ÓÔ

                                                                                                                             
                                  

Al-khwarizmi  
                                                                                                                                               Engineering 
                                                                                                                                                    Journal 

Al-Khwarizmi Engineering Journal, Vol.3, No2, pp 67 -86 (2007)  

Correlation for fitting multicomponent vapor-liquid equilibria data and 
prediction of azeotropic behavior

Dr. Khalid Farhod Chasib Al-Jiboury
Chemical Engineering Department

University of Technology

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Abstract
Correlation equations for expressing the boiling temperature as direct function of liquid 

composition have been tested successfully and applied for predicting azeotropic behavior of 
multicomponent mixtures and the kind of azeotrope (minimum, maximum and saddle type) using 
modified correlation of Gibbs-Konovalov theorem. Also, the binary and ternary azeotropic point have 
been detected experimentally using graphical determination on the basis of experimental binary and 
ternary vapor-liquid equilibrium data.

In this study, isobaric vapor-liquid equilibrium for two ternary systems: “1-Propanol – Hexane –
Benzene” and its binaries “1-Propanol – Hexane, Hexane – Benzene and 1-Propanol – Benzene” and the 
other ternary system is “Toluene – Cyclohexane – iso-Octane (2,2,4-Trimethyl-Pentane)” and its binaries 
“Toluene – Cyclohexane, Cyclohexane – iso-Octane and Toluene – iso-Octane” have been measured at 
101.325 KPa. The measurements were made in recirculating equilibrium still with circulation of both the 
vapor and liquid phases. The ternary system “1-Propanol – Hexane – Benzene” which contains polar 
compound (1-Propanol) and the two binary systems “1-Propanol – Hexane and 1-Propanol – Benzene” 
form a minimum azeotrope, the other ternary system and the other binary systems do not form 
azeotrope.

All the data passed successfully the test for thermodynamic consistency using McDermott-Ellis 
test method (McDermott and Ellis, 1965).

The maximum likelihood principle is developed for the determination of correlations parameters 
from binary and ternary vapor-liquid experimental data which provides a mathematical and 
computational guarantee of global optimality in parameters estimation for the case where all the 
measured variables are subject to errors and the non ideality of both vapor and liquid phases for the 
experimental data for the ternary and binary systems have been accounted.

The agreement between prediction and experimental data is good. The exact value should be 
determined experimentally by exploring the concentration region indicated by the computed values.

Keywords: Vapor-Liquid Equilibria, Azeotropic Behavior, Multicomponent system

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Introduction

The term azeotrope means “nonboiling by 
any means” (Greek: a - non, zeo - boil, tropos -
way/mean), and denotes a mixture of two or more 
components where the equilibrium vapor and 
liquid compositions are equal at a given pressure 
and temperature. Systems which do not form 
azeotropes are called zeotropic.

Departures from Raoult’s law 
frequently manifest themselves in the formation of 
azeotropes, azeotropes are liquid mixtures 
exhibiting maximum or minimum boiling points 
that represent, respectively, negative or positive 
deviations from Raoult’s law. Also constant-
boiling mixtures are referred to as azeotropes. 
They occur between certain combinations of the 
components of the mixture or mixtures of close-
boiling species of different chemical types.
Azeotropes occur frequently between compounds 
whose boiling points differ by less than about 30 
oC. If the vapor and liquid are of the same 
composition, the two-phase mixture is called a 
homogeneous azeotrope. The constant-boiling 
mixtures that occur with a vapor and two or more 
dense phases are called heterogeneous azeotropes
(This is because in order to get the liquid to phase 
separate into two liquid phases the components 
must be highly repulsive) (Hoffman, 1964).

The y-x curve crosses the y = x line (the 
45o line) at the azeotropic composition, in the 
region below this intersection with the diagonal, 
the equilibrium vapor is richer in one component 
than the liquid, above this intersection, the vapor 
is poorer in this component than the 
corresponding liquid from which it comes.

For the isobaric system the temperature 
versus composition curves coincide at a minimum 
at the azeotropic composition. Such azeotropic 
systems frequently occur when the two 
components are dissimilar functionally and the 
boiling points are not greatly different of about 25 
oC for most systems, although much greater 
differences are noted in some systems (mixture in 
which there is strong repulsion between the 
different species), because less thermal energy has 
to be introduced to get molecules to enter the 
vapor phase to overcome the attraction in the 
liquid.

A high-boiling azeotrope differs from the 
low-boiling azeotrope in that the temperature 
versus composition curves of the isobaric plot 

exhibit maximum, the high-boiling azeotrope is 
less common than the low-boiling azeotrope, it 
generally occurs between components whose 
molecules are somewhat attracted to each other 
(mixture in which there is strong attraction 
between the different species), because additional 
thermal energy has to be introduced to get 
molecules to enter the vapor phase to overcome 
the attraction in the liquid.

For the T-x-y diagrams if the dew and 
bubble curves touch at some intermediate 
composition where the coexisting phases have the 
same composition. Such a point is known as an 
azeotrope and although the two phases coexist 
there at the same temperature, pressure and 
composition, it is distinguished from a critical 
point by a difference in the density of the phases 
(Harold, 1970).

An analysis of the structural properties of 
vapor-liquid equilibrium (VLE) diagrams 
provides a fundamental understanding of the 
highly nonideal thermodynamic behavior of 
azeotropic mixtures. The possibility to graphically 
represent the VLE depends on the number of 
components in the mixture. Phase diagram 
analysis is an excellent tool for gaining insights 
into the complex behavior of nonideal ternary 
mixtures.

Ternary vapor-liquid equilibrium (VLE) 
diagrams provide a graphical tool to predict 
qualitatively the feasible separations for 
multicomponent azeotropic mixtures before 
detailed simulation or experimental study of their 
distillation (Walas, 1985).

The main difference for ternary and 
multicomponent mixtures is that an azeotropic 
point is not necessarily an absolute extreme 
(minimum or maximum point) of the boiling 
temperature at isobaric condition, but it may be a 
local extreme (saddle).

The tendency of a mixture to form an 
azeotrope depends on two factors: 
(a) The difference in the pure component 

boiling points.
(b) The degree of nonideality.

The closer the boiling points of the pure 
components and the less ideal mixture, the greater 
the likelihood of an azeotrope. The important 
points about azeotropes are:
 They limit the separation achievable by 

distillation because even with infinite trays, 

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there is no thermodynamic driving force to 
separate beyond the azeotrope.
 The azeotropic point can be moved, for 

example by lowering the pressure, so that the 
desired composition may be obtained.. In fact, 
for some systems, the azeotrope might not 
even form if the pressure is changed enough 
and the azeotrope can be broken.
 Azeotropes can be used to help separations. 

Adding a third component to a system which 
forms a minimum boiling point azeotrope with 
one of the original components can be used to 
entrain that component, allowing for it to be 
vaporized away and recovered (Marc et al., 
1998).

Theory of azeotropic predictions

The ability to predict azeotropic behavior 
becomes more and more important and complex 
when the number of components in the mixture 
increase. This prediction is tested successfully by 
applying the Gibbs-Konovalov theorem 
(Malesinski, 1965). They developed correlation 
equations for expressing boiling temperature of 
the vapor-liquid equilibria data, isobaric 
conditions, as a function of liquid composition, 
where

   







 




1

111

...
2

N N

ij

N

i
iijiijiji

i
ii xxCxxBAxxTxT jjj
O

    (1)

Where Toi is the boiling temperature of pure 
component i in oC, and N number of component in 
the mixture.

This equation is useful for obtaining 
isothermals and for exploring the azeotropic 
behavior of binary and ternary mixtures. The 
coefficient Aij, Bij, and Cij are binary or ternary 
parameters which are determined directly from the 
binary or ternary data. The increase in the number 
of parameters increases the prediction accuracy.

And another correlation for prediction 
azeotropic behavior of the binary and ternary 
mixtures, and the kind of azeotrope (minimum, 
maximum and saddle type), will be discussed 
below:

The Gibbs-Konovalov theorem 
(Malesinski, 1965) for multicomponent systems 
states that the following conditions are fulfilled at 

the azeotropic point where xi = yi at constant 
pressure:

0
,





















j
xPix

T
i = 1,2,…, N…... (2)

The index x`j means that the relationship T
is given in terms of all mole fractions as 
independent variables, except xj. The 
differentiation is carried out by keeping all the 
mole fractions constant except xi. For a ternary 
mixture, the application of (2) namely 
    0

33 ,2,1
  xPxP xTxT to the ternary form of 

(1) yield that:

  021 213323223122112112  TTxxxxxxxxxx
OO

….. (3)

  05.0221 32212132312 3312111  TTxxxxxxxxxx
OO

.(4)

Where

xxx 213 1 
…………………………………….. (5)

    ...2  xxCxxBA jj iijiijijk ……………… (6)

      



  ...432

32
xxExxDxxCB jjj iijiijiijijk 

                                                                            (7)
For 

K = 1 : ij = 12 and � = 1…………….… 
(8)

K = 2 : ij = 13 and � = 2………………. 
(9)

K = 3 : ij = 23 and � = 
1………………(10)

Equations (3) and (4) are nonlinear 
equations for the unknowns x1 and x2 and if a 
ternary mixture exhibits azeotropic behavior, the 
solution for x1 and x2 should lie within 0-1. 

For binary mixtures, one obtains from 
  0

2,1
 xPxT that:

    0221 2111111 2  TTxxx OO ……. (11)

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Equation (11) is solved numerically and an 
azeotrope exists if solution x1 is within the 
interval 0<x1<1.

The use of optimization technique for 
determining the parameters of the correlation by 
means of isobaric vapor-liquid equilibria requires 
minimization of an objective function. The 
objective function is:









 

m

i

cal
i

obs
i TTFO

1

2

.
…………………  (12)

In a multicomponent mixture under 
isobaric conditions, there is a possibility for an 
azeotrope with a minimum boiling point, with a 
maximum boiling point and a saddle point 
azeotrope. The thermodynamic conditions 
determining the kind of the multicomponent 
azeotrope for isobaric equilibria may be derived as 
follows: (2) indicates that the boiling temperature 
of a multicomponent mixture may be expressed as 
a function of the liquid composition, namely, 
T(x1, x2, …, xN-1). Designating by T and x the 
coordinates of the azeotropic point and expanding 
T into a Taylor series around the azeotropic point 
gives:

      xxxx
xx

TxxTxxT kkjj
kjj k

NN















  




  
2

2
1

,...,,..., 1111

  (13)

Where terms beyond the second derivative 
were neglected and according to (2) the first 
derivatives vanish at the azeotropic point. 
Equation (13) is known as the quadratic form and 
the kind of azeotrope depends on the following 
determinant:

a

aaa

aaa
aaa

x
T

xx
T

xx
T

xx
T

x
T

xx
T

xx
T

xx
T

x
T

N

NNNN

N

N

NNN

N

N

1

1,12,11,1

1,22221

1,11211

2

2

2

2

1

2

2

2

2
2

2

12

2
1

2

21

2

1

2

111

1

1
2










































...
............

...

...

...

............

...

...

                                                                           (14)

Where the derivatives are taken at the 
azeotropic point. For simplicity, the principal 

minors of the determinant aN-1 are designated as 
follows:

a
aaa
aaa
aaa

aaa
aa

33

333231

232221

131211

22
2221

1211                  ; 
  (15)

On the basis of theorems related to 
quadratic forms and by considering (13), (14) and 
(15), the followings can be concluded concerning 
the possible kinds of azeotropes (Table 1):

Table 1. Conditions for determination of kinds
      of multicomponent azeotropes

Minimum in
T - x

Maximum in
T - x

Neither maximum 
or minimum in T 

– x, namely 
Saddle type 

behavior

011 a

022 a

033 a

044 a
.
.
.

011 a

022 a

033 a

044 a
.
.
.

Neither one of the 
conditions in the 

two left hand 
columns is 
fulfilled.

Plotting calibration curves

Refractive index and density 
measurements at 30 oC were chosen as rapid and 
accurate methods for analysis of equilibrium 
mixtures. Calibration curves analyses of all binary 
systems are usually represented graphically by 
plotting refractive index against the compositions.

Combination of refractive index and 
density measurements were employed for the 
analysis of the ternary systems graphically, if the 
concentration of one of the constituents is kept 
constant while the other two are varied. This was 
done by interpolation of previously determined 
refractive index-composition and density-
composition charts.

The schematic diagram of the calibration 
curve of the ternary system 1-Propanol – Hexane 
– Benzene is illustrated in figure 1.a, and the 
calibration curve of the ternary system Toluene –
Cyclohexane – iso-Octane is illustrated in figure 
1.b, this is a graphical method for analysis of the 
ternary system. Another very accurate method 
used for analysis is by the use of computer 
program designed and developed by using 

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Mathcad software for the purpose of finding the 
compositions of the ternary or binary systems at a 
given refractive index and density

1 - PROPANOL

HEXANE BENZENE
0 25 50 75 100

25

50

75

10
0

0

0

25

50

100

75

1.
48

5

1.
48

0

1.
47

5

1.
47

0

1.
46

5

1.
46

0

1.
45

5

1.
45

0

1.
44

5

1.
44

0

1.
43

5

1.
43

0

1.
42

5

1.
42

0

1.
41

5

1.
41

0

1.
40

5

1.
40

0

1.
39

5

1.
39

0

1.
38

5

1.
38

0

1.
37

5

R
E

F
R

A
C

T
IV

E
IN

D
E

X

0.
84

6

0.
85

2

0.
85

80.
84

0
0.

83
4

0.
82

8
0.

82
2

0.
81

6
0.

81
0

0.
80

4
0.

79
8

0.
79

2
0.

78
6

0.
78

0
0.

77
4

0.
76

8
0.

76
2

0.
75

6
0.

75
0

0.
74

4
0.

73
8

0.
73

2
0.

72
6

0.
72

0
0.

71
4

0.
70

8
0.

70
2

0.
69

6
0.

69
0

0.
68

4
0.

67
8

0.
67

2
0.

6
66

0.
66

0

D
E

N
S

IT
Y

Fig. 1.a. Triangular  plot  of  Refractive  index  –
                Density – Composition (Calibration 
               curve) of the ternary system 1-propanol 
–
              Hexane – Benzene at 30 oC, in which
              lines of constant refractive index and 
              constant density are plotted.

TOLUENE

CYCLOHEXANE iso-OCTANE
0 25 50 75 100

25

50

75

10
0

0

0

25

50

100

75

0.
84

0

0.8
34

0.8
28

0.7
74

0.7
68

0.7
800

.78
60.7

920.
79

80.8
040.8

10
0.8

160
.82

2

0.7
62

0.7
56

0.7
50

0.7
44

0.7
38

0.7
32

0.7
26

0.7
20

DE
NS

IT
Y

0.7
14

0.7
08

0.7
02

0.6
96

0.6
90

1.4
15

1.4
10

1.4
05

1.4
00

1.3
95

1.4
20

1.4
25

1.4
30

1.4
35

1.4
40

1.4
45

1.4
50

1.4
55

1.4
60

RE
FRA

CT
IVE

IND
EX

1.4
65

1.4
70

1.4
75

1.390

Fig. 1.b. Triangular  plot  of  Refractive  index  –   
               Density  – Composition (Calibration
              curve) of the ternary system Toluene–
               Cyclohexane-iso-Octane at  30 oC, in
              which lines  of  constant  refractive 
              index and constant density are plotted.

Expermintal work

Apparatus and procedure
The experimental determination of vapor-

liquid equilibrium data are performed using a 
circulating equilibrium apparatus, Fig. 2, in which 
the gas and liquid phases of a mixture are 
circulated continuously at constant pressure. The 
apparatus used was an all-glass recirculating still. 
This equipment has a side-heating unit which 
ensures complete mixing of the liquid mixture. 
The design also prevented liquid drop entrainment 
and partial condensation in the vapor phase. The 
equilibrium temperature were measured by a 
digital thermometer with an accuracy of ±0.1 oC. 
In each experiment, the liquid mixture was heated 
in a recirculating still at fixed pressure of 101.3 
KPa. Equilibrium was reached after 1h where the 
temperature of the liquid and vapor phases were 
constant (the boiling point had become stable) and 
their difference was within 0.1 oC. Samples of the 
equilibrium phases were taken at small volumes 
and were analyzed by measuring densities and 
refractive indices at 30 oC.

Fig.2. Flow diagram for equilibrium apparatus.

system selection

Ternary system I

The ternary system I consists of 1-
Propanol, Hexane and Benzene and its binaries.

One of the problems in the field of 
extractive distillation is to find a quantitative 
method of assessing solvents, in term of the 
physical properties of the constituents, in order to 

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!!ÔÏ

select the most efficient solvent for a particular 
process for separating closely boiling compounds.

The system studied in this work was 
composed of Hexane and Benzene. These 
hydrocarbons are difficult to separate because of 
closeness of boiling points. 1-Propanol was used 
as a solvent, to determine experimentally the 
effect of polar components on the separation of 
binary systems. Also, these hydrocarbons having 
different structures comes from different groups 
and form non-ideal solution, hexane (straight 
chain) from alkanes group and benzene from 
aromatics group and 1-propanol from alcohols 
group.
Ternary system II

The ternary system II consists of Toluene, 
Cyclohexane and iso-Octane (2,2,4-Trimethyl-
pentane) and its binaries.

Aromatic hydrocarbons have long been 
known to show non-ideal characteristics when 
mixed with other type of hydrocarbons. In the 
commercial production of toluene from 
petroleum, the toluene must be separated from 
hydrocarbons having about the same boiling point, 
thus the separation of Toluene as a pure 
compound by ordinary fractionation is very 
difficult. Also, the importance of using iso-Octane
(2,2,4-Trimethyl-pentane) as the central 
component in an investigation of orientation 
effects in the pure liquid components.

The system studied in this work was 
composed of Toluene, Cyclohexane and iso-
Octane (2,2,4-Trimethyl-pentane). These 
hydrocarbons having different structures come 
from different groups and form non-ideal solution, 
cyclohexane and iso-octane from alkanes group 
and toluene from aromatics group.

Published data for constituent binary 
systems 1-propanol – hexane, hexane – benzene, 
1-propanol – benzene, and toluene – cyclohexane
were reported in the literature (Hala et al., 1968; 
Ohe, 1989; Hirata et al., 1975). To our knowledge, 
no isobaric or isothermal VLE experimental data 
have been reported previously in the literature for 
the binary systems cyclohexane – iso-octane and 
toluene – iso-octane, and for the two ternary 
systems 1-propanol – hexane – benzene, and 
toluene – cyclohexane – iso-octane.

Possible experimental work errors
Systematic errors in the experiments can 

be ruled out since all the analytical methods were 
carefully controlled (rechecking the analytical 
procedure) and all the measurements were 
duplicated. The value of the experimental results 
depends on the care employed by the 
experimentalist in eliminating sources of error. 
Obtaining good experimental data requires 
appreciable experimental skill, experience and, 
patience.

Possible causes of the deviation were due 
to the fact that the mixtures were not ideal 
mixtures or due to inaccuracy of refractometer, 
thermometer or barometer or incorrect reading off 
of their values. The limit of error in reading the 
refractometer was one in the fifth place, and did 
not vary noticeably for various mixtures.

The estimated precision of the equilibrium 
mixture composition measurements was ±0.001 
mole fraction for the liquid phase and ±0.005 
mole fraction for the vapor phase. The estimated 
uncertainties in the equilibrium temperature and 
pressure were ±0.1 oC and 0.5 mmHg, 
respectively.

Care was taken to assure that equilibrium 
really existed, that the temperature and pressure 
were measured at the position where equilibrium 
existed, and that the taking of samples for analysis 
did not disturb the equilibrium appreciably.

Thermodynamic consistency test
One of the greatest arguments in favor of 

obtaining redundant data is the ability to assess 
the validity of the data by means of a 
thermodynamic consistency test. The consistency 
of the experimental data was examined to provide 
information on the thermodynamic plausibility or 
inconsistency and to recognize any deviations of 
the measured values.

According to McDermott-Ellis test 
method, two experimental points a and b are 
thermodynamically consistent if the following 
condition is fulfilled (McDermott and Ellis, 1965):

D < Dmax   (16)

The local deviation D is given by

  



N

i
iaibibia xxD

1

lnln 
  (17)

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In this method, it is recommended the use 
of a fixed value of 0.01 for Dmax if the accuracy in 
the measurement of the vapor and the liquid mole 
fraction is within 0.001. The local maximum 
deviation, Dmax, due to experimental errors, is not 
constant, and is given by

 















 

N

i ibibiaia
ibiamax yyxyx

xxD
1

1111

            







NN

i
ibia

i
iaib

P

P
xxx

11

lnln2 

            















 

N

bi a
ibia ttt

xx
1

11 …………  (18)

table 2. Results of Thermodynamic
  Consistency Test.

System D Dmax
Thermodynamic 

Consistency 
Test

S
ys

te
m

 I

1-Propanol –
Hexane

0.0243 0.026 Pass

Hexane – Benzene 0.0161 0.021 Pass

1-Propanol –
Benzene

0.0274 0.029 Pass

1-Propanol –
Hexane – Benzene

0.0321 0.035 Pass

S
ys

te
m

 I
I

Toluene –
Cyclohexane

0.0196 0.023 Pass

Cyclohexane – iso-
Octane

0.0153 0.018 Pass

Toluene – iso-
Octane

0.0178 0.024 Pass

Toluene –
Cyclohexane – iso-

Octane
0.0246 0.026 Pass

The conclusion can be drawn that all the data are 
thermodynamically consistency

Statistical measurement and analysis of 
dispersion

To know the applicability and accuracy of 
any proposed correlation it is very important to 
know how this correlation fits the experimental 
data which is done by comparing the obtained 
results from the proposed correlation with the 
experimental data.

The various measurement of dispersion or 
variation are available, the most common being 

the Mean Overall Deviation and Average Absolute 
Deviation.
The Mean Overall Deviation “mean D %” is a 
more tangible element indicating the overall 
goodness of the fit of the data by the correlation 
and it reads:

100%   1 






n
M

MM

Dmean

n

i

ii
obsd
i

calcdobsd

.......  (19)

Average Absolute Deviation “AAD”

n

MM
AAD

n

i
ii
calcdobsd





 1
……………… (20)

where M is an intensive property and n is the 
number of data point.

These equations are used to calculate 
Mean Overall Deviation “mean D%” and Average 
Absolute Deviation “AAD” of experimental results 
of binary and ternary systems.

Azeotropic predictions

The wide concentration field which must 
be investigated experimentally in order to detect 
an azeotrope in the ternary mixtures, reflects the 
importance of the correlation to be able to predict 
azeotropic behavior.

The boiling temperature of the mixture 
was correlated with the liquid composition by the 
following equation:

   







 




1

111

...
2

N N

ij

N

i
iijiijiji

i
ii xxCxxBAxxTxT jjj
O

                                                                            (1)
Where Toi is the boiling temperature of pure 
component i in oC, and N number of component in 
the mixture.

This equation is useful for obtaining 
isothermals and for exploring the azeotropic 
behavior of binary and ternary mixtures. The 
coefficient Aij, Bij, and Cij are binary or ternary 
parameters which are determined directly from the 
binary or ternary data. The increase in the number 
of parameters increases the prediction accuracy.

For exploring the azeotropic behavior of 
binary and ternary mixtures equation (1) was 
solved simultaneously with equations 3 – 10. An 

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azeotrope existed if solution x1 is within the 
interval 0<x1<1. If the computations did not yield 
a solution which corresponds to a true azeotrope, 
the system would not exhibit azeotropic behavior.

The following experiment was performed 
in order to check the validity of the predicted 
values. Ternary solutions with the calculated 
compositions were prepared and introduced in the 
recirculating still. The system was boiled until
equilibrium was achieved. The analysis yielded 
the equilibrium compositions of the vapor and 
liquid. It was seen that the liquid composition 
deviated very small from the vapor composition, 
the computed values did not correspond to true 
azeotropes. On the other hand, the true value of 
the azeotropic point was identical to the solution 
or was located in their vicinity. It was observed 
that the azeotropic boiling temperature was very 
accurately predicted than the composition. The 
reason for this behavior is due to the fact that 
variations in the temperatures in the vicinity of the 
azeotropic point are usually relatively small in 
comparison to the concentration (Anderson et al., 
1978).

discussion of results

In order to assess the reliability of 
experimental results, the equilibrium curves were 
plotted and compared with those from literature 
(Hala et al., 1968; Ohe, 1989; Hirata et al., 1975), 
as shown in figures 3 to 8. It may be seen that 
there is good agreement between the experimental 
data of this work and the experimental data from 
literature and confirms the reliability of this work.

The fact that the equilibrium curves for 
both binaries 1- Propanol – Hexane and 1-
Propanol – Benzene cross the 45o line indicates 
that they form azeotropes.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Hexane in Liquid

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M
ol

e
F

ra
ct

io
n

of
H

ex
an

e
in

V
ap

o
r

Equilibrium Curve

Experimental data

Literature data



Fig. 3. Vapor - Liquid equilibrium Diagram in
               Binary system of 1- Propanol – Hexane.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Hexane in Liquid

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0
M

ol
e

F
ra

ct
io

n
of

H
ex

an
e

in
V

ap
o

r

Equilibrium Curve

Experimental data

Literature data



Fig. 4. Vapor - Liquid Equilibrium Diagram in
              Binary system of Hexane – Benzene.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Benzene in Liquid

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M
o

le
F

ra
ct

io
n

of
B

en
ze

ne
in

V
ap

o
r

Equilibrium Curve

Experimental data

Literature data



Fig. 5. Vapor - Liquid Equilibrium Diagram in
               Binary System of 1 - Propanol –

               Benzene.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Cyclohexane in Liquid

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M
ol

e
F

ra
ct

io
n

o
f

C
y

cl
oh

ex
an

e
in

V
ap

o
r

Equilibrium Curve

Experimental data

Literature data



Fig. 6. Vapor - Liquid equilibrium Diagram in
               Binary system of Toluene –
                Cyclohexane.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Cyclohexane in Liquid

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M
ol

e
F

ra
ct

io
n

o
f

C
y

cl
oh

ex
an

e
in

V
ap

o
r

Equilibrium Curve

Experimental data

Fig. 7. Vapor – Liquid Equilibrium Diagram in 
               Binary system of Cyclohexane - iso-

               Octane.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of iso-Octane in Liquid

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0
M

o
le

F
ra

ct
io

n
of

is
o

-O
ct

an
e

in
V

ap
or

Equilibrium Curve

Experimental data

Fig. 8. Vapor - Liquid Equilibrium Diagram in 
                Binary System of Toluene - iso-Octane.

The T-x-y diagrams were plotted at 
isobaric condition, figures 9 to 14. All those 
experimental results were compared with 
literature. It may be seen that there is good 
agreement between the experimental data of this 
work and the data from literature.

The binary mixtures of toluene –
cyclohexane and toluene – iso-octane are nearly 
ideal solution. The other binary mixtures showed 
positive deviations.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Hexane

60

65

70

75

80

85

90

95

100

T
E

M
PE

R
A

T
U

R
E

,
C

o

T-x-y Curves

Liquid phase experimental data

Liquid phase literature data

Vapor phase experimental data

Vapor phase literature data





Fig. 9. Boiling Point - Composition Curve for
           1 - Propanol – Hexane at  760 mmHg.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Hexane

66

68

70

72

74

76

78

80

82

T
E

M
P

E
R

A
T

U
R

E
,

C

o

T-x-y Curves

Liquid phase experimental data

Liquid phase literature data

Vapor phase experimental data

Vapor phase literature data





Fig. 10. Boiling Point - Composition Curve for
                Hexane – Benzene at 760 mmHg.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Benzene

74

76

78

80

82

84

86

88

90

92

94

96

98

T
E

M
P

E
R

A
T

U
R

E
,

Co

T-x-y Curves

Liquid phase experimental data

Liquid phase literature data

Vapor phase experimental data

Vapor phase literature data





Fig. 11. Boiling Point - Composition Curve for
                1 - Propanol – Benzene at 760 mmHg.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Cyclohexane

80

85

90

95

100

105

110

115

T
E

M
P

E
R

A
T

U
R

E
,

C

o

T-x-y Curves

Liquid phase experimental data

Liquid phase literature data

Vapor phase experimental data

Vapor phase literature data





Fig. 12. Boiling Point - Composition Curve for
                 Toluene – Cyclohexane at  760 mmHg.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of Cyclohexane

80

82

84

86

88

90

92

94

96

98

100

T
E

M
P

E
R

A
T

U
R

E
,

C

o

T-x-y Curves

Liquid phase experimental data

Vapor phase experimental data





Fig. 13. Boiling Point - Composition  Curve  for
               Cyclohexane - iso-Octane at 760 mmHg.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Mole Fraction of iso-Octane

98

100

102

104

106

108

110

112

T
E

M
P

E
R

A
T

U
R

E
,

Co

T-x-y Curves

Liquid phase experimental data

Vapor phase experimental data





Fig. 14. Boiling Point - Composition Curve for
                 Toluene - iso-Octane at 760 mmHg.

The experimental data for binary system 
were found to be in close agreement with the 
reported values of the literature.

There was no evidence of erroneous data 
at the entire concentration range because no 
scattering points were observed and all the points 
of data fall on smooth curves, showing the 
experimental values to be consistent. The 
difference between the experimental points and 
the smoothing curves are within the analytical 

accuracy (the estimated precision of the 
equilibrium mixture composition measurements 
was ±0.001 mole fraction for the liquid phase and 
±0.005 mole fraction for the vapor phase. The 
estimated uncertainties in the equilibrium 
temperature was ±0.1 oC). The average scatter of 
the experimental points from the best line is less 
than that of the previous investigations in the 
literature.

For system I the experimental data for all 
the three binaries show that they are non-ideal in 
nature. When studying the equilibrium of hexane-
benzene, the observation made in this 
investigation was that no separation obtainable at 
concentration above 97 mole % hexane. The 1-
propanol-hexane and 1-propanol-benzene systems 
show minimum boiling azeotropes.

For system II the toluene-cyclohexane 
system does not form an azeotrope, but it seems 
that there is a tendency toward azeotrope 
formation at high cyclohexane concentration (at 
low-boiling ends of the T-x-y diagram). The 
cyclohexane – iso-octane system shows nearly 
ideal behavior. The toluene – iso-octane system 
does not form an azeotrope, but it seems that there 
is a tendency toward azeotrope formation at high 
iso-octane concentration. Table 5 indicates the 
azeotropic composition and temperature of these 
binaries.

The distribution of the experimental vapor-
liquid equilibrium data for ternary systems in the 
composition triangle diagram may be readily 
appreciated. These distributions were presented in 
figures 15 and 16. Each equilibrium composition 
is shown in figures 15 and 16 by an arrow, with 
the flight end representing the composition of the 
liquid phase and the arrowhead that of the vapor 
phase.

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!!ÔÕ





1 - PROPANOL

HEXANE BENZENE
0 25 50 75 100

25

50

75

10
0

0

0

25

50

100

75

















 










































 






























Fig. 15. Composition diagram for the ternary 
             mixture 1 - Propanol + Hexane +
             Benzene   flight end of arrows (●)
             indicate equilibrium concentrations of 
             the liquid phase at  760 mmHg  and the 
             arrowheads (►) corresponding vapor
             phase compositions.

TOLUENE

CYCLOHEXANE iso - OCTANE
0 25 50 75 100

25

50

75

10
0

0

0

25

50

100

75














































 





 








 



























































  
 



    




 























 









 



























 

 



Fig. 16. Composition diagram for the ternary
              mixture Toluene  + Cyclohexane  +  iso-

             Octane flight end of arrows (●) indicate
            equilibrium concentrations of the liquid
             phase at  760 mmHg and the arrowheads
             (►) corresponding vapor phase

             compositions.

Figure 17 shows equilibrium isotherms on 
the liquid phase composition diagram obtained 

from the experimental data by graphical 
interpolation.

Hexane
( 68.7   o C )

Benzene 
( 80.1  o C )

  1 - Propanol 
( 97.2   o C )

68
.86

8
o C

70
.13

7 
o C

7
1.4

05 oC
72.674 oC

73.942 oC
75.211 oC

76.479 oC

77.747 o
C

79.016 o
C

81.553 o
C

84.089 o
C

Fig. 17. Composition diagram for the liquid
              phase of the ternary mixture 1 – Propanol
             + Hexane  +  Benzene, showing constant
              boiling temperature contours at 760 
              mmHg.

Figure 17 shows a sharp temperature drop 
occurs from the hexane – benzene binary as small 
amounts of 1-propanol are added. The drop is less 
severe at high hexane concentrations. The interior 
of the surface is nearly flat at points reasonably 
removed from the pure substances. The flat 
interior also rises rapidly as pure 1-propanol is 
reached. A trough with only slight curvature 
connecting the two binary azeotropes (1-propanol 
– hexane and 1-propanol – benzene). The surfaces 
rising from the trough toward pure 1-propanol and 
also toward the hexane – benzene binary show 
slightly outward curvature.

The ternary system I shown in figure 18 
exhibits a ternary azeotrope. The azeotropic 
composition and temperature of the two binaries 
are shown in the same figure. In this system, 
however, there is a point, 0, where boiling 
temperature is lower than any of the boiling 
temperatures of the three pure compounds. This 
composition is known as the ternary azeotrope. 
The lines of constant boiling temperature 
surrounding the azeotropic composition 0, and as 
one proceeds away from 0 the successive 

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!!ÔÖ

isotherms that are intersected represent increasing 
boiling temperatures.

Hexane
( 68.7      oC )

Benzene
( 80.1    oC )

1-Propanol
( 97.2     oC )

o

o

o 76   
o C

65.8    
o 

C

67.2   
o 

C

68
.5

  
o C

68
.2

  
o C67

.9
01

 
o C

Fig. 18. Composition diagram for the liquid
              phase of the Ternary  mixture 1 –
             Propanol + Hexane + Benzene, showing
              constant   boiling   temperature  contours 
              and the location of the ternary and binary
             azeotrope at 760 mmHg.

In order to determine the coordinates of 
the ternary azeotrope, figure 19 was drawn. It is a 
spatial plot of the region where the azeotropic 
point might be located (a crater-like shape or 
valley).

Hexane
1-Propanol

T
em

pe
ra

tu
re

 o C

68.7  oC

97.2  oC

80.1  oC

67.2         o C

Fig. 19. Spatial plot of the ternary system shows
               the region of the ternary azeotrope.

No ternary azeotrope was detected 
experimentally in the ternary system II as shown 
in figures 20 and 21, and the prediction method 
did not discover any maximum or minimum with 
the interior of the phase surface, hence, the system 
does not contain ternary azeotrope. Also, the 
shape of curves indicates that the system does not 
exhibit azeotropic behavior.

Isothermals for both ternary systems were 
obtained as shown in figures 17 and 20. The 
prominent fact which may be concluded from the 
behavior of the isothermals in figure 17 is the 
existence of a ternary minimum boiling point 
azeotrope in the 1 - Propanol + Hexane + Benzene 
system. On the other hand, it may be concluded 
from figure 20 that the system Toluene + 
Cyclohexane + iso-Octane does not exhibit a 
ternary azeotrope.

Cyclohexane
( 80.4  oC )

iso-Octane
( 99  oC )

Toluene
( 110.9   oC )

82.862 oC

84.124 oC

85.386 oC

86.648 oC

87.910 oC

89.171 oC

90.433 oC

91.695 oC

92.957 oC

94.219 oC

95.481 oC

96.743oC

 98.005oC
 99.267 oC

 100.529 oC

 103.052 oC

 105.576 oC

Fig. 20. Composition diagram for the liquid phase 
             of the ternary mixture Toluene +
              Cyclohexane + iso-Octane, showing
             constant boiling temperature contours at 
              760 mmHg.

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T
em

p
er

at
u
re

 o C

Toluene

Cyclohexane

iso-Octane

110.9    
o
C

99   
o
C

80.4   
o
C

Fig. 21. Spatial plot of the ternary system Toluene
             + Cyclohexane + iso-Octane.

A comparison of calculated and 
experimental temperature results using equation 1 
is given in figure 22, the other results of binary 
and ternary systems are illustrated in figures 23 to 
29.

If there is complete agreement between 
experimental and predicted temperature values, all 
the points would fall on the 45-degree line. It 
seem from figure 22 that the equation 1 is useful 
for obtaining isothermals and for exploring the 
azeotropic behavior for binary and ternary system 
because the deviation of all the points from the 
45o line is small. The comparison with 
experimental temperature values indicates that the 
calculations show rather good agreement with the 
data as illustrated in figures 22 to 29.

The value of the parameter of equation 1 
and comparison between experimental and 
calculated temperature for binary systems are  
illustrated in table 3. And, The value of the 
parameter of equation 1 and comparison between 
experimental and calculated temperature for 
ternary systems are illustrated in table 4.

68 70 72 74 76 78 80 82
T ( C )

68

70

72

74

76

78

80

82

T
(

C
)

o
ca

lc
.

o
expt.

           Fig. 22. Comparison of calculated and
             experimental temperature results for
             Hexane – Benzene system

65 70 75 80 85 90 95 100
T ( C )

65

70

75

80

85

90

95

100

T
(

C
)

o
ca

lc
.

o
expt.

             Fig. 23. Comparison of calculated and
             experimental temperature results for 1-
              Propanol – Hexane system.

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75 80 85 90 95 100
T ( C )

75

80

85

90

95

100

T
(

C
)

o
ca

lc
.

o
expt.

Fig. 24. Comparison of calculated and
             experimental temperature results for 1-
              Propanol – Benzene system.

80 84 88 92 96 100 104 108 112
T ( C )

80

84

88

92

96

100

104

108

112

T
(

C
)

o
ca

lc
.

o
expt.

Fig. 25. Comparison of calculated and
             experimental temperature results
             for Toluene – Cyclohexane system

80 82 84 86 88 90 92 94 96 98 100
T ( C )

80

82

84

86

88

90

92

94

96

98

100

T
(

C
)

o
ca

lc
.

o
expt.

Fig. 26. Comparison   of calculated   and 
              experimental temperature results for 

              Cyclohexane – iso-Octane system

98 100 102 104 106 108 110 112
T ( C )

98

100

102

104

106

108

110

112

T
(

C
)

o
ca

lc
.

o
expt.

Fig. 27. Comparison of calculated  and 
             experimental temperature results for
             Toluene – iso-Octane system.

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65 70 75 80 85 90 95
T ( C )

65

70

75

80

85

90

95

T
(

C
)

o
ca

lc
.

o
expt.

           Fig. 28. Comparison of  calculated  and
              experimental temperature results for 1-
              Propanol – Hexane – Benzene system

80 85 90 95 100 105 110
T ( C )

80

85

90

95

100

105

110

T
(

C
)

o
ca

lc
.

o
expt.

         Fig. 29. Comparison  of  calculated  and 
              Experimental temperature results for
              Toluene – Cyclohexane – iso-Octane
              system

As the table 3 indicates that the equation 1 
gives values that have an overall average 
absolute deviations (AAD) and overall 
percentage mean deviations for all six binary 
systems of 0.271 and 0.323 respectively, for 138 
data points.

As the table 4 indicates that the equation 1 
gives values that have an overall average absolute 
deviations (AAD) and overall percentage mean 

deviations for all two ternary systems of 0.781 and 
0.959 respectively, for 256 data points.

Comparison between experimental and 
literature [90] and calculated values using 
equation 1 and equations 3 to 10 of the azeotropic 
point for the binary systems are illustrated in table 
5, and for the ternary systems were illustrated in 
table 6.

It seems from tables 5 and 6 that the calculated 
temperature and composition of azeotropic point 
of binary and ternary systems and detecting the 
type of azeotropic point by using equation 1 with 
equations 3 to 10 gives good results comparing 
with the experimental and literature data.

Table 3. The value of the parameter of equation
                   (1) and comparison between
                   experimental and calculated 
                   temperature for binary systems.

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Table 4. The value of the parameter of equation
                   (1) and comparison between
                   experimental and calculated
                   temperature for ternary systems

Parameters

System I System II

O
ve

ra
ll

 E
rr

or

1-Propanol + 
Hexane + 
Benzene

Toluene + 
Cyclohexane 
+ iso-Octane

A12 -54.125 -18.591

A13 -32.434 -27.409

A23 -11.598 -7.2043

B12 -26.423 -17.344

B13 -18.348 -16.695

B23 6.6892 13.539

C12 -56.826 -15.174

C13 -11.838 -89.674

C23 -22.179 -58.457

D12 -68.378 -20.796
D13 1.0346 3.7671
D23 -31.647 -74.527

E12 63.0217 52.524

E13 -10.659 -25.473

E23 112.729 186.163

No. of data 
points

121 135 256

Mean D % 1.169 0.749 0.959

AAD 0.869 0.692 0.781

7 Conclusion
Based on this study, the following conclusions 

can be made:

1. The recirculating apparatus used with all 
accessories proved to be very suitable for 
obtaining VLE data.

2. All the data obtained are 
thermodynamically consistent because they 
passed the thermodynamic consistency test of 
McDermott-Ellis test method.

3. A binary azeotrope is an extremal point of 
the binary mixture temperature functions. 
While a ternary azeotrope is not necessary a 
global temperature extreme for the ternary 
mixture, thus being a saddle point of the 
temperature surface.

4. Ternary mixtures may exhibit distinctly 
different isotherm maps depending on the 
existence of azeotropes, their type (minimum-, 
maximum- or saddle boiling), their location, 
and the relative order of the boiling points of 
the pure components.

5. Equation 1 which is a modified correlation 
of Gibbs-Konovalov theorem is very 
important for obtaining isothermals and for 
exploring the azeotropic behavior of binary 
and ternary systems.
(a) Using equation 1 for the six 

binary systems of 138 data point 
give an overall average absolute 
deviations (AAD) and an overall 
percentage mean deviations of 
0.271 and 0.323 respectively.

(b) Using equation 1 for the two 
ternary systems of 256 data point 
give an overall average absolute 
deviations (AAD) and an overall 
percentage mean deviations of 
0.781 and 0.959 respectively.

Table 5. Comparison between experimental and literature and calculated values of the azeotropic
                   point for the binary systems

Binary system
System I System II

1-Propanol(1) + 
Hexane(2)

Hexane(1) 
+ 

Benzene(2)

1-Propanol(1) 
+ Benzene(2)

Toluene(1)+ 
Cyclohexane(2)

Cyclohexane(1)+
iso-Octane(2)

Toluene(1) + 
iso-Octane(2)

Experimental

Temperature 
oC

65.8 - 76.0 - - -

x1 0.109 - 0.236 - - -
x2 0.891 - 0.764 - - -

Literature

Temperature
oC

65.73 - 76.2 - - -

x1 0.114 - 0.225 - - -

x2 0.886 - 0.775 - - -

Calculated

Temperature 
oC

65.512 - 75.743 - - -

x1 0.146 - 0.217 - - -
x2 0.854 - 0.783 - - -

Azeotropic type
Minimum 

boiling
No 

azeotropy
Minimum 

boiling
No

azeotropy
No

azeotropy
No 

azeotropy

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Table 6. Comparison between experimental and literature and calculated values of the azeotropic 
point for the ternary systems

Ternary system
System I System II

1-Propanol (1) + Hexane (2) + Benzene (3) Toluene(1)+ Cyclohexane(2) + iso-Octane(3)

Experimental

Temperature 
oC 67.4 -

x1 0.342 -

x2 0.625 -
x3 0.033 -

Literature

Temperature
oC

No data No datax1
x2
x3

Calculated

Temperature 
oC

67.23 -

x1 0.346 -
x2 0.628 -
x3 0.026 -

Azeotropic type Minimum Boiling Azeotropy No azeotropy

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References

Anderson, T. F., Abrams, D. S., Grens, E. A., 
“Evaluation of parameters for Nonlinear 
Thermodynamic Models”, AIChE J., 24, 20, 
1978.

Hala, E., Pick, J., Fried, V., and Vilim O., 
“Vapor-Liquid Equilibrium”, 2nd ed., Pergamon 
press, London, 1968.

Harold, R. N., “Phase Equilibrium in Process 
Design”, Wiley-Interscience Publisher, New 
York, 1970.

Hirata, M., Hoen, S., and Nagahama, K., 
“Computer Aided Data Book of Vapor-Liquid 
Equilibria”, Kodansha Limited, Tokyo, 1975.

Hoffman, E. J., “Azeotropic and Extractive 
Distillation”, Interscience Publisher, New York, 
1964.

Malesinski, W., “Azeotropy and other Theoretical 
Problems of Vapor-Liquid Equilibrium”, 
Interscience, PWN, New York, 1965.

Marc, J. A., Martin, J. P. and Thamas, F. T., 
“Thermophysical Properties of Fluid, An 
introduction to their prediction”, Imperial College 
Press, first reprint, 1998.

McDermott, C., Ellis, S. R. M., “A 
Multicomponent Consistency Test”, Chem. Eng. 
Sci., 20, 293, 1965.

Ohe, S., “Vapor-Liquid Equilibria Data Book”, 
Elsevier Scientific Publishing Company, 
Netherlands, 1989.

Walas, S. M., “Phase Equilibria in Chemical 
Engineering”, Butterworth Publishers, London, 
1985.

  
  

  
  
  
  
  
  
  
  
  
  

  
  

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عددة وللتنبؤ بحالة سائل لألنظمة المت –عالقات ریاضیة لتمثیل بیانات توازن بخار 
األیزوتروب

  خالــد فرھــود. د
قسم الھندسة الكیمیاویة
  الجامعة التكنولوجیة 

  
  :الخالصة
عالقات ریاضیة للتعبیر عن درجة حرارة غلیان الخالئط لألنظمة المتعددة كدالة مباشرة لتركیز السائل تم   

أقل درجة (متعددة وكذلك الیجاد نوع األیزوتروب اختبارھا بنجاح وطبقت للتنبؤ بحالة األیزوتروب للخالئط ال
- Gibbs(باستخدام العالقات المطورة لنظریة ) حرارة، أعلى درجة حرارة، نوع ثابت درجة الحرارة

Konovalov ( كذلك نقطة األیزوتروب لألنظمة الثنائیة و الثالثیة تم التحقق منھا بصورة عملیة باستخدام التعیین
  .سائل لألنظمة الثنائیة والثالثیة –لبیانات العملیة لتوازن بخار البیاني على أساس ا

-١"ثالثیة وھيسائل بثبوت الضغط الجوي ألثنین من األنظمة ال –في ھذه الدراسة تم قیاس توازن بخار   
بنزین و  –ھكسان ، ھكسان  –بروبانول -١" وكذلك األنظمة الثنائیة التابعة لھ وھي " بنزین –ھكسان  -بروبانول 

ثالثي  – ٢،٢،٤(ایزو أوكتان  –ھكسان حلقي  –تلوین "وكذلك للنظام الثالثي األخر وھو "بنزین –بروبانول -١
 –ایزو أوكتان و تلوین  –ھكسان حلقي، ھكسان حلقي  –تلوین (عة لھ وھي وكذلك األنظمة الثنائیة التاب)" مثیل بنتان

  .كیلوباسكال ١٠١.٣٢٥تم قیاسھا جمیعًا في ظروف ) ایزو أوكتان
النظام الثالثي . القیاسات تمت في برج التوازن الدوار الذي یتم من خاللھ تدویر كال الطورین البخار والسائل  

وكذلك اثنین من األنظمة الثنائیة " بروبانول-١"الذي یحتوي مركب قطبي وھو " بنزین –ھكسان  -بروبانول -١"
النظام . تكون حالة األیزوتروب نوع أقل درجة حرارة" بنزین –بروبانول -١ھكسان و  –بروبانول -١"وھي 

  .الثالثي األخر وكذلك األنظمة الثنائیة الباقیة ال تكون حالة األیزوتروب
تازت بنجاح اختبار الصحة والدقة من الناحیة الثرمودینامیكیة باستخدام طریقة االختبار كل النتائج العملیة اج  

  لـ
 )McDermott-Ellis.(  

استخدمت إلیجاد ثوابت ) Maximum Likelihood Principle(طریقة االختیار األفضل المسماة   
ائیة والثالثیة وھذه الطریقة  تضمن من الناحیة سائل لألنظمة الثن –العالقات باستخدام البیانات العملیة لتوازن البخار 

الریاضیة والحسابیة االختیار األفضل لقیم الثوابت في العالقات حیث إنھا تعمل في حالة كل المتغیرات المقاسة تكون 
  .معرضھ للخطأ وكذلك عندما تكون حالة الألمثالیة في كال الطورین البخار والسائل لألنظمة الثالثیة والثنائیة

القیم الصحیحة یجب أن تعین تجریبیًا بواسطة . التقارب بالقیم جید بین البیانات المتنبأ بھا والتي تم قیاسھا عملیًا
  .استكشاف منطقة التركیز التي یتم حسابھا بواسطة النتائج المحسوبة من ھذه العالقات الریاضیة

  

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