, 6c 215 – 231 and 6d 250 – 272.
Figure
24. Á_{i}^{} vs. X for the methanol (1) + water (2)
binary at 10 [K] < T – 273.15 < 115 [K] calculated using the
Margules equation, xaxis: X,
yaxis: Á_{1}^{} and Á_{2}^{}, () Á_{1}^{},
() Á_{2}^{}, data are cited
from ref. 2, part 1 pages 38 – 73, 1a 49 – 57, 1b 29 – 33 and 1c 57 – 99.
Figure
25. Á_{i}^{} vs. X for the ethanol (1) + water (2)
binary at 10 [K] < T – 273.15 < 130 [K] calculated using the van
Laar equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 1 pages 157 – 196, 1a 117 – 155, 1b 83 – 108
and 1c 181 – 253.
Figure 26. Twoliquid
phase forming Á_{i}^{} vs. X for the acrylonitrile
(1) + water (2)
binary at 0 [K] < T – 273.15 < 100 [K] calculated using the van
Laar equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 1 pages 38 – 73, 1a 49 – 57, 1b 29 – 33 and 1c
57 – 99.
Figure
27. Á_{i}^{} vs. X for the acetaldehyde
(1) + water (2) binary at 0 [K] < T – 273.15 < 100 [K]
calculated using the Margules equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 1 pages 83 – 94, 1a 78 – 81, 1b 38 – 40 and 1c
134.
Figure 28. Á_{i}^{}
vs. X for the water (1) + n,
ndimethylformamide (2)
binary at 30 [K] < T – 273.15 < 100 [K] calculated using the
Margules equation,
xaxis: X, yaxis: Á_{1}^{}
and Á_{2}^{}, () Á_{1}^{}, () Á_{2}^{}, data are cited from ref.
2, part 1c pages 389 – 401.
Figure 29. Á_{i}^{}
vs. X for the water (1) + n,
ndimethylacetamide (2)
binary at 20 [K] < T – 273.15 < 80 [K] calculated using the
Margules equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 1a pages 319 – 322.
Figure 210. Maximum
azeotropic Á_{i}^{} vs. X for the acetone (1) +
chloroform (2) binary at 0 [K] < T – 273.15 < 55 [K] calculated
using the van Laar equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 3b pages 9 – 24, 3c 108 – 113 and 3+4 87 –
125.
Figure
211. Twoliquid phase forming Á_{i}^{} vs. X for the
2butanone (P)
+ water (2) binary at 0 [K] < T – 273.15 < 120 [K] calculated
using the van Laar equation, xaxis: X,
yaxis: Á_{1}^{} and Á_{2}^{}, () Á_{1}^{},
() Á_{2}^{}, data are cited
from ref. 2, part 1 pages 359, 1a 271, 1b 206 – 210 and ref. 3, part 1 page
217.
Figure
212. Minimum azeotropic Á_{i}^{} vs. X for the ethanol
(P)
+ 2,2,4 trimethylpentane (2) binary at 0 [K] < T – 273.15 < 76
[K] calculated using the van Laar equation,
xaxis: X, yaxis: Á_{1}^{} and Á_{2}^{},
() Á_{1}^{}, () Á_{2}^{},
data are cited from ref. 2, part 2a pages 501 – 504, 2c 467 and 2h 501.
Conclusion
for Chapter 2
Using constant temperature binary VLE and LLE data,
the present manual demonstrates that the infinite dilution activity
coefficients are expressed with power functions of the critical ratios.
Chapter 3 How
to Predict Infinite Dilution Activity Coefficients
Introduction
The present manual provides a pure
prediction method for infinite dilution activity coefficients of binary systems,
in which (i) vapor pressures are calculated using the Clapeyron equation, (ii)
experimental infinite dilution activity coefficient data are correlated using
the power functions of critical ratios, (iii) partial molar excess entropy and
enthalpy are calculated from the power functions, (iv) the exponents of the
power function are calculated using the linear approximation of partial molar
excess enthalpy, (v) the universal line for exponents is established, (vi) the
universal line for prepower constants is presented and finally (vii) a
criterion judging the data quality of infinite dilution activity coefficients
is presented.
Models
and Calculation Manual for Chapter 3
Relationships between temperature
and the infinite dilution activity coefficients
The infinite dilution activity coefficient of component i in a binary
system is related with the critical ratio, X, as follows (S. Kato,
MTMSf24):
lnÁ_{i}^{}
= ln a_{Xi }+ b_{Xi }ln X (i=
1, 2) @ (31)
The critical ratio, X, is
defined as follows:
X = (p_{1s} + p_{2s}) /
(P_{C1}+p_{2s}) @ (32)
The constants a_{Xi} and b_{Xi}
are system dependent. It was shown that Equation (31) holds at temperatures
covering T < T_{Ci} using 2700 binary VLE and LLE
systems (S. Kato, MTMfS24).
Partial molar excess quantity definition
The partial molar excess enthalpy
of component i is defined as follows:
h_{i}^{E} = R[ÝlnÁ_{i}^{}/Ý(1/T)] (33)
Using Equations
(1) to (3), b_{Xi} is related with h_{i}^{E}
as follows:
b_{Xi} lnX
= Ćh_{i}^{E}^{}/RT (34)
The temperature
parameter, Ć, is defined as follows:
Ć=T
lnX / [ÝlnX /Ý(1/T)] (35)
It should be
noted that parameters, Ć and lnX, are temperature dependent. Moreover,
they are pure substance parameters. Meanwhile, partial molar excess entropy is defined
as follows:
lnÁ_{i}^{} = h_{i}^{E
}/ RT – s_{i}^{E}/R (36)
Using Equations
(31) to (36), the ratio of entropy to enthalpy is expressed as follows:
(s_{i}^{E}/R)/(
h_{i}^{E}/RT) = Ć ln a_{Xi} / (b_{Xi}
ln X) + ( 1 – Ć) (37)
Equations (34)
and (37) are starting equations for the Á_{i}^{}pure
prediction. They include two parameters, a_{Xi} and b_{Xi},
to be purely predicted.
The
linear approximation of exponents b_{X1}
and b_{X2}
The present investigation uses the following linear
approximation:
b_{Xi} = c_{1}
h_{i}^{E}/RT + c_{2}Ć+ c_{3}
ln X + c_{4}_{@}_{ }(38)
The parameters c_{1} to c_{4} are
constants. Each term in the right side of Equation (38)
is the consisting quantity of b_{Xi} shown in Equation (34). Eliminating
h_{i}^{E}/RT from Equations. (34) and (38), b_{Xi}
is given as follows:
b_{Xi,pre}=(c_{2}Ć+c_{3
}ln X+c_{4})/(1c_{1 }ln X/Ć) @(i = 1, 2) @@@@(39)
In the pure
prediction process, first, the T, Ć and lnX values of a Á_{1}^{}predicting
binary system are determined. Using data fitting, it is possible to regress the
c_{1} to c_{4} constants to the experimental b_{X1},
h_{1}^{E}/RT, Ć and lnX values of the Á_{1}^{}predicting
binary system. Constants c_{1} to c_{4} for the
second component are similarly determined using experimental b_{X2},
h_{2}^{E}/RT, Ć and lnX values. Moreover,
the c parameters in Equation (38) must satisfy the following
requirement;
Ýb_{Xi}/ÝT = 0 (310)
Meanwhile, the
parameter c_{1} is a linear function of 1/(h_{i}^{E}/RT).
Furthermore, constants, c_{2} to c_{4}, linearly increase
with increasing h_{i}^{E}/RT as follows:
c_{1} = k_{11}/(h_{i}^{E}/RT)+k_{21 }(311)
c_{2} = k_{21}h_{i}^{E}/RT+k_{22 } (312)
c_{3} = k_{31}h_{i}^{E}/RT+k_{32 } (313)
c_{4} = k_{41}h_{i}^{E}/RT+k_{42 } (314)
Parameters k_{ij}
denote constants. In addition, kvalues are determined using the
following quadratic function, if binaries are chosen form the 2700 systems, in
which their Ć values are close to that value of Á_{i}^{}predicting
system.
k_{ij} = l (lnX)^{2} + m lnX
+ n (i
= 1, 2, 3, 4, j=1, 2) (315)
The parameters l,
m and n are constants. For the use of pure prediction, the k_{ij}
values must be tabulated, in advance, in a kvalue parameter table for b_{X1}.
The kvalue parameter table, supplied from the TC Lines JP, includes
2700 binary systems, in which each system has k_{ij} values
determined using parameter fittings to b_{X1} experimental
values. Empirically, experimental b_{Xi} values are related with
prediction values, b_{Xi,pre}, as follows:
b_{xi,pre} = żb_{Xi} + Ŕ (i= 1, 2) (316)
Parameters, ż
and Ŕ are universal constants. Their values are empirically determined using
experimental b_{Xi} values and predicted b_{Xi,pre}
values. It should be stressed that pure prediction is possible, because the
constants are universal. Two components must have the same universal constants.
The
pure prediction of b_{X1}
and b_{X2}
The pure prediction of b_{X1}
consists of following steps: (i) fix the system temperature, T, and a
binary system forming a Á_{i}^{}predicting system, (ii)
calculate Ć using Equation (35) and lnX using Equation (32). (iii)
determine constants l, m and n in Equation (315) using
data regression to k_{ij} obtained from the kvalue
parameter table for b_{X1}. Binaries are arbitrarily chosen from
the table including 2700 systems, if they have Ć values close to that of the Á_{i}^{}predicting
system, (iv) assume a b_{X1} value, (v) determine h_{1}^{E}/RT
value using Equation (34), (vi) determine c_{1} to c_{4}
values using Equations (311) to (314), (vii) calculate b_{X1,pre}
value using Equation (39), (viii) calculate b_{X1} using
Equation (316), finally, (ix) repeat the trialanderror process between (iv)
and (viii) steps until the same value of b_{X1} is obtained. The
same procedure is applied to determine b_{X2} value using a kvalue
parameter table for b_{X2}, separately determined. The
calculation process is pure prediction, because the experimental values of b_{X1}
and b_{X2} are not used in the calculation procedures.
The
linear approximation of prepower constants a_{X1} and a_{X2}
Following the linear approximation
used for the b_{Xi} exponent, the present investigation assumes
the prepower constant, a_{Xi}, as follows:
lna_{Xi} /b_{Xi}= d_{1}
[(s_{i}^{E}/R)/( h_{i}^{E}/RT)]+
d_{2}Ć+ d_{3}lnX + d_{4} (i=1,
2)
(317)
The parameters d_{1}, d_{2} d_{3}
and d_{4} denote constants. Each term in the right side of
Equation (17) is the consisting quantity of a_{Xi} shown in
Equation (37). Eliminating T s_{i}^{E}/ h_{i}^{E}
in Equations (37) and (317), a_{Xi} is given as follows:
lna_{Xi,pre} /b_{Xi} =
[d_{1}(1Ć) + d_{2}Ć+ d_{3}lnX
+ d_{4}]/(1+d_{1}Ć/lnX) (i = 1, 2) (318)
Similarly to b_{Xi} cases, parameters
d_{1} to d_{4} are linear function of T s_{i}^{E}/h_{i}^{E}
as follows:
d_{1} = k_{11}/( T s_{i}^{E}/h_{i}^{E})+k_{21 }@@@@@ (319)
d_{2} = k_{21} T s_{i}^{E}/h_{i}^{E}
+k_{22 } (320)
d_{3} = k_{31} T s_{i}^{E}/h_{i}^{E}+k_{32 } (321)
d_{4} = k_{41} T s_{i}^{E}/h_{i}^{E}+k_{42 } (322)
The same
relationships, Equations (315) and (316), are applied to this case.
The
pure prediction of a_{X1}
and a_{X2}
The same procedure determining b_{X1}
and b_{X2} are applied to the cases of lna_{X1}/b_{X1}
and lna_{X2}/b_{X2}. It should be noted that b_{X1}
and b_{X2} are predetermined without using a_{X1}
and a_{X2}. The Marquardt method is used for
the optimization of parameters c_{1} to c_{4} and d_{1} to d_{4}.
Calculations are performed using programing codes written in Perl, an
interpreter language.
Data Sources
Vapor
pressures
Vapor pressures are calculated using the
Clapeyron equation as follows:
p_{is}
= P_{Ci}exp(C_{Clap1}T_{Clap}^{3}
+ C_{Clap2}T_{Clap}2 + C_{Clap3}T_{Clap}) (323)
T_{Clap}
= 1 – 1/T_{ri} (324)
Clapeyron temperature, T_{Clap},
is advantageous, because it allows to define hypothetical liquids (S. Kato,
AIChE J. 51 (2005) 32753285). Vapor pressure data are cited from ref. 5
which uses the Wagner equation covering temperatures T < T_{Ci}.
Furthermore, the Antoine equations are cited from ref. 2 and ref. 3.
Constant
temperature binary VLE and LLE data
Constant temperature VLE and LLE
data are cited from ref. 2. Furthermore, constant temperature LLE data are
cited from ref. 3. They include a total number of 2700 binary systems. Each
binary system includes three or more datasets at different temperatures.
Infinite dilution activity coefficients are calculated as follows: the Margules
equation is used for nonazeotropic mixtures, because Margules parameters are
reported in ref. 2. The van Laar equation is used for azeotropic and twoliquid
phase forming mixtures with the parameters reported in ref. 2. For LLE data,
van Laar parameters are determined using LLE data reported in ref. 3.
Meanwhile, the Wilson equation is used for alcoholhydrocarbon mixtures along
with the distinction criteria identified in Chapter 1, in which the Wohl
equation is used.
Results and
Discussion for Chapter 3
Vapor
pressures calculated using the Clapeyron equation
In Figure
31, P_{ri }is
plotted vs. T_{Clap} for ethanol at temperatures 20 [K] < T
– 273.15 < 120 [K]. Figure 31 shows that the Clapeyron equation
satisfactorily correlates vapor pressures calculated using the Wagner equation.
Furthermore, Figure 31 demonstrates that reduced pressure almost linearly
increases with increasing Clapeyron temperature, passing through the origin.
Infinite
dilution activity coefficients correlated using the power function of critical
ratio
In Figure
32a, Á_{1}^{}
and Á_{2}^{} are plotted vs. X for the benzene (1) +
heptane (2) binary. In Figure 32a, bold solid lines represent power function
correlations. Figure 32a demonstrates that power function correlations are
satisfactory. In Figure
32b, Á_{1}^{}
and Á_{2}^{} are plotted vs. 1/T for the benzene (1) +
heptane (2) binary. Figures 32a and 32b show that correlation errors are the
same. Furthermore, Figure 32a includes dotted lines, representing power
functions satisfying ideal solution behavior at the critical temperature of
benzene. Figure 32a shows that, for hydrocarbon mixtures, power function
correlations satisfying ideal solution approximation at the critical
temperature of lowerboiling point substance are acceptable.
The
values of partial molar excess enthalpy and entropy at infinite dilution
It is partial molar excess quantities that provide
pure prediction methods for infinite dilution activity coefficients. In Figure 33a, h_{1}^{E}/R and h_{2}^{E}/R
are plotted vs. X for the ethanol (1) + water (2) binary at temperatures
10 [K] < T  273.15 < 200 [K]. Figure 33a shows that partial
molar excess enthalpies calculated using Equations (1), (2) and (3) vary with
increasing temperature. The constants, a_{Xi} and b_{Xi},
are correlated using Á_{i}^{} values calculated from the van Laar equation,
because the ethanol + water binary is classified into an azeotropic mixture. In
Figure 33b, s_{1}^{E}/R and s_{2}^{E}/R
are plotted vs. X for the ethanol (1) + water (2) binary at temperatures
10 [K] < T  273.15 < 200 [K]. Figure 33b shows that partial
molar excess entropies calculated using Equations (31), (32), (33) and (36)
vary with increasing temperature. It should be noted that enthalpies and
entropies are constant, if Á_{i}^{}= a exp(b/T)type
temperature effects are used.
Entropyenthalpy
compensation rule
The pure prediction of infinite
dilution activity coefficients uses partial molar excess entropyenthalpy
compensation rule. In Figure
34, s_{1}^{E}/R
and s_{2}^{E}/R are plotted vs. h_{1}^{E}/RT
and h_{2}^{E}/RT at temperatures 40 [K] < T
 273.15 < 120 [K]. Figure 34 shows that partial molar excess
entropyenthalpy compensation rule almost holds for the hexane (1) + heptane
(2) binary, because correlation lines are passing through the origin. However,
the methanol (1) + water (2) binary is remote from the compensation rule,
because correlation lines deviate from the origin. Of the 2700 binaries
compiled in ref. 2 and ref. 3, a very limited number of binaries satisfies the
partial molar excess entropyenthalpy compensation rule. Figure 34
demonstrates that partial molar excess entropyenthalpy compensation rule
cannot be a universal relationship.
The
preparation of kvalue parameter table for b_{X1}, b_{X2}, a_{X1}
and a_{X2}
To demonstrate the preparation
procedure of the kvalue parameter table for b_{X1}, the
ethanol (1) + water (2) binary is selected as a Á_{i}^{}predicting
binary system. First, system temperature is fixed at 274.15 [K]. The values of
Ć from Equation (35) and lnX from Equation (32) are 0.3798 and 7.943,
respectively. Next, using data regression, c_{1} to c_{4}
constants are determined from Equation (38), in which b_{X1} =
0.0726, h_{1}^{E}/RT = 1.519, Ć = 0.3798 and lnX
= 7.934 of the Á_{i}^{}predicting binary system are used.
Determined constants are listed in Table
31. In the calculation, the
experimental value of b_{X1} from Equation (31) is determined
from lnÁ_{1}^{} vs. lnX plot using experimental data
cited from ref. 2. The van Laar equation is used in this case, because the
ethanol + water system is classified into an azeotropic mixture at 274.15 K.
The value of h_{1}^{E}/RT is calculated using
Equation (33) or Equation (34). Next, a Âclosest binary is selected using
the following filter, Â:
(Ć_{k} – Ć_{pre})/ Ć_{pre} <
Â (325)
(lnX_{k
} lnX_{pre})/lnX_{pre} < Â (326)
The suffix k
denotes kth binary selected from the 2700 a_{Xi}, b_{Xi}registered
binary systems. The suffix pre denotes Á_{i}^{}predicting
binary, the ethanol (1) + water (2) binary in this case. In the case of ethanol
(1) + water (2) binary, the Âclosest binary was the ethanol (1) +
1propanol (2) system at the smallest Â (=
0.013). That means two binaries, Á_{i}^{}predicting binary
and Âclosest binary, are selected satisfying Â
0.013. In the next step, c_{1} to c_{4}
constants of the Âclosest binary are determined using b_{X1}, h_{1}^{E}
values of the Âclosest binary system. These values are included in Table 31
with the determined c_{1} to c_{4} values.
Finally, using Equations (311) to (314), k_{ij} values are
determined by solving simultaneous two linear equations including c_{i
}(i_{ }= 1, 4) values and h_{i}^{E}
values of the Á_{i}^{}predicting binary and Âclosest binary.
Determined k_{ij }values of the ethanol (1) + water (2) binary
are listed in Table 31. The procedures are repeated at temperatures covering 1
[K]
T – 273.15
T_{max}, in which T_{max}
denotes the maximum temperature providing the existence of two binaries, Á_{i}^{}predicting
binary and Âclosest binary. The value of T_{max} was 45 +
273.15 [K] for the ethanol (1) + water (2) binary. In this temperature range, b_{X1,pre}
is constant. Preparation procedures for b_{X2}, a_{X1}
and a_{X2} are the same with that of b_{X1}.
The
determination of universal lines
In Figure
35, k_{11} is
plotted versus lnX at 274.15 [K] using 75 binary systems satisfying (Ć_{k} – Ć_{pre})/ Ć_{pre} <
0.05 at Ć_{pre}= 0.3798 for the ethanol (1) + water (2) Á_{i}^{}predicting
binary. Figure 35 shows that data are very scattering. The other k_{ij}
values show similar scattering. However, the following modifications on k_{21}
and k_{22} greatly improve the convergence of the universal
lines, that is, Equation (316):
k_{21}
= (Ć/lnX  k_{12} – k_{31}lnX  k_{41})/Ć (327)
k_{22
}= (k_{11} + k_{32}lnX + k_{42})/Ć (328)
Equation
(327) is obtained from the equality in the coefficients of h_{i}^{E},
Ć/lnX = k_{12} + k_{21}Ć + k_{31}lnX
+ k_{41}. Furthermore, Equation (328) arises from the
intercepts, k_{11} + k_{22}Ć + k_{32}lnX
+ k_{42} = 0. In Table 32, to establish the universal lines, 25 binaries are
selected, because they have a large number of VLE and LLE datasets. In Figure 36, using binaries shown in Table 32, b_{X1pre}
is plotted versus experimental b_{X1}. The following converged
universal line is established:
b_{Xipre}
= b_{Xi} (i=1, 2) (329)
The same
converged line is obtained between b_{X2pre} and b_{X2}
as shown in Figure 36. In Figures 37, experimental b_{X1} and b_{X2}
are directly compared with predicted values using the universal line, Equation
(329). The pure prediction is satisfactory, because prediction errors are less
than 1 % at b_{Xi} > 0.2.
A similar calculation process is
applied to lna_{Xi}/b_{Xi}. In Figure 38, the following universal line for lna_{Xi}/b_{Xi}
is shown:
lna_{Xipre}/b_{Xi}
= (lna_{Xi}/b_{Xi}) (i=1, 2) (330)
The values of k_{ij}
are scattering again in this case. However, the following modifications on k_{21}
and k_{22} greatly improve the universal line:
k_{21}
=  (lnX/Ć+ k_{12} + k_{31}lnX + k_{41})/Ć (331)
k_{22}
= (1Ć)lnX/Ć k_{11} – k_{32}lnX – k_{42})/Ć (332)
Equation (331)
is obtained from the equality in the coefficients of Ts_{i}^{E}/h_{i}^{E},
lnX/Ć= k_{12} + k_{21}Ć + k_{31}lnX
+ k_{41}. Furthermore, Equation (332) arises from the
intercepts, k_{11} + k_{22}Ć + k_{32}lnX
+ k_{42} = (1 Ć)lnX/Ć. In Figure
39, experimental a_{X1}
and a_{X2} are directly compared with the values predicted using
the universal line, Equation (330), and the modifications, Equations (331)
and (332). The pure prediction is satisfactory, because prediction errors are
less than 1 %.
The
establishment of VLE and LLE data quality tests
Using the pure prediction
procedures of a_{Xi} and b_{Xi }(i= 1, 2), Á_{i}^{}
is calculated from Equation (31). In
Figure 310, experimental Á_{i}^{}
is directly compared with predicted for the 25 binaries shown in Table 32 at
274.15 [K]. The agreement between the two is encouraging. Pure predictions
often provide data quality judgement criteria. Therefore, the Á_{i}^{}pure
prediction procedure is applied to 100 binaries selected from the 2700 binary
systems including 25 binaries in Table 32. A criterion, }30 %, was obtained,
because experimental@Á_{i}^{} values are identical with
predicted Á_{i}^{} values within the criterion which satisfies
the statistical }95 % data reliability limit. Finally, the present
investigation proposes the following criteria for the VLE and LLE data quality
judgement:
(Á_{i}^{}_{exp}
 Á_{i}^{}_{pre})/ Á_{i}^{}_{exp}
< 0.3 (i=1,2) (333)
Suffix exp
denotes experimental data. Suffix pre denotes predicted data. Applying Equation
(333) to the 25 binary systems in Table 32, the tetrachloromethane
+ benzene binary provided the nonreliable values of both Á_{1}^{} and Á_{2}^{}.
Figure Captions for Chapter 3
Figure 31.
The vapor pressures of ethanol at 10 [K] < T – 273.15< 120 [K], xaxis: T_{Clap }(=11/T_{ri}), yaxis: P_{ri} (=p_{is}/P_{Ci}), () calculated from the Wagner equation using
parameters cited from ref. 4, () correlated using the Clapeyron equation.
Figure 32a.
Á_{i}^{} vs. X for the benzene (1) + heptane (2)
binary, xaxis:
X, yaxis:
Á_{1}^{} and Á_{2}^{}, () benzene, (Ą)
heptane, () power function correlations, (. . .) power
function correlations with ideal solutions at X = 1, Á_{1}^{}
and Á_{2}^{}are calculated using the Margules equation fitted
to the constant temperature VLE data cited from ref. 2, part 6b 120157, part
6c 464472 and part 6e 574587.
Figure 32b.
Á_{i}^{} vs. 1/T for the benzene (1) + heptane (2)
binary, xaxis: 1/T, yaxis:
Á_{1}^{} and Á_{2}^{}, () benzene, (Ł)
heptane, () exponential function correlations, Á_{1}^{}
and Á_{2}^{} are calculated using the Margules equation fitted
to the constant temperature VLE data cited from ref. 2, part 6b 120157, part
6c 464472 and part 6e 574587.
Figure 33a. h_{i}^{E}/R
vs. X for the ethanol (1) + water (2) binary at 10 [K] < T 
273.15 < 200 [K], xaxis: X, yaxis:
h_{1}^{E}/R
and h_{2}^{E}/R [K]. ()
ethanol, (Ł) water, Equations (31), (32) and (33)
are used with a_{X1} = 7.21, b_{X1} = 0.0726, a_{X2}
= 2.25, b_{X2} = 0.0248, Á_{1}^{} and Á_{2}^{}
data are calculated using the van Laar equation fitted to the constant
temperature VLE data cited from ref. 2, part 1 150196, part 1a 116155, part
1b 83111 and part 1c 176253.
Figure 33b. s_{i}^{E}/R
vs. X for the ethanol (1) + water (2) binary at 10 [K] < T 
273.15 < 200 [K], xaxis: X, yaxis:
s_{1}^{E}/R
and s_{2}^{E}/R. ()
ethanol, (Ł) water, Equations (31), (32), (33)
and (36) are used with a_{X1} = 7.21, b_{X1} =
0.0726, a_{X2} = 2.25, b_{X2} = 0.0248, Á_{1}^{}
and Á_{2}^{} data are calculated using the van Laar equation
fitted to the constant temperature VLE data cited from ref. 2, part 1 150196,
part 1a 116155, part 1b 83111 and part 1c 176253.
Figure 34.
s_{i}^{E}/R vs. h_{i}^{E}/RT
at 40 [K] < T  273.15 < 120 [K], xaxis:
h_{1}^{E}/RT and h_{2}^{E}/RT, yaxis: s_{1}^{E}/R and s_{2}^{E}/R, (♦) s_{1}^{E}/R vs. h_{1}E^{}/RT
for the hexane (1) + heptane (2) binary, (Ą)
s_{2}^{E}/R vs. h_{2}^{E}/RT
for the hexane (1) + heptane (2) binary, (Ł)
s_{1}^{E}/R vs h_{1}^{E}/RT
for the methanol (1) + water (2) binary and (~)
s_{2}^{E}/R vs h_{2}^{E}/RT
for the methanol (1) + water (2) binary, Á_{1}^{} and Á_{2}^{}
data are calculated using the Margules equation cited from ref. 2.
Table 31.
The values of k_{ij} for the ethanol (1) + water (2) system at
274.15 [K].
@ 
Á_{i}^{}predicting
binary system 
Âclosest binary
system 
@ 
ethanol (1) + water
(2) 
ethanol (1) +
1propanol (2) 
Ć 
0.3798 
0.3751 
lnX 
7.943 
8.024 
b_{X1} 
0.0726 
0.1143 
h_{1}^{E}/RT 
1.519 
2.445 
c_{1} 
0.70488 
0.70488 
c_{2} 
2.92333 
2.9233 
c_{3} 
0.01008 
0.01008 
c_{4} 
0.82153 
0.82152 
k_{11} 
0.0744 

k_{21} 
0.7225 

k_{31} 
0.0323 

k_{41} 
0.2646 

k_{12} 
0.7353 

k_{22} 
1.1570 

k_{32} 
0.0891 

k_{42} 
0.1744 
@ 
Figure 35.
lnX vs. k_{11} for the binaries satisfying (Ć_{k}
– Ć_{pre})/ Ć_{pre} < 0.05 with Ć_{pre} = 0.3798
for the ethanol (1) + water (2) binary at T = 274.15 [K], xaxis:
lnX, yaxis:
k_{11}, () k_{11}, () correlation using Equation (315).
Table
32. 25 binary systems used for the establishment of the universal lines.
N_{t} 
N_{P} 
N_{T} 
first comp.
(1) + second comp. (2) 
148 
70 
78 
ethanol +
water 
107 
34 
73 
water +
acetic acid 
93 
43 
50 
methanol +
water 
74 
52 
22 
benzene +
cyclohexane 
74 
40 
34 
benzene +
ethanol 
68 
34 
34 
acetone + water 
66 
37 
29 
methanol +
benzene 
64 
48 
16 
tetrachloromethane
+ benzene 
62 
31 
31 
benzene +
heptane 
59 
40 
19 
acetone +
chloroform 
58 
30 
28 
acetone +
methanol 
55 
17 
38 
2propanol +
water 
48 
23 
25 
1propanol +
water 
45 
36 
9 
1,4dioxane +
water 
44 
10 
34 
formic acid +
water 
42 
28 
14 
acetone +
benzene 
41 
17 
24 
heptane +
toluene 
41 
26 
15 
benzene +
1propanol 
40 
38 
2 
methanol +
cyclohexane 
39 
18 
21 
water +
1butanol 
38 
19 
19 
tertbutanol
+ water 
36 
25 
11 
ethanol +
heptane 
35 
17 
18 
ethyl acetate
+ ethanol 
35 
19 
16 
methyl
tertbutyl ether + methanol 
35 
16 
19 
2butanone +
watr 
N_{t}: total
number of VLE and LLE datasets in ref. 2 and 3 N_{P}: number of
constant pressure datasets 

N_{T}: number of
constant temperature datasets 
Figure 36.
The universal line for b_{X1} and b_{X2} obtained
using 25 binaries listed in Table 2, xaxis: b_{X1} and b_{X2}, yaxis: b_{X1pre} and b_{X2pre}, () b_{X1pre} calculated using
Equations (39) and (311) to (315), () the universal line for b_{Xi}: b_{Xipre}
= b_{Xi}, () b_{X2pre}.
Figure 37.
Comparison between experimental b_{Xi} and predicted b_{Xipre}
calculated using the universal line, Equation (329), for the 25 binaries
included in Table 2, xaxis:
experimental
b_{Xi}, yaxis:
predicted
b_{Xipre}, () b_{X1}
vs. b_{X1pre}, () b_{X2}
vs. b_{X2pre} and (yellow solid line) diagonal line.
Figure 38.
The universal line for lna_{X1}/b_{X1} and lna_{X2}/b_{X2}
obtained using 25 binaries listed in Table 32, xaxis: lna_{X1}/b_{X1}
and lna_{X2}/b_{X2}, yaxis: lna_{X1pre}/b_{X1}
and lna_{X2pre}/b_{X2}, () lna_{X1pre}/b_{X1}
calculated using Equations (318) to (322), (331) and (332), () the
universal line for lna_{Xi}/b_{Xi}: lna_{Xipre}/b_{Xi}
= lna_{Xi}/b_{Xi}, ()
lna_{X2pre}/b_{X2}.
Figure 39.
Comparison between experimental a_{Xi} and predicted a_{Xipre}
calculated using the universal line, Equation (330), for the 25 binaries
included in Table 32, xaxis:
experimental
a_{Xi}, yaxis:
predicted
a_{Xipre}, () a_{X1}
vs. a_{X1pre}, () a_{X2}
vs. a_{X2pre} and (yellow solid line) diagonal line.
Figure 310.
Establishment of VLE and LLE data quality judgement criteria calculated using
the 25 binary systems shown in Table 32 at 274.15 [K]. xaxis: experimental
Á_{i}^{}, yaxis
Á_{i}^{}
predicted using pure prediction method for a_{Xi} and b_{Xi}, () Á_{1}^{},
() Á_{2}^{}, (yellow solid
line) diagonal line, (blue solid line) +30% reliability line, (purple solid
line) 30% reliability line.
Conclusion
for Chapter 3
The present manual provides a pure prediction method for Á_{i}^{}
values. Furthermore, a criterion for data quality judgement, Equation (333),
has been established.
Nomenclature
for Chapter 3
a_{Xi} prepower
function constant for component i defined in Eq. (31)
b_{Xi} exponent
of power function for component i defined in Eq. (31)
C_{Clap1}, C_{Clap2}, C_{Clap3} constants defined in the Clapeyron Eq. (312)
c_{1}, c_{2}, c_{3}, c_{4} constants
defined in Eq. (38)
d_{1}, d_{2}, d_{3}, d_{4} constants
defined in Eq. (310)
h_{i}^{E} partial
molar excess enthalpy of component i at infinite dilution
P_{Ci} critical
pressure of component i
P_{C1} critical
pressure of component 1
p_{is} vapor
pressure of component i
P_{ri} reduced
pressure of component i
p_{1s} vapor
pressure of component 1
p_{2s} vapor
pressure of component 2
R gas
constant
s_{i}^{E} partial
molar excess entropy of component i at infinite dilution
T system
temperature
T_{Ci} critical
temperature of component i
T_{Clap} Clapeyron
temperature defined in Eq. (313)
T_{ri} reduced
temperature (= T/T_{Ci}) of component i
X critical
ratio defined in Eq. (32)
Á_{i}^{} infinite
dilution activity coefficient of component i
Ć temperature
parameter defined in Eq. (35)
Chapter 4@How to
Define Hypothetical Liquids
Introduction
Liquids and vaporliquid
equilibria (VLE) exist at the temperature T_{C1}
T. The Equationofstates approaches have
been commonly used for the analyses of highpressure VLE. However, a very
limited number of investigations tried to apply thermodynamics to binary VLE
analyses at temperatures, T_{C1}
T < T_{C2}, because
hypothetical liquids are needed above the critical temperature of lower
boilingpoint substances. The reduced temperature of the component i,
ln(p_{is}/P_{Ci}), is proportional to the
Clapeyron temperature factor, 11/T_{ri}, at the temperature
range, 0.7T_{Ci} < T < T_{Ci}.
Therefore, it is natural to extend the proportionality to the temperature range
T_{Ci }< T, which defines the stable vapor pressure
values of the hypothetical liquids of pure substance i. It is rational
to apply the Wohl activity coefficient equation to the highpressure VLE
analyses, because the Wohl equation most rigidly satisfy the GibbsDuhem
equation (S. Kato, Fluid Phase Equilib. 297 (2010) 192  199).
The present manual clarifies that
highpressure binary Px and xy VLE data can be satisfactorily
correlated using the three parameter Wohl equation and hypothetical liquids.
Models and Calculation Manual for Chapter 4
The activity coefficient model
Using activity coefficients, the phase
equilibria are described as follows (ref. 5, R. C. Reid, J. M. Prausnitz, B. E.
Poling, The Properties of Gases and Liquid 1987 Chap. 8 MacGrawHill):
Py_{i}_{ }= Á_{i}x_{i}p_{is}
(i
= 1, 2)
(41)
Á_{i} = Á_{i}^{(Pa)}
(Ó_{is}/Ó_{i}^{V}) exp[V_{i}^{L}(PP_{a})/RT]
exp[V_{i}^{0}(Pp_{is})/RT]
(42)
Even at
the highpressure VLE, the present investigation assumes that the pressure
effects on fugacity coefficients cancel out. Therefore, using the Wohl equation
(K. Wohl, Trans. AIChE 42 (1946) 215  249), the activity coefficients are
given as follows:
lnÁ_{1} = Ć_{2}^{2}
[A + 2(BCA)Ć_{1}]
(43)
lnÁ_{2} = Ć_{1}^{2}
[B + 2(A/CB)Ć_{2}]
(44)
Ć_{1} = Cx_{1}/(Cx_{1}
+ x_{2}) (45)
Ć_{2} = 1 – Ć_{1
}(46)
The
parameters A, B and C are system dependent. Their values
are constant at 0 < x_{1} < 1. The Wohl equation
originally uses molar volume ratios as the values of C.
Model parameter determination
The system pressure is calculated using
Equations (41), (43) and (44) as follows:
P = Á_{1}x_{1}p_{1s}
+ Á_{2}x_{2}p_{2s} (47)
The parameter values of A, B
and C are determined using the following objective function:
OF_{P}=(1/n)°(P_{k,exp}
– P)/P_{k,exp} (48)
The
index k varies from 1 to n. Using constant temperature Px
data, A, B and C values minimizing the objective function,
OF_{P}, are determined. Meanwhile, the vapor phase mole fraction
of component 1 is given as follows:
y_{1} = Á_{1}x_{1}p_{1s}/P (49)
A different set of A, B and C
is determined using the following objective function:
OF_{y} = (1/n)  (y_{1k,exp}
– y_{1})/y_{1k,exp } (410)
The
Marquardt method is used for the determination of parameters A, B
and C. Calculations are performed using programs written in Perl, an
interpreter language.
Hypothetical liquid definition
Following the Clapeyron plots proposed by
Kato (S. Kato, AIChE J. 51 (2005) 32753285), the vapor pressure of the
hypothetical liquid is calculated as follows:
ln(p_{is}/P_{Ci})
= h(11/T_{ri}) (411)
The
constant, h, is determined from the linear plot, ln(p_{is}/P_{Ci})
vs. (11/T_{ri}), passing through the origin. Using the vapor
pressure data covering 0.7T_{Ci} < T < T_{Ci},
the proportionality line is extended to T > T_{ci},
which defines vapor pressure values of the hypothetical liquids.
Data Sources
Vapor pressures
Vapor pressure values were calculated using
the Wagner equation. Wagner parameters are compiled in the literature^{5}
Highpressure VLE data
Highpressure VLE data are cited
from Knappfs datasets^{4} compiled in Dechema Chemistry Data Series.
Results and Discussion for Chapter 4
The vapor pressure of hypothetical liquid
In Figure 41, ln(p_{is}/P_{Ci})
is plotted vs. 11/T_{ri} for four substances at temperatures
covering 0.7T_{Ci }< T < T_{Ci}.
Figure 41 demonstrates that vapor pressure values can be satisfactorily
correlated with straight lines passing through the origin. Therefore, the
straight lines are extended above critical to determine h values of
hypothetical liquids. In Table 41, h values determined from Eq.
(411) are listed. It should be noted that the h values of hydrocarbons
are close to six.
The correlation of highpressure Px and xy
data
In Figure 42a, experimental Px and Py data are plotted vs. x_{1}
and y_{1}, respectively, for the helium4 + nitrogen binary at
121.74 K. Figure 42a includes a correlated line
determined by minimizing OF_{P }values. The minimized value,
that is, the average relative deviations defined in Equation (412) is 0.28 %
for the Px correlation.
(ARD)_{P}=
(1/n)°(P_{k,exp} – P)/P_{k,exp}_{min} (412)
In Figure
42b, experimental y_{1} data are plotted vs. x_{1}
for the helium4 + nitrogen binary at 121.74 K. Figure 42b includes a
correlated line determined by minimizing OF_{y }values. The
minimized value, that is, the average relative deviations defined in Equation (413)
is 0.27 % for the xy correlation.
(ARD )_{y}= (1/n)°(y_{1k,exp}
– y_{1})/y_{1k,exp}_{min} (413)
Figure 42b
shows a phasesplitting trend appearing in the liquid phase, because y_{1}
has the maximum value. The three parameter Wohl equation provides better
correlation results than two parameter equations including the Margules, van
Laar, Wilson, NRTL and UNIQUAC equations. Figures 42a and 42b show that the
hypothetical liquid correlations using the three parameter Wohl equations are
satisfactory.
In Figure 43, the values of A
are plotted vs. critical ratios, X, defined as follows:
X = (p_{1s} + p_{2s})/(P_{C1}
+ p_{2s}) (414)
As
shown in Equations (43) and (44), A and B are respectively
related with infinite dilution activity coefficients as follows:
A = lnÁ_{1}^{} (415)
B = lnÁ_{2}^{} (416)
Using
low and near critical temperature VLE and LLE data, Kato (S. Kato, MTMS24)
demonstrated that infinite dilution activity coefficients are expressed using
the power function of X at X 1. Similarly to lowboiling point VLE,
Figure 43 demonstrates that the hypothetical liquids and Px data
provide the power function of X. Moreover, Figure 43 shows that A
values determined from xy data converge to a smooth line. Therefore,
hypothetical liquids and the Wohl equation are advantages, because i) the
satisfactory correlations of Px and xy data over critical are obtained,
further, ii) Aparameter prediction using the line shown in Figure 43
is possible. The Aparameter increases with increasing X.
However, the increasing trends appearing in helium4 hypothetical liquids do
not appear in the other hypothetical liquids.
In Figure 44, A/B is
plotted vs. C for the helium4 + component 2 system including nitrogen
and carbon monoxide as a component 2 at 77 K < T < 122 K. Figure 44
shows that C = A/B holds for the xy data. As shown
in Equations (43) and (44), the Wohl equation is identical with the van Laar
equation, if C = A/B holds. Using lowpressure VLE and LLE data, it
is shown in Chapter 1 that the van Laar equation represents azeotropic or
twoliquid phase forming behavior. Therefore, Figure 44 demonstrates that the xy
data reflect azeotropic or twoliquid forming behavior represented by the van
Laar equation. If C = 1 holds, the Wohl equation is identical with the
Margules equation. However, the parameter C widely varies from 4 to 4 as
shown in Figure 44.
In Figure 45, experimental
Px and Py data are plotted vs. x_{1} and y_{1},
respectively, for the hydrogen + methane binary at 144.26 K. Figure 45 shows
that correlations are satisfactory. In Figure 46, A values are
plotted vs. X for the hydrogen + component 2 system including nitrogen
and methane as a component 2 at 77 K < T < 174 K. Figure 46 shows
that the increasing trends of A values are very clear. In Figure 47,
A/B is plotted vs. C for the hydrogen + component 2 including
nitrogen and methane at 77 K < T < 174 K. Figure 47 demonstrates
that the van Laar equation represents the Px data, because C = A/B
holds. Meanwhile, the xy data reflect different factors, because A/B
values deviate from the line representing C = A/B.
In Figure 48, experimental
Px and Py data are plotted vs. x_{1} and y_{1},
respectively, for the carbon dioxide + propane binary at 277.59 K. Figure 48
shows that Px and Py correlations fairly well represent the beak
behavior appearing near x_{1} = 1. Weighing objective functions
may improve the correlations. In Figure 49a, A is plotted vs. X
for the carbon dioxide + propane binary at 244. K < T < 345 K.
Figure 49a demonstrates that, at X = 1, the A values are
continuous at the critical point of lowerboiling point substance, carbon
dioxide, which means that the two assumptions, Equation (42) and hypothetical
liquids, are rational and convenient. In Figure 49b, B is
plotted vs. X for the carbon dioxide + propane binary at 244. K < T
< 345 K. Figure 49b shows that the decreases in B (= lnÁ_{2}^{})
at X > 1 are striking, which means that the hypothetical liquid
component, carbon dioxide, controls the total pressure satisfying P ŕ Á_{1}^{}x_{1}p_{1s}.
In Figure 49c, C is plotted vs. X for the carbon dioxide +
propane binary at 244. K < T < 345 K. Figure 49c shows that B
and C have similar trends. In Figure 410, A/B is
plotted vs. C for the carbon dioxide + propane binary at 244. K < T
< 345 K. Figure 410 shows that C = A/B holds for the Px
data at the limited range, C > 0.4, where xy data are not
found in the literature. Both Px and xy data deviate from the C
= A/B line at C < 0.4.
In Figure 411, A is
plotted vs. X for the 25 lowerboiling point substances using Px
data. Table 42 lists the 25 binaries. Figure 411 shows that, at X
> 1, A decreases with increasing X. It is important to note
that, at X = 1, the hypothetical liquids behave as ideal solutions,
because the values of A (= lnÁ_{1}^{}) are close to
zero.
Figure Captions for Chapter 4
Figure 41.
vapor pressures calculated from the Wagner equation between 0.7 < T_{ri}
< 1, xaxis: 11/T_{ri},
yaxis: ln(p_{is}/P_{Ci}), () helium4, () hydrogen, ()
nitrogen, () carbon dioxide, () linear correlations passing through the
origin.
Table 41. The values of h for seven
substances.
Substance 
h 

Helium4 
3.39 

Hydrogen 
4.35 

Nitrogen 
5.60 

Carbon
Dioxide 
6.56 

Methane 
5.48 

Ethane 
5.92 

Propane 
6.20 
Figure 42a. Pxy relationships for the helium4 + nitrogen binary at
121.74 K;
xaxis: x_{1}
and y_{1}, yaxis:
P [bar], ()
experimental Px data, ()
correlated Px data with A=2.24, B=12.3 and C=0.318,
() experimental Py
data, ()
correlated Py data with A=1.32, B=18.2 and C=0.221;
data are cited from ref. 4, page 199.
Figure 42b.
xy relationships for the helium4 + nitrogen binary at 121.74 K; xaxis: x_{1},
yaxis: y_{1}, () experimental data cited from ref. 4, () correlated line with A=1.32, B=18.2
and C=0.221.
Figure 43. Relationships between X
and A calculated for the helium4 + nitrogen binary at 77 K < T
< 122 K, xaxis: X, yaxis:
A, ()
determined from Px data, () a power function correlating A values from Px
data, () determined
from xy data.
Figure 44.
Relationships between C and A/B calculated for the
helium4 + component 2 including nitrogen and carbon monoxide at 77 K < T
< 122 K, xaxis:
C, yaxis: A/B, () determined from Px
data, ()
correlation of A/B values determined from the Px data, () determined from xy
data, ()
correlation of A/B values determined from the xy data;
data are cited from ref. 4 in pages 199 – 204.
Figure 45.
Pxy relationships for the hydrogen + methane
binary at 144.26 K; xaxis: x_{1} and
y_{1}, yaxis: P [bar], () experimental Px
data, ()
correlated Px data with A=0.780, B=2.5e7 and C=0.00011,
() experimental Py
data, () correlated Py data with A=0.316,
B=15.8 and C=0.461; data are cited from ref. 4 in page 221.
Figure 46.
Relationships between X and A calculated for the hydrogen +
component 2 including nitrogen and methane at 77 K < T < 174 K, xaxis: X, yaxis:
A, ()
determined from Px data for the hydrogen + nitrogen binary, () determined from Px
data for the hydrogen + methane binary; data are cited from ref. 4 in pages 210
– 227.
Figure 47.
Relationships between C and A/B calculated for the
hydrogen + component 2 including nitrogen and methane at 77 K < T
< 174 K, xaxis:
X, yaxis: A/B, () determined from Px
data, ()
correlation of A/B values determined from the Px data, () determined from xy
data, ()
correlation of A/B values determined from the xy data;
data are cited from ref. 4 in pages 210 – 227.
Figure 48.
Pxy relationships for the carbon dioxide + propane binary at 277.59 K; xaxis: x_{1} and
y_{1}, yaxis: P [bar], () experimental Px
data, ()
correlated Px data with A=0.491 B=0.693 and C=0.708,
() experimental Py
data, () correlated Py data with A=0.367,
B=5.79 and C=0.280; data are cited from ref. 4 in page 589.
Figure 49a. Relationships between X
and A calculated for the carbon dioxide + propane binary at 244. K < T
< 345 K, xaxis: X, yaxis:
A, ()
determined from Px data, () a power function correlating A values from Px
data, () determined
from xy data.
Figure 49b.
Relationships between X and B calculated for the carbon dioxide +
propane binary at 244. K < T < 345 K, xaxis: X, yaxis:
B, ()
determined from Px data, () determined from xy data.
Figure 49c.
Relationships between X and C calculated for the carbon dioxide +
propane binary at 244 K < T < 345 K, xaxis: X, yaxis:
C, ()
determined from Px data, () determined from xy data.
Figure 410.
Relationships between C and A/B calculated for the carbon
dioxide + propane including nitrogen and methane at 244 K < T <
345 K, xaxis: C,
yaxis: A/B, () determined from Px data, () determined from xy
data, ()
correlation of A/B values determined from xy data; data
are cited from ref. 4 in page 589.
Figure 411.
Relationships between X and A calculated for the lowerboiling
point component 1 + component 2 binary, xaxis: X, yaxis:
A, ()
determined from Px data, data are cited from ref. 4; component 1
includes 25 substances excluding helium4 and hydrogen.
Table 42. Binary systems used in Figure 411.
Data are cited from ref. 4.
Component
1 
component
2 
N_{2} 
CO,
Ar,, CO_{2}, methane to heptane, benzene 
CO 
methane
to propane 
Ar 
O_{2},
methane 
Methane 
C_{2}H_{2},
propylene, CO_{2}, propane to heptane, benzene to mxylene, 
C_{6}H_{12},
mcresol 

CF_{4} 
fluoroform 
C_{2}H_{2} 
ethane
to heptane, acetylene, CO, CO_{2} 
CO_{2} 
ethane,
nitrousoxide, difluoromethane, propylene, propane to decane,
dichlorofluoromethane, isobutene, 1butene, 2methylbutane, 
diethylether,
methylacetate, methanol, C6H6, C6H12, H2O, toluene, 

Ethane 
propylene,
isobutene, diethylether, acetone, methylacetate, methanol, 
C_{6}H_{6},
C_{6}H_{12} 

Acetylene 
propylene 
Chlorotrifluoromethane 
dichlorodifluoromethane 
Carbonylsulfide 
propane 
Propylene 
propane,
1butene, isobutene, 2methylbutane, ethanol, C_{6}H_{6} 
Chloropentafluoroethane 
chlorodifluoromethane 
Chlorodifluoromethane 
dichlorodifluoromethane 
NH_{3} 
H_{2}O 
1Butene 
1,3butadiene,
butane 
1,3butadiene 
butane 
Pentane 
C6H6 
2,2Dimethylbutane 
1pentanol 
2,3Dimethylbutane 
1pentanol 
2Methylpentane 
1pentanol 
Methanol 
H_{2}O 
Hexane 
C_{6}H_{6},
isopropylalcohol, 1pentanol 
Perfluorobenzene 
C_{6}H_{6} 
C_{6}H_{12} 
C_{6}H_{6} 
C_{6}H_{6} 
heptane 
Conclusion for Chapter 4
It is natural and convenient to use
hypothetical liquids and an approximation, Equation (42), because
highpressure binary Px and xy VLE data can be satisfactorily
correlated using the three parameter Wohl equation and hypothetical liquids.
Nomenclature for Chapter 4
A, B, C Wohl parameters defined in Eqs. (43) and (44)
(ARD)_{P}
average relative deviation of P defined in Eq. (412)
h constant
defined in Eq. (411)
OF_{P} objective
function defined in Eq. (47)
OF_{y} objective
function defined in Eq. (410)
n number
of data points
P system
pressure
P_{a} reference
pressure
P_{Ci} critical
pressure of component i
P_{k,exp} k th experimental total pressure data
p_{is} vapor
pressure of component i
R gas
constant
T system
temperature
T_{Ci} critical
temperature of component i
T_{C1} critical
temperature of component 1
T_{C2} critical
temperature of component 2
T_{ri} reduces
temperature of component i
V_{i}^{L} partial molar volume of component i^{ }
V_{i}^{0} molar
volume of pure liquid i at T
X critical
ratio defined in Eq. (413)
x_{1} liquid
phase mole fraction of component 1
x_{2} liquid
phase mole fraction of component 2
y_{i} vapor
phase mole fraction of component i
y_{1} vapor
phase mole fraction of component 1
y_{1k,ezp} k
th experimental vapor phase mole fraction of component 1
Á_{i} activity
coefficient of component i in the liquid phase
Á_{2} activity
coefficient of component 2 in the liquid phase
Á_{i}^{(Pa)} acti vity
coefficient of component i in the liquid phase at P = P_{a}
Ó_{is} fugacity
coefficient of the saturated vapor I at p_{is}
Ó_{i}^{V} vapor
phase fugacity coefficient of component I in the mixture at P
Ć_{1} parameter
defined in Eq. (45)
Ć_{2} parameter
defined in Eq. (46)
Superscript
(P) reference
pressure at P
infinite
dilution
Subscript
min minimization
Literature
cited
1)
Prausnitz J. M.,
Molecular Thermodynamics of FluidPhase Equilibria; PrenticeHall: Upper Saddle
River, NJ, 1960.
2) J.
Gmehling, U. Onken, Vapor–Liquid Equilibrium Data Collection, DECHEMA
Chemistry Data Series, Vol. I, parts 1 to 8a, DECHEMA:
Frankfurt, 1977 – 2001, including Landolt Bornstein (1960, 1972, 1975) and
other data collections shown in part 1, pages XLIV and XLV.
3) J.M.
Sorensen, W. Arlt, Liquid–Liquid Equilibrium Data Collection, Binary
Systems, DECHEMA Chemistry Data Series, Vol. V, parts 1 to 4, DECHEMA:
Frankfurt, 1979, including data collections shown in part 1, page XX.
4) Knapp, H.; Doring, R.; Oellrich, L.;
Plocker, U.; Prausnitz, J. M. VaporLiquid Equilibria for Mixtures of Low
Boiling Substances. DECHEMA Chemistry Data Series, Vol. VI; DECHEMA: New York,
1982.
5) R.C.
Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquid 1987
MacGrawHill.