J. Chem. Thermodynamics 2002, 34, 413–422
doi:10.1006/jcht.2000.0822
Available online at http://www.idealibrary.com on
Excess enthalpies of (di-n-butyl ether +
2,2,4-trimethylpentane + heptane, or octane) at
the temperature 298.15 K
Ding-Yu Peng,a George C. Benson,b and Benjamin C.-Y. Lu
Department of Chemical Engineering, University of Ottawa, Ottawa,
Ontario, Canada K1N 6N5
Measurements of excess molar enthalpies at the temperature 298.15 K in a flow
microcalorimeter, are reported for the ternary mixtures [x1 {CH3 (CH2 )3 }2 O +
x2 (CH3 )3 CCH2 CH(CH3 )2 + (1 − x1 − x2 )CH3 (CH2 )ν CH3 ] where ν = 5 and 6.
Smooth representations of the results are described and used to construct constant-enthalpy
contours on Roozeboom diagrams. It is shown that useful estimates of the ternary enthalpies can be obtained from the Liebermann–Fried model, using only the physical properc 2002 Elsevier Science Ltd. All rights reserved.
ties of the components and their binary mixtures. KEYWORDS: excess enthalpy; ternary mixture; di-n-butyl ether; 2,2,4-trimethylpentane;
n-alkanes; Liebermann–Fried model
1. Introduction
In a recent paper (1) from our laboratory measurements were described of excess molar
enthalpies at T = 298.15 K for the two ternary mixtures consisting of the oxygenate
ethyl tert-butylether (ETBE), 2,2,4-trimethylpentane (TMP), and either n-heptane (nC7)
or n-octane (nC8). To extend that investigation, we have made similar measurements for
the analogous mixtures in which the ETBE was replaced by di-n-butyl ether (DNBE).
2. Experimental
The alkanes used in the present investigation were the same as in our earlier work. (1) The
DNBE, with mole fraction purity > 0.990, was obtained from Aldrich Chemical Company.
Apart from partial degassing, all of the components were used without further purification.
Densities ρ(T = 298.15 K), measured in an Anton-Paar digital densimeter, were (763.99,
687.93, 679.93, and 698.80) kg · m−3 for the DNBE, TMP, nC7, and nC8, respectively.
These results agree within <0.1 per cent with values in the literature. (2, 3)
a Visiting Professor from the Department of Chemical Engineering, University of Saskatchewan, Saskatoon,
Saskatchewan, Canada S7N 5C9.
b To whom correspondence should be addressed.
0021–9614/02
c 2002 Elsevier Science Ltd. All rights reserved.
414
D.-Y. Peng, G. C. Benson, and B. C.-Y. Lu
E
TABLE 1. Experimental excess molar enthalpies Hm,12
for [x1 {CH3 (CH2 )3 }2 O
+ (1 − x1 )(CH3 )3 CCH2 CH(CH3 )2 ] at the temperature 298.15 K
x1
E
Hm,12
E
Hm,12
x1
J · mol−1
J · mol−1
E
Hm,12
x1
J · mol−1
x1
E
Hm,12
J · mol−1
0.0501
23.49
0.3002
98.56
0.5504
113.23
0.8001
71.34
0.1000
43.90
0.3504
106.14
0.6003
109.17
0.8500
56.69
0.1503
62.01
0.4004
111.27
0.6506
102.76
0.9000
39.99
0.1993
76.97
0.4499
113.82
0.7002
94.36
0.9501
20.54
0.2501
88.58
0.4994
114.23
0.7504
83.94
TABLE 2. Parameters h k and standard deviations s a for the represenE (i < j) at T = 298.15 K by equation (2)
tation of Hm,i
j
Component
j
h1
h2
h3
h4
s/(J · mol−1 )
DNBE
TMPb
458.11
22.17
12.51
13.33
0.26
DNBE
nC7c
475.32
35.68
20.31
TMP
nC7d
44.03
−0.80
0.103
DNBE
nC8e
568.4
25.37
1.4
TMP
nC8d
104.49
−0.032
i
1.88
0.52
0.10
i
n {H E (calc) − H E (expt)}2 /(n − p) 1/2 , where n is the
m
1 m
number of experimental values and p is the number of parameters h k ; b present
work; c Reference 6; d Reference 7; e Reference 8.
a Defined by s =
hP
An LKB flow microcalorimeter (Model 10700-1), thermostatted at T = (298.150 ±
0.003) K, was used to measure the excess molar enthalpies HmE . Details of the equipment
and the operating procedure have been described previously. (4, 5) In studying the ternary
E
mixtures, the excess molar enthalpy Hm,1+23
was determined for several pseudo-binary
mixtures in which component 1, (DNBE) was added to binary mixtures of components 2
(TMP) and 3 (either nC7 or nC8), having fixed compositions. For this purpose, binary
mixtures with selected values of x2 /(1 − x1 − x2 ) were prepared by mass from partially
degassed samples of the components. Measurements were made on a Mettler balance with
E
a sensitivity of 0.01 mg. The excess molar enthalpy Hm,123
of the ternary mixture was then
obtained from the relation:
E
E
E
Hm,123
= Hm,1+23
+ (1 − x1 )Hm,23
,
(1)
E
where Hm,23
is the excess molar enthalpy of the particular binary mixture. Over most of
the composition range, the errors of the excess molar enthalpies and the mole fractions of
the final ternary mixtures are estimated to be <0.005 · |HmE | and <5 · 10−4 , respectively.
E of (DNBE + TMP + nC7 or nC8) at T = 298.15 K
Hm
415
130
120
110
H Em, 1+23 /(J . mol – 1)
100
90
80
70
60
50
40
30
20
10
0
0.0
0.2
0.4
x1
0.6
0.8
1.0
E
FIGURE 1. Excess molar enthalpies Hm,1+23
for [x1 {CH3 (CH2 )3 }2 O + x2 (CH3 )3 CCH2 CH(CH3 )2
+ (1 − x1 − x2 )CH3 (CH2 )5 CH3 ] at the temperature 298.15 K plotted against mole fraction x1 .
Experimental results: 1, x2 /(1 − x1 − x2 ) = 0.3341; , x2 /(1 − x1 − x2 ) = 1.0000; ∇,
x2 /(1 − x1 − x2 ) = 3.0005; , x1 + x2 = 1. Curves: . . ., x2 = 0, Reference 6; —–, calculated
E given in
from the representation of the results by equations (3) and (4), using the ternary term Hm,T
the footnote of table 3; - - - -, estimated by the Liebermann–Fried model.
◦
3. Results and discussion
E (i
Hm,i
j
Excess molar enthalpies
< j), at T = 298.15 K, for four of the five constituent binary mixtures of present interest, have been reported previously: {DNBE(1) + nC7(3)}, (6)
{TMP(2) + nC7(3)}, (7) {DNBE(1) + nC8(3)}, (8) and {TMP(2) + nC8(3)}. (7) The experE
imental values of x1 and Hm,12
for {DNBE(1) + TMP(2)} are listed in table 1. The smoothing function:
E
−1
Hm,i
j /(J · mol ) = x i (1 − x i )
m
X
h k h1 − 2xi ik−1 ,
(2)
k=1
was fitted to the results by the method of least-squares with all points weighted equally. The
values of the coefficients h k , obtained from the analysis, are listed in table 2, along with the
standard deviation s of the representation. Also included in table 2 are the representations
E
(6–8)
of Hm,i
j for the other constituent binaries.
The experimental results for the two ternary mixtures are reported in tables 3 and 4,
E
where values of Hm,1+23
are listed against the mole fraction x1 of DNBE. Also included
E
in those tables are the corresponding values of Hm,123
, calculated from equation (1). The
416
D.-Y. Peng, G. C. Benson, and B. C.-Y. Lu
E
TABLE 3.
Experimental excess molar enthalpies Hm,1+23
at the temperature 298.15 K for the addition of DNBE to (TMP + nC7) mixtures to form
[x1 {CH3 (CH2 )3 }2 O + x2 (CH3 )3 CCH2 CH(CH3 )2 + (1−x1 −x2 )CH3 (CH2 )5 CH3 ], and
E
E
values of Hm,123
calculated from equation (1) using the smooth representation of Hm,23
by equation (2)
x1
E
Hm,1+23
J · mol−1
a
E
Hm,123
J · mol−1
x1
E
Hm,1+23
J · mol−1
a
E
Hm,123
J · mol−1
x1
E
Hm,1+23
J · mol−1
a
E
Hm,123
J · mol−1
E /(J · mol−1 ) = 8.19
x2 /(1 − x1 − x2 ) = 0.3341, Hm,23
0.0500
24.80
32.58
0.4000
115.24
120.15
0.6996
95.80
98.26
0.1000
46.55
53.92
0.4497
117.29
121.80
0.7495
84.83
86.88
0.1498
65.06
72.02
0.4997
117.65
121.75
0.7983
71.77
73.42
0.2000
80.34
86.89
0.5500
115.66
119.35
0.8502
56.61
57.84
0.2502
91.70
97.84
0.6002
110.99
114.26
0.8998
39.73
40.55
0.2999
104.03
109.76
0.6496
104.73
107.60
0.9499
20.79
21.20
0.3494
109.27
114.60
E /(J · mol−1 ) = 11.01
x2 /(1 − x1 − x2 ) = 1.0000, Hm,23
0.0501
24.40
34.86
0.3996
112.95
119.56
0.6996
94.56
97.87
0.1001
45.16
55.07
0.4499
116.05
122.11
0.7501
83.68
86.43
0.1499
64.30
73.66
0.5001
116.05
121.55
0.7999
70.88
73.08
0.2002
78.54
87.34
0.5499
113.84
118.79
0.8500
55.31
56.96
0.2499
90.88
99.14
0.5999
109.63
114.03
0.9001
38.73
39.83
0.2995
101.12
108.83
0.6497
103.34
107.20
0.9500
19.95
20.50
0.3500
109.05
116.20
E /(J · mol−1 ) = 8.33
x2 /(1 − x1 − x2 ) = 3.0005, Hm,23
0.0501
23.31
31.22
0.4004
112.24
117.23
0.7002
94.04
96.54
0.1001
43.90
51.40
0.4504
114.87
119.45
0.7507
83.26
85.34
0.1500
62.64
69.72
0.4998
115.12
119.29
0.8017
69.69
71.34
0.1999
76.25
82.91
0.5502
113.03
116.78
0.8500
56.24
57.49
0.2505
88.86
95.10
0.6005
108.72
112.05
0.8999
39.97
40.80
0.2997
98.57
104.40
0.6503
102.42
105.33
0.9499
20.53
20.95
0.3504
106.58
111.99
E
a Ternary term for representation of H E
−1
m,1+23 by equations (3) and (4): Hm,T /(J · mol ) =
x1 x2 {(1 − x1 − x2 )/(1 − x1 + x2 )}(43.38 + 75.01x1 − 129.55x2 − 253.08x12 ), s = 0.46 J · mol−1 .
E
results for Hm,1+23
are plotted in figures 1 and 2. Also plotted in those figures are the
results given in table 1 for DNBE(1) + TMP(2), which correspond to the case x1 + x2 = 1
and curves for x2 = 0, calculated from equation (2) with the coefficients given in table 2.
E
E
In all cases, the maximum values of Hm,1+23
and Hm,123
occur near x1 = 0.5. At constant
E
x1 , Hm,1+23 decreases as x2 /(1 − x1 − x2 ) increases. This behaviour is similar to that
E of (DNBE + TMP + nC7 or nC8) at T = 298.15 K
Hm
417
E
TABLE 4.
Experimental excess molar enthalpies Hm,1+23
at the temperature 298.15 K for the addition of DNBE to (TMP + nC8) mixtures to form
[x1 {CH3 (CH2 )3 }2 O + x2 (CH3 )3 CCH2 CH(CH3 )2 + (1 − x1 − x2 )CH3 (CH2 )6 CH3 ],
and
E
E
values of Hm,123
calculated from equation (1) using the smooth representation of Hm,23
by
equation (2)
E
Hm,1+23
x1
J · mol−1
a
E
Hm,123
J · mol−1
x1
E
Hm,1+23
J · mol−1
a
E
Hm,123
J · mol−1
x1
E
Hm,1+23
J · mol−1
E /(J · mol−1 ) = 19.70
x2 /(1 − x1 − x2 ) = 0.3342, Hm,23
45.40
0.4002
127.15
138.97
0.7002
108.52
a
E
Hm,123
J · mol−1
0.0498
26.68
0.1000
0.1500
0.1998
50.19
71.21
88.67
67.92
87.96
104.43
0.4501
0.5002
0.5501
130.54
131.87
129.71
141.37
141.72
138.57
0.7501
0.8001
0.8499
95.87
81.42
63.98
100.79
85.36
66.94
114.43
0.2499
101.30
116.08
0.6004
125.15
133.02
0.9000
45.32
47.29
0.2998
0.3500
112.69
121.50
126.48
134.31
0.6498
118.05
124.95
0.9500
24.00
24.99
E /(J · mol−1 ) = 26.12
x2 /(1 − x1 − x2 ) = 1.0004, Hm,23
50.02
0.4000
119.81
135.48
0.6005
117.11
71.13
0.4000
120.50
136.17
0.6502
110.45
127.55
119.59
0.0500
0.1001
25.20
47.62
0.1497
66.98
89.19
0.4496
122.55
136.93
0.7006
101.26
109.08
0.2000
0.2491
0.2501
83.73
96.14
95.84
104.63
115.75
115.43
0.4503
0.5001
0.5002
122.48
123.05
123.72
136.84
136.11
136.78
0.7501
0.7992
0.8497
90.10
76.78
60.24
96.63
82.02
64.17
0.2997
107.16
125.45
0.5500
121.72
133.48
0.9000
41.49
44.10
0.3000
106.71
124.99
0.5501
121.36
133.11
0.9501
23.39
24.69
0.3501
115.00
131.98
E /(J · mol−1 ) = 19.68
x2 /(1 − x1 − x2 ) = 3.0003, Hm,23
0.0500
23.98
42.68
0.4001
114.61
126.42
0.7493
86.45
91.38
0.0999
0.1499
0.1999
45.37
64.48
79.81
63.09
81.21
95.56
0.4507
0.5013
0.5498
117.14
118.05
116.52
127.95
127.86
125.38
0.7997
0.8506
0.8996
73.04
57.72
40.70
76.98
60.66
42.68
0.2502
0.2998
0.3505
91.60
101.76
109.63
106.36
115.54
122.41
0.5997
0.6491
0.6996
112.13
105.93
97.12
120.01
112.84
103.03
0.8998
0.9500
40.63
20.14
42.60
21.12
E
a Ternary term for representation of H E
−1
m,1+23 by equations (3) and (4): Hm,T /(J · mol ) = x 1 x 2 {(1 −
2
−1
x1 − x2 )/(1 − x1 + x2 )}(15.47 − 276.18x1 − 104.14x2 + 131.68x1 ), s = 0.48 J · mol .
found for the analogous mixtures containing ETBE. (1) In the present work, the decreases
are relatively small for the nC7 mixtures, but are more significant for those containing nC8.
E
The values of Hm,1+23
were represented as a sum of binary terms (9) with an added
ternary contribution:
E
E
E
E
Hm,1+23
= [x2 /(1 − x1 )]Hm,12
+ [(1 − x1 − x2 )/(1 − x1 )]Hm,13
+ Hm,T
,
(3)
418
D.-Y. Peng, G. C. Benson, and B. C.-Y. Lu
150
140
130
120
110
H Em, 1+23 /(J . mol – 1)
100
90
80
70
60
50
40
30
20
10
0
0.0
0.2
0.4
x1
0.6
0.8
1.0
E
FIGURE 2. Excess molar enthalpies Hm,1+23
for [x1 {CH3 (CH2 )3 }2 O + x2 (CH3 )3 CCH2 CH(CH3 )2
+ (1 − x1 − x2 )CH3 (CH2 )6 CH3 ] at the temperature 298.15 K plotted against mole fraction x1 .
Experimental results: 1, x2 /(1 − x1 − x2 ) = 0.3342; , x2 /(1 − x1 − x2 ) = 1.0004; ∇,
x2 /(1 − x1 − x2 ) = 3.0003; , x1 + x2 = 1. Curves: . . ., x2 = 0, Reference 8; —–, calculated
E given in
from the representation of the results by equations (3) and (4), using the ternary term Hm,T
the footnote of table 4; - - - -, estimated by the Liebermann–Fried model.
◦
TABLE 5. Values of the interaction parameters Ai j and
A ji , standard deviations s, and isobaric thermal expansivities
α p at T = 298.15 K, in Liebermann–Fried model calculations for [x1 {CH3 (CH2 )3 }2 O + x2 (CH3 )3 CCH2 CH(CH3 )2 +
(1 − x1 − x2 )CH3 (CH2 )5 CH3 ]
and
[x1 {CH3 (CH2 )3 }2 O +
x2 (CH3 )3 CCH2 CH(CH3 )2 + (1 − x1 − x2 )CH3 (CH2 )6 CH3 ]
α p /kK−1
Component
i
DNBE
j
Ai j
A ji
TMP
0.9024
1.0137
s
i
j
0.47
1.126a
1.197b
1.256a
(J · mol−1 )
DNBE
nC7
0.8841
1.0288
0.75
1.126a
TMP
nC7
0.9830
1.0071
0.24
1.197b
1.256a
1.41
1.126a
1.164a
0.12
1.197b
1.164a
DNBE
TMP
nC8
nC8
0.9057
0.9901
0.9900
0.9892
a Benson et al.; (6) b Rajagopal and Subrahmanyam. (14)
E of (DNBE + TMP + nC7 or nC8) at T = 298.15 K
Hm
419
TMP
100
20
30
40
50
6
70
80
90
0
10
a
0
11
11
4.6
11
.01
0
11
10
117
115
0
10
90
80
70
60
500
4 0
3
20
119.0
nC7
10
12
2
12 .6
2
12 .0
1
12 .0
0.0
DNBE
TMP
100
20
30
40
50
6
70
80
90
0
10
b
0
11
11
.25
11
1
12 21.1
12 1.0
0.0
0
11
0
10
90
10
119.5
119
117
115
80
70
60
500
4 0
3 0
2
10
nC7
5.0
DNBE
E
FIGURE 3. Contours for constant values of Hm,123
/(J · mol−1 ) for [x1 {CH3 (CH2 )3 }2 O
+ x2 (CH3 )3 CCH2 CH(CH3 )2 + (1 − x1 − x2 )CH3 (CH2 )5 CH3 ] at the temperature 298.15 K. a, calE from
culated from the representation of the experimental results by equations (1) to (4) with Hm,T
the footnote of table 3; b, estimated by the Liebermann–Fried model.
E
where the values of Hm,i
j were calculated from the appropriate smoothing functions. The
form:
E
Hm,T
/(J · mol−1 ) = x1 x2 {(1 − x1 − x2 )/(1 − x1 + x2 )}(c0 + c1 x1 + c2 x2
+ c3 x12 + c4 x1 x2 + c5 x22 + · · ·),
(4)
which was adopted for the latter contribution, is similar to the form used by Morris
et al. (10) with an extra skewing factor (1 − x1 + x2 )−1 inserted. Values of the coefficients
ci were obtained from least-squares analyses in which equations (3) and (4) were fitted
420
D.-Y. Peng, G. C. Benson, and B. C.-Y. Lu
TMP
100
20
30
40
50
6
70
80
90
a
0
10 5
10
0
11
26
4.6
.12
11
0
11
5
10 0
10
90
80
70
60
500
4 0
3 0
2
10
10
20
nC8
142.2
142
141
140
135
130
125
120
115
14
14 2.4
2.3
DNBE
TMP
100
20
30
40
50
6
70
80
90
0
10 5
10 0
11
26
5.0
11
.15
0
11
5
10 0
10
90
80
70
60
500
4 0
3 0
2
142.2
142
141
140
135
130
125
120
115
nC8
10
20
14
14 2.34
2.2
5
10
b
DNBE
E
FIGURE 4. Contours for constant values of Hm,123
/(J · mol−1 ) for [x1 {CH3 (CH2 )3 }2 O
+ x2 (CH3 )3 CCH2 CH(CH3 )2 + (1 − x1 − x2 )CH3 (CH2 )6 CH3 ] at the temperature 298.15 K. a, calE from
culated from the representation of the experimental results by equations (1) to (4) with Hm,T
the footnote of table 4; b, estimated by the Liebermann–Fried model.
E
E
to the values of Hm,1+23
in tables 3 and 4. The resulting forms for Hm,T
are given in the
footnotes of those tables, along with the standard deviation s for each representation. The
E
solid curves for Hm,1+23
in figures 1 and 2 were calculated from equation (3) using these
representations.
E
Equations (1) to (4) were also used to calculate the constant Hm,123
contours plotted on
the Roozeboom diagrams in figures 3(a) and 4(a). In both of these, some of the contours
E
do not extend to the edges of the triangle, but indicate a rise of Hm,123
to an internal
maximum. It is evident from a comparison of figures 3(a) and 4(a) that the area associated
E of (DNBE + TMP + nC7 or nC8) at T = 298.15 K
Hm
421
with this rise is more extensive for the nC7 mixture. This is attributable to the fact that the
E
E
maximum value of Hm,12
for DNBE(1) + TMP(2) is closer to the maximum of Hm,13
for
DNBE(1) + nC7(3), than to that for DNBE(1) + nC8(3).
Recently, (11) it was found that the Liebermann–Fried model, (12, 13) can be extended
to provide estimates of the thermodynamic properties of multicomponent mixtures,
using only the properties of the pure components and interaction parameters derived
from analyses of the excess enthalpies of their constituent binaries. This approach was
investigated for the present ternary mixtures. Reference can be made to the work of Wang
et al. (11) for the equations used in this application.
The values of the Liebermann–Fried interaction parameters Ai j and A ji for the constituent binaries are given in table 5. These were obtained by fitting the Liebermann–Fried
E
formula for Hm,i
j to the primary experimental results for the excess molar enthalpies, as
given in table 1 and References 6 to 8. Also included in that table are values of the standard
deviations s achieved in the fitting process, and values (6, 14) of the isobaric thermal expansivities α p , used in evaluating the contributions due to different sizes of the molecules.
E
Estimates of Hm,1+23
, derived from the Liebermann–Fried model, are shown as dashed
curves in figures 1 and 2. In both cases, it can be seen that the theory predicts correctly the
order of the three experimental curves and their positions relative to the curves for the two
constituent binaries. The root-mean-square deviations for the 57 points in table 3 and the
64 points in table 4 are (1.25 and 0.88) J · mol−1 , respectively.
E
Constant Hm,123
contours, estimated on the basis of the model, are shown on the
Roozeboom diagrams in figures 3(b) and 4(b). It is clear from a comparison of the two
E
parts in each figure, that the Liebermann–Fried model provides useful estimates of Hm,123
for both of the present mixtures, including predictions of the location and magnitude of the
internal maximum.
The financial support of the Natural Sciences and Engineering Research Council of Canada
(NSERC) is gratefully acknowledged.
REFERENCES
1. Peng, D.-Y.; Benson, G. C.; Lu, B. C.-Y. Thermochim. Acta 2001, 372, 1–9.
2. TRC—Thermodynamic Tables—Non-Hydrocarbons. Thermodynamic Research Center:
The Texas A&M University System, College Station TX 77843-3111: 1988, Table 23-2-1(1.2115)-a, dated 31 December 1965.
3. TRC—Thermodynamic Tables—Hydrocarbons. Thermodynamic Research Center: The Texas
A&M University System, College Station TX 77843-3111: 1988, Table 23-2-(1.203)-a dated
30 April 1956;Table 23-2-(1.101)-a, page 1, dated 31 October 1977.
4. Tanaka, R.; D’Arcy, P. J.; Benson, G. C. Thermochim. Acta 1975, 11, 163–175.
5. Kimura, F.; Benson, G. C.; Halpin, C. J. Fluid Phase Equilib. 1983, 11, 245–250.
6. Benson, G. C.; Luo, B.; Lu, B. C.-Y. Can. J. Chem. 1988, 66, 531–534.
7. Peng, D.-Y.; Horikawa, Y.; Wang, Z.; Benson, G. C.; Lu, B. C.-Y. J. Chem. Eng. Data 2001, 46,
237–238.
8. Jiménez, E.; Franjo, C.; Segade, L.; Legido, J. L.; Paz Andrade, M. I. Fluid Phase Equilib. 1997,
133, 179–185.
9. Tsao, C. C.; Smith, J. M. Chem. Eng. Prog. Symp. Ser. No. 7 1953, 49, 107–117.
10. Morris, J. W.; Mulvey, P. J.; Abbott, M. M.; Van Ness, H. C. J. Chem. Eng. Data 1975, 20,
403–405.
422
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11. Wang, Z.; Peng, D.-Y.; Benson, G. C.; Lu, B. C.-Y. J. Chem. Thermodynamics 2001, 33,
1181–1191.
12. Liebermann, E.; Fried, V. Ind. Eng. Chem. Fundam. 1972, 11, 350–354.
13. Liebermann, E.; Fried, V. Ind. Eng. Chem. Fundam. 1972, 11, 354–355.
14. Rajagopal, E.; Subrahmanyam, S. V. J. Chem. Thermodynamics 1974, 6, 837–876.
(Received 4 October 2000; in final form 2 November 2000)
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