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VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 667
F
rom the point of view of separation processes involving
vapour-liquid systems (e.g. fractional distillation), it is
important to know if an azeotrope exists in a particular
system. If complete experimental vapour-liquid equilibrium data
are available, this information is known explicitly. If, however,
the data are embedded in a correlation for liquid-phase activity
coefficients (γi) (or, equivalently, for the excess molar Gibbs
function (gE)), or if the data come from a model for γi or gE, this
information is only known implicitly. Models may include a
solution model such as a regular solution (Hildebrand et al.,
1970), or a group-contribution method based on molecular
structure, such as UNIFAC (Fredenslund et al., 1975), for use
when no experimental or model parameters are available.
The possible existence of an azeotrope may be considered for
a system either at constant temperature (T) or at constant
pressure (P). The former may be more important from a theoretical
On Criteria for Occurrence of Azeotropes
in Isothermal and Isobaric Binary Systems
Ronald W. Missen
Department of Chemical Engineering and Applied Chemistry, University of Toronto, 200 College Street,
Toronto, ON, Canada M5S 3E5
point of view, but since separation processes involving vapour-
liquid systems operate essentially at constant pressure, this
situation is more important from this practical point of view.
Attention has been paid in the literature to the isothermal case,
but we address both situations here.
These considerations lead to the question of obtaining criteria
for the occurrence of azeotropes in general. Some guidance in
the literature for isothermal systems comes from the “Bancroft
rule” (King, 1969) and from criteria developed by Prigogine and
Defay (1954) for regular solutions. The former, which states, in
effect, that if two species have the same saturation vapour
pressure p* (at T′, say), then there is an azeotrope at T′, is
self-evident, since any departure from ideality in such a case
leads to an azeotrope. Comprehensive criteria for isothermal
systems are given by Brandini (1974) in terms of activity coeffi-
cients at infinite dilution (γ∞). We include here a different
Criteria are developed for the occurrence of azeotropes in binary nonelectrolyte systems for both isothermal and isobaric situations in terms of
liquid-phase activity coefficients at infinite dilution (γ∞). In the case of isothermal systems at temperature T, for a positive azeotrope, γ∞
lvc ≥ p*mvc/
p*lvc, where lvc refers to the less volatile component, mvc to the more volatile component, and p* to saturation vapour pressure at T; for a
negative azeotrope, γ∞
mvc ≤ p*lvc/p*mvc. In the case of isobaric systems at pressure P, for a minimum-boiling azeotrope, γ∞
lvc(TBmvc) ≥ P/p*lvc(TBmvc),
where TB refers to the boiling point at P; for a maximum-boiling azeotrope, γ∞
mvc(TBlvc) ≤ P/p*mvc(TBlvc). The criteria are also given in terms of the
parameters of selected correlations for the excess molar Gibbs function (gE). Examples of the use of the criteria are provided. Various methods
that generate values of γ∞ can be used in conjunction with the criteria, for example, in screening procedures.
Des critères basés sur les coefficients d’activité en phase liquide pour une dilution infinie (γ∞) ont été mis au point pour prédire l’apparition des
azéotropes dans les systèmes non électrolytiques binaires dans des conditions isothermes et isobares. Dans le cas des systèmes isothermes à la
température T, pour une azéotrope positive, γ∞
lvc ≥ p*mvc/p*lvc , où lvc réfère au composant moins volatil, mvc au composant plus volatil et p* à la
pression de vapeur saturante à T; pour une azéotrope négative, γ∞
mvc ≤ p*lvc/p*mvc . Dans le cas de systèmes isobares à la pression P, pour une
azéotrope d’ébullition minimum, γ∞
lvc(TBmvc) ≥ P/p*lvc(TBmvc) , où TB réfère au point d’ébullition à P; pour une azéotrope d’ébullition maximum,
γ∞
mvc(TBlvc) ≤ P/p*mvc(TBlvc). On donne également les critères pour les paramètres de corrélations sélectionnées pour la fonction molaire d’excès de
Gibbs (gE). Des exemples d’utilisation des critères sont donnés. Diverses méthodes donnant des valeurs de γ∞ peuvent être utilisées avec ces
critères, par exemple dans les méthodes de tamisage.
Keywords: isothermal azeotrope criteria; isobaric azeotrope criteria; binary systems
668 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005
development of these together with a graphical interpretation.
We also give a corresponding new treatment for isobaric systems.
Examples to illustrate application of the criteria developed are
provided.
ISOTHERMAL CRITERIA FOR OCCURRENCE
OF AZEOTROPES
Qualitative Considerations
Figure 1 illustrates schematically P(x1) relationships for a fixed-
T plane intersecting a P-T-x1 saturated-liquid surface, where x1is
mole fraction of component 1. It shows P(x1) for various degrees
of departure from Raoult’s law (A, for an ideal system) in both
positive and negative directions. Curves B, C, and D show
increasing degrees of positive deviation: B for no azeotrope and
D for a positive azeotrope (at the maximum, where x1 = y1 (not
shown, the mole fraction in the coexisting vapour phase,
whether actual or virtual)); C represents the boundary between
these two types of behaviour, as the indication of incipient
occurrence of a positive azeotrope with respect to increasing
extent of nonideality. (For still greater positive deviation than in
D, single liquid-phase stability eventually breaks down, and
increasing degrees of partial miscibility set in, leading ultimately
to complete immiscibility, the greatest degree of positive
deviation from Raoult’s law.). Correspondingly, curves E and G
(negative azeotrope at the minimum) illustrate increasing
degrees of negative deviation, with F representing incipient
negative azeotrope formation with respect to increasing nonide-
ality. (There is no corresponding single-phase stability breakdown
in this direction.)
Assumptions
We assume the following:
(A1) The system is a binary nonelectrolyte system of
components 1 and 2 that are completely miscible in the
liquid phase.
(A2) Information for γi or gE is for a saturated liquid phase,
actually or potentially in equilibrium with a vapour
phase.
(A3) The γi or gE data refer to Raoult’s law ideality; that is, γi
→ 1 as xi → 1.
(A4) The γi or gE data refer either to isothermal or to isobaric
(case to follow) conditions.
(A5) The dependence of γi or gE on P is negligible, but the
dependence on T must be allowed for.
(A6) The pressure is sufficiently low that the Poynting correc-
tion (Sandler, 1999) is equal to one.
Criteria in Terms of Activity Coefficients at Infinite
Dilution
The activity coefficients at infinite dilution are designated γ1
∞ (at
x1 = 0) and γ2
∞ (at x2 = 0) for components 1 and 2, respectively.
Our goal is to obtain criteria in terms of these quantities.
In the absence of any fugacity corrections at low pressure
(assumption (A6)), the total pressure at T is given by
P = γ1(T,x1)x1p1* (T) + γ2(T,x1)x2p2* (T) (1)
From Equation (1), on differentiation at constant T, we obtain
∂
∂
= +
∂
∂
− +
∂
∂
P
x
p x p
x
p x p
x1
1 1 1 1
1
1
2 2 2 2
2
1
* * * *γ
γ
γ
γ
(2)
= 0 (at an azeotrope) (3)
Consider incipient positive azeotrope formation with component
1 as the more volatile component (p1* > p2* at T). This occurs
as x1 → 1 at C in Figure 1. In such a case, x2 → 0, γ1 → 1, γ2 →
γ2
∞, x1p1* ∂γ1/∂x1 = x1p1*γ1 ∂lnγ1/∂x1 → p1* ∂lnγ1/∂x1 → 0, and
x2p2* ∂γ2/∂x1 → 0. From Equations (2) and (3), we conclude that
a criterion for incipient formation of a positive azeotrope is
γ2
∞ = p1*/p2* ( > 1). Instead of component 1 being the more
volatile component, an arbitrary designation may have component
2 in this role. The situation depicted in Figure 1 is then reversed
(although not shown). Incipient azeotrope formation occurs as
x1 → 0 or x2 → 1, in which case, γ1 → γ1
∞ and γ2 → 1. From
Equations (2) and (3), the criterion for an incipient azeotrope is
γ1
∞ = p2*/p1* ( > 1). In either case, if the component designa-
tions are changed to more volatile component (mvc) and less
volatile component (lvc), instead of components 1 and 2, the
criterion for an incipient positive azeotrope can be written as
one equation:
p*2
P
X1
F
T CONSTANTT CONSTANTT CONST
p*1
0 1
G
E
A
B
D
C
Figure 1. Schematic representation of P(x1) at fixed T showing positive
and negative deviations from Raoult’s law.
A Raoult’s law
B Positive deviation, no azeotrope
C Positive deviation, incipient positive azeotrope
D Positive deviation, positive azeotrope at maximum
E Negative deviation, no azeotrope
F Negative deviation, incipient negative azeotrope
G Negative deviation, negative azeotrope at minimum
VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 669
γlvc
mvc
lvc
p
p
∞
=
*
*
criterion for
incipient positive
azeotrope formaation










(4)
As the degree of positive departure from ideality increases, in
the sense of curve D in Figure 1, azeotrope formation occurs. In
every case, γ∞
lvc increases relative to the value in Equation (4).
Thus, a criterion for the existence of a positive azeotrope is
γlvc
mvc
lvc
p
p
∞
≥






*
*
criterion for
positive azeotrope (5)
If γ∞
lvc < p*mvc/p*lvc, no azeotrope exists.
Similarly, consider incipient negative azeotrope formation as
x1 → 0 at F in Figure 1, with component 1 as the mvc (or the
reverse as x1 → 1, if component 2 is the mvc). Considerations
based on Equations (2) and (3) corresponding to those above
lead to the following criterion for an incipient negative
azeotrope:
γmvc
lvc
mvc
p
p
∞
=
*
*
criterion for
incipient negative
azeotrope formaation










(6)
As the degree of negative departure from ideality increases in
the sense of curve G in Figure 1, azeotrope formation occurs. In
every case, γ∞
mvc decreases relative to the value in Equation (6).
Table 1. Expressions for γ1
∞ and γ2
∞ from selected gE equations
gE equation γ1
∞ γ2
∞
Redlich-Kister, Equation (8)
Van Laar
Wilson
NRTL
exp ( ) exp
exp exp
1
1
1
1
0 0
1
RT
g
RT
g
A B
k
K
k
k
k
K
k
= =
∑ ∑−
















Λ 22
21
21
12
21 12 12 12 1
1
1
1exp( ) exp( )
exp exp - exp
− −
+ ( ) 
Λ
Λ
Λ
τ τ α τ τ 2
2 21 12 2
1+ ( ) τ α τexp -
Table 2. Examples of use of criteria (5) and (7) for positive and negative azeotropes
System T/ºC mvc dev. gE criterion (5) criterion (7) ref.
corr.a γ∞
lvc P*mvc pos. γ∞
lvc P*lvc neg.
P*lvc azeo. P*mvc azeo.
n-C5H12 (1) 40 (1) + VL 4.07 1.52 Y ← N/A → (a)
+ 40 (1) + W 4.02 1.52 Y ← N/A → (a)
propionaldehyde (2) 40 (1) + NRTL 3.85 1.52 Y ← N/A → (a)
n-butyraldehyde (1) 45 (1) + VL 3.05 2.30 Y ← N/A → (a)
+ 45 (1) + W 3.12 2.30 Y ← N/A → (a)
n-C7H16 (2) 45 (1) + NRTL 3.04 2.30 Y ← N/A → (a)
isobutyraldehyde (1) 45 (1) + VL 2.72 3.41 N ← N/A → (a)
+ 45 (1) + W 2.76 3.41 N ← N/A → (a)
n-C7H16 (2) 45 (1) + NRTL 2.73 3.41 N ← N/A → (a)
DMA (1) + CH3OH (2) 40 (2) – RK ← N/A → 0.406 0.019 N (b)
CH3F (1) + HCl (2) –90.8 (2) – RK ← N/A → 0.267 0.668 Y (c)
C6F6(1) + C6H6(2) 60 (2) ± RK 1.20 1.04 Y 0.898 0.979 Y (d)
(a) Eng and Sandler (1984); (b) Zielkiewicz (2003); (c) Senra et al. (2002); (d) Gaw and Swinton (1968)
aRK (Redlich-Kister); VL (van Laar); W (Wilson)
Figure 2. Regions of occurrence of positive and negative azeotropes in
binary systems at fixed T according to criteria (5) and (7).
(note change of variables in lower left and upper right fields)
670 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005
a gE expression is available, but is outside the scope of this
paper.
As noted above, Brandini (1974) presents essentially the same
criteria as (5) and (7) for isothermal systems, but expresses them
in terms of components 1 and 2, and provides no graphical
interpretation corresponding to Figure 2. The designation as
components 1 and 2 requires four statements rather than the two
in criteria (5) and (7).
Criteria in Terms of Parameters of gE Correlations
As noted above, the information required to apply the criteria for
existence of azeotropes may be embedded in equations for the
excess molar Gibbs function (gE). In Table 1, expressions are
provided for γ1
∞ and γ2
∞ from four such equations: Redlich-Kister
(1948), van Laar (Carlson and Colburn, 1942), Wilson (1964),
and NRTL (Renon and Prausnitz, 1968). The van Laar results are
readily apparent from the Carlson-Colburn forms of the activity
coefficients. The Redlich-Kister equation is an expansion in
terms of composition that can be represented by:
g x x g x xE
k
K
k
k
= −
=
∑1 2
0
1 2( ) (8)
The results in Table 1 can be derived from Equation (8).
Examples of Use of Criteria
Table 2 shows randomly selected examples of the use of criteria
(5) and (7) to indicate presence or absence of an azeotrope. The
first nine entries come from the work of Eng and Sandler (1984)
on three aldehyde systems (column 1) at the T given in column
2. In each of these cases, component (1) is the more volatile
component (mvc), and the deviation from ideality is positive.
For each system, they compared the fitting of several gE correla-
tions (corr.), including van Laar (VL), Wilson (W) and NRTL
equations. The use of criterion (5) is applicable here, and
columns 6 to 8 give the results – azeotropes in the first two
systems, but no azeotrope in the third, as consistently shown by
each gE correlation. The next two entries (Zielkiewicz, 2003;
Senra et al., 2002) refer to negative deviation and the use of
criterion (7) in conjunction with the RK equation. The last entry
in Table 2 is for the rare case of a double azeotrope, in the
hexafluorobenzene + benzene system at 60°C (Gaw and
Swinton, 1968). This is the situation in which both a positive
and a negative azeotrope occur at a particular T. In this case,
both criteria (5) and (7) are required, and they indicate the
existence of the two azeotropes observed. In all 12 cases in
Table 2, the results of application of criterion (5) or (7) agree
with experimental results.
ISOBARIC CRITERIA FOR OCCURRENCE
OF AZEOTROPES
Qualitative Considerations
Figure 3 illustrates schematically T(x1) relationships for fixed-P
planes intersecting a P-T-x1 saturated-liquid surface. The
behaviour labelled by C′, D′, F′, and G′ corresponds to that
labelled by C, D, F, and G in Figure 1. Thus, D′ represents a
minimum-boiling azeotrope at fixed P, which corresponds to a
positive azeotrope at fixed T. Similarly, G′ represents a maximum-
boiling azeotrope, which corresponds to a negative azeotrope.
The behaviour indicated at C′ and F′ represents incipient
formation (with respect to increasing extent of nonideality) of
Thus, a criterion for the existence of a negative azeotrope is
γmvc
lvc
mvc
p
p
∞
≤






*
*
criterion for
negative azeotrope
(7)
If γ∞
mvc > p*lvc/p*mvc, no azeotrope exists.
Figure 2 shows the results expressed by the criteria of (5) and
(7) in graphical form. In the lower left (closed) field, the
diagonal line representing incipient azeotrope formation divides
the regions of negative azeotrope occurrence and no azeotrope
according to (7). Similarly, the continuation of the diagonal line
through the upper right (open-ended) field divides the regions of
positive azeotrope occurrence and no azeotrope according to (5).
The horizontal axis at γ = 1 represents ideal (Raoult’s law)
behaviour, and forms one border of each “no azeotrope” field.
The vertical axis at a pressure ratio = 1 represents the Bancroft
rule, and forms one border of each azeotrope field. In a model
sense, Figure 2 indicates the extent of nonideality, in terms of γ∞
lvc
or γ∞
mvc that can be tolerated to avoid the occurrence of an
azeotrope for a given ratio of vapour pressures.
In Figure 2, the indefinite extension of criterion (5) for a
positive azeotrope can be deceiving. Assumption (A1) notwith-
standing, increasing nonideality, represented by increasing γ∞
lvc,
leads eventually to a miscibility gap in the liquid phase. This
cannot be predicted by criterion (5), and, if suspected, must be
tested independently by a stability analysis. This is facilitated if
TB1
C'
D'
X1
G'
F'
T
P CONSTANTP CONSTANTP CONST
TB2
0 1
Figure 3. Schematic representation of T(x1) at fixed P for various
situations:
D′ minimum-boiling azeotrope
C′ incipient formation of minimum-boiling azeotrope
G′ maximum-boiling azeotrope
F′ incipient formation of maximum-boiling azeotrope
VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 671
minimum-boiling and maximum-boiling azeotropes, respec-
tively. TB1 and TB2 are the boiling points of components 1 and 2,
respectively, at P. The more volatile component (mvc) has the
lower boiling point: TB1 < TB2.
Criteria in Terms of Activity Coefficients at Infinite
Dilution
The activity coefficient intercepts at x1 = 0 and x1 = 1, γ1
∞ and
γ2
∞, respectively, again represent the activity coefficients at
infinite dilution. Our goal is to obtain criteria for the occurrence
of azeotropes in terms of these quantities.
From Equation (1), which represents P(T,x1), with dP = 0, we
form the derivative
∂
∂
= −
+
∂
∂
− +
∂
∂
∂
T
x
p x p
x
p x p
x
x p1
1 1 1 1 1
1
1
2 2 2 2 2
2
1
1 1 1
γ γ
γ
γ γ
γ
γ
* *
ln
* *
ln
*
lln *
*
ln *γ
γ γ
γ
γ1
1 1
1
2 2 2
2
2 2
2
∂
+ +
∂
∂
+
T
x
dp
dT
x p
T
x
dp
dT
(9)
= 0 (at an azeotrope) (10)
where (Smith and Missen, 1991)
∂
∂
= −
l
T
h
RT
i i
E
2
nγ
(11)
where
–
hi
E is the excess partial molar enthalpy of species i, and R
is the gas constant.
Corresponding to the isothermal case, consider first incipient
formation of a minimum-boiling azeotrope as x1 → 1 at C′ in
Figure 3. Then, x2 → 0, T → TB1, γ1 → 1, γ2 → γ2
∞ (TB1),
–
hi
E →
0, and Equations (9) and (10) reduce to
−
−
=
∞
p T T p T
dp
dT
B1 B1 2 B1 2 1
1
*( ) ( ) *( )
*
γ
0 (12)
From Equation (12), since dp1*/dT ≠ 0, we obtain the following
criterion for incipient formation of a minimum-boiling azeotrope
with component 1 as the mvc:
γ2 1
1 1
2 1 2 1
∞
= =( )
*( )
*( ) *( )
T
p T
p T
P
p T
B
B
B B
(13)
If component 2 is arbitrarily designated as the mvc, then the
criterion for incipient formation becomes
γ1 2
2 B2
1 B2 1 B2
∞
= =( )
*( )
*( ) *( )
T
p T
p T
P
p T
B (14)
Equations (13) and (14) can be combined into one criterion by
using designations mvc and lvc instead of 1 and 2:
Table 4. Examples of use of criterion (18) for maximum-boiling azeotropes
System P/ TB1 (P)/ TB2 (P)/ mvc gE γ∞
mvc P
kPa ºC ºC corr.a (TBlvc) P*mvc(TBlvc) azeo ref.
1-butanol (1) 101.3 117.7 77.1 (2) VL 0.44 0.31 N (a)
+ 101.3 117.7 77.1 (2) W 0.40 0.31 N (a)
1-butylamine (2) 101.3 117.7 77.1 (2) NRTL 0.43 0.31 N (a)
2-butanol (1) 101.3 99.6 77.1 (2) VL 0.41 0.51 Y (b)
+ 101.3 99.6 77.1 (2) W 0.35 0.51 Y (b)
1-butylamine (2) 101.3 99.6 77.1 (2) NRTL 0.42 0.51 Y (b)
CH3OH (1) + diethylamine (2) 97.3 63.5 53.8 (2) M3 0.37 0.72 Y (c)
CH3OH (1) + 1-butylamine (2) 97.3 63.5 76.6 (1) M2 0.79 0.61 N (c)
(a) Dominguez et al. (1997); (b) Dominguez et al. (2002); (c) Nakanishi et al. (1967)
aVL (van Laar); W (Wilson); M2, M3 (two-constant, three-constant Margules)
Table 3. Examples of use of criterion (16) for minimum-boiling azeotropes
System P/ TB1 (P)/ TB2 (P)/ mvc gE γ∞
lvc P azeo ref.
kPa ºC ºC corr.a (TBmvc) P*lvc(TBmvc)
1-butanol (1) 101.3 117.7 68.8 (2) VL 8.6 7.9 Y (a)
+ 101.3 117.7 68.8 (2) W 12.4 7.9 Y (a)
n-hexane (2) 101.3 117.7 68.8 (2) NRTL 6.7 7.9 Nb (a)
2-butanol (1) 101.3 99.6 68.8 (2) VL 6.8 3.6 Y (b)
+ 101.3 99.6 68.8 (2) W 8.3 3.6 Y (b)
n-hexane (2) 101.3 99.6 68.8 (2) NRTL 5.8 3.6 Y (b)
n-hexane (1) 101.3 68.8 77.1 (1) VL 1.83 1.36 Y (a)
+ 101.3 68.8 77.1 (1) W 1.83 1.36 Y (a)
1-butylamine (2) 101.3 68.8 77.1 (1) NRTL 1.82 1.36 Y (a)
CH3OH (1) + triethylamine (2) 97.3 63.5 88.3 (1) M2 2.09 2.37 N (c)
C3H7OH (1) + H2O (2) 101.3 87.8 100.0 (1) VL 3.10 1.57 Y (d)
(a) Dominguez et al. (1997); (b) Dominguez et al. (2002); (c) Nakanishi et al. (1967); (d) Carlson and Colburn (1942)
aVL (van Laar); W (Wilson); M2 (two-constant Margules)
bsee text
672 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005
γlvc B mvc
lvc B mvc
T
P
p T
∞
=( )
( )*
criterion for
incipient formation of
mminimum boiling azeotrope−










(15)
Since the actual occurrence of a minimum-boiling azeotrope,
as in curve D′ in Figure 3, results from a greater positive
departure from ideality (that is, a greater value of γ∞
lvc) than
represented in Equation (15), a criterion for this is
γlvc B mvc
lvc B mvc
T
P
p T
∞
≥( )
( )*
criterion for
minimum - boiling
azeotrrope










(16)
If γ∞
lvc(TBmvc) < P/p*lvc(TBmvc), there is no azeotrope.
Next, consider incipient formation of a maximum-boiling
azeotrope as x1 → 0 at F′ in Figure 3. Or, conversely, if component
2 is arbitrarily designated as the mvc, incipient formation occurs
as x2 → 0 (not shown in Figure 3). In either case, it occurs as
xmvc → 0. Arguments similar to those above for a minimum-
boiling azeotrope lead to the following criterion for incipient
formation:
γ∞
mvc B lvc
mvc B lvc
T
P
p T
( )
( )*
=
criterion for incipient
formation off
maximum - boiling azeotrope










(17)
Since the actual occurrence of a maximum-boiling azeotrope, as
in curve G′ in Figure 3, results from a greater negative departure
from ideality (that is, a smaller value of γ∞
mvc)than represented in
Equation (17), a criterion for this is
γmvc B lvc
mvc B lvc
T
P
p T
∞
≤( )
( )*
criterion for
maximum - boiling
azeotrrope










(18)
If γ∞
mvc(TBlvc) > P/p*mvc(TBlvc), there is no azeotrope.
Figure 4 shows the results expressed by criteria (16) and (18)
in graphical form. In the lower left (closed) field, the diagonal
line representing incipient azeotrope formation divides the
regions of maximum-boiling azeotrope occurrence and no
azeotrope according to (18). Similarly, the continuation of the
diagonal line through the upper right (open-ended) field divides
the regions of minimum-boiling azeotrope occurrence and no
azeotrope according to (16). Figure 4 is similar to Figure 2, but
the variables and azeotrope fields have different significance.
In the use of Figure 4 or criterion (16) for a minimum-boiling
azeotrope, the possible occurrence of a liquid-phase miscibility
gap for sufficiently large values of γ∞
lvc(TBmvc) must be consid-
ered, as discussed above for a positive azeotrope. Criterion (16)
cannot predict this.
In the use of criteria (16) and (18), values of γ∞
lvc(TBmvc) and
γ∞
lvc(TBlvc) are the ones naturally obtained from experimental
vapour-liquid equilibrium data at P. If, however, γ∞
lvc and γ∞
mvc are
obtained otherwise, for example, from gE(T′), their values,
γ∞
lvc(T′) and γ∞
lvc(T′) must be adjusted to give the values required
for criteria (16) and (18). This temperature adjustment is done
by means of Equation (11). Thus, for γ∞
lvc
ln
γ
γ
lvc B mvc
lvc
T
T
lvc
ET
T
h T
T
dT
B mvc
∞
∞ ′
∞
′








= − ∫
( )
( )
( )1
R 2 (19)
and for γ∞
mvc
ln
γmvc B lvc
mvc
T
T
mvc
ET
T
h T
T
dT
B lvc
∞
∞ ′
∞
′








= − ∫
( )
( )
( )
³
1
R 2 (20)
where
–
hlvc
E∞
and
–
hE∞
mvc are the excess partial molar enthalpies at
infinite dilution of lvc and mvc, respectively. The excess enthalpy
(heat of mixing) compilations of Christensen et al. (1982, 1988)
and of Christensen et al. (1984) and Gmehling and Holderbaum
(1989,1991) are useful for this purpose.
Criteria in Terms of Parameters of gE Correlations
As noted for isothermal systems, since γi
∞ can be obtained from
gE correlations, the criteria for the occurrence of azeotropes in
isobaric systems can be applied using the parameters of the gE
correlations, as given in Table 1. The last three of these are used
in the examples of applications of the criteria in the following
section.
Examples of Use of Criteria
Tables 3 and 4 show randomly selected examples of the use of
criteria (16) and (18), respectively, to indicate presence or
absence of azeotropes. In these tables, vapour pressures required
were calculated from Antoine constants provided by Boublík et
al. (1984) for the butanols, and by Nakanishi et al. (1967) for
methanol and the amines.
In Table 3, the first nine entries for three systems come from
the work of Dominguez et al. (1997, 2002) for positive deviations
from ideality for systems involving butanols, n-hexane, and
1-butylamine (column 1) at 101.3 kPa (column 2). The boiling
points (TB) of the components are listed in columns 3 and 4. The
MINIMUM - BOILING
AZEOTROPE
NO
AZEOTROPE
P CONSTANT
γlvcγlvcγ∞(TBmvc)
P/plvc*(TBmvc)
MAXIMUM -
BOILING
AZEOTROPE
NO
AZEOTROPE
γmvγmvγc∞(TBlvc)
P/pmvc (TBlvc)0 1P/p0 1P/p *0 1*(T0 1(TBlvc
0 1
Blvc)0 1)
1
Figure 4. Regions of occurrence of minimum-boiling and maximum-
boiling azeotropes in binary systems at fixed P, according to criteria
(16) and (18).
(note change of variables in lower left and upper right fields)
VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 673
mvc is indicated next, followed, in order, by the gE (or γ) correla-
tion (corr.) used to fit experimental equilibrium data, the value
of γ∞
lvc at TBmvc, the value of the ratio P/p*lvc(TBmvc), and the
conclusion as to whether an azeotrope occurs. For each of these
three systems (nine entries), the authors fitted the van Laar
(VL), Wilson (W), and NRTL equations, among others, to the
experimental data, and provided resulting values of γ∞
lvc(TBmvc)
In each case except one, the existence of a minimum-boiling
azeotrope is indicated, in agreement with experimental results.
The one apparent exception is for the third entry. In this case,
however, the NRTL estimate of γ∞
lvc(TBmvc) (together with two
others not listed here) is low in comparison with the estimates
from the van Laar and Wilson equations (graphical extrapolation
indicates a value ≥ 9, in line with these last two). (This example
shows that the experimental data from VLE measurements
across the composition range do not always provide for a good
statistical fit at the extremes for γ1
∞ and γ2
∞ for a given correla-
tion.) The last two entries (Nakanishi et al., 1967; Carlson and
Colburn, 1942) similarly illustrate situations in which no
azeotrope and an azeotrope, respectively, occur.
In Table 4, the first six entries for two systems come from the
work of Dominguez et al. (1997, 2002) for negative deviations
from ideality. For the system 1-butanol (1) + 1-butylamine (2),
the conclusion is that there is no azeotrope. For the system 2-
butanol (1) + 1-butylamine (2), the conclusion is the opposite
– there is a maximum-boiling azeotrope. Both of these conclu-
sions agree with the experimental result. The last two entries,
from the work of Nakanishi et al. (1967), also provide one
example in which an azeotrope occurs and one in which there
is no azeotrope.
CONCLUSION
Any method that generates values of activity coefficients at
infinite dilution, γ1
∞ and γ2
∞, in a binary system can be used to
determine whether an azeotrope exists in the system at a
specified (constant) T, in accordance with Figure 2 (based on
criteria (5) and (7)); or in the system at a specified (constant) P
in accordance with Figure 4 (based on criteria (16) and (18)).
(For isothermal systems, the criteria are essentially the same as
those given by Brandini (1974).) In a model sense, Figure 2 or
Figure 4 shows the extent of nonideality that can be tolerated to
avoid an azeotrope for a given pressure ratio; the greater the
ratio, the greater the extent of nonideality allowable, consistent
with qualitative considerations.
Gmehling et al. (1994) have described various experimental
methods for determining γ∞, including the use of gas-liquid
chromatography and ebulliometry (as well as from VLE measure-
ments). Malanowski and Anderko (1992) note these and
describe two estimation methods. Applications should prove
useful for screening for various purposes.
ACKNOWLEDGEMENT
Financial assistance has been received from the Natural Sciences
and Engineering Research Council of Canada.
NOMENCATURE
A,B parameters in Van Laar equation
gE excess Gibbs function (J mol-1)
gk parameter in Redlich-Kister equation (8)
–
hi
E excess partial molar enthalpy of component i (J mol-1)
p* saturation vapour pressure (kPa)
P pressure (kPa)
R gas constant, 8.3145 (J mol-1 K-1)
T temperature (K or °C)
y mole fraction in vapour phase
x mole fraction in liquid phase
Greek Symbols
γ activity coefficient
Λ12, Λ21 parameters in Wilson equation
τ12, τ21 parameters in NRTL equation
Superscripts
∞ at infinite dilution
Subscripts
B boiling point
i component i
lvc less volatile component
mvc more volatile component
1, 2 component 1, 2
REFERENCES
Boublík, T., V. Fried and E. Hála, “The Vapour Pressures of
Pure Substances,” 2nd ed., Elsevier, Amsterdam, 1984.
Brandini, V., “Use of Infinite-Dilution Activity Coefficients for
Predicting Azeotrope Formation at Constant Temperature and
Partial Miscibility in Binary Liquid Mixtures,” Ind. Eng.
Chem. Fundam., 13, 154– 156 (1974).
Carlson, H. C. and A. P. Colburn, “Vapor-Liquid Equilibria of
Nonideal Solutions,” Ind. Eng. Chem., 34, 581– 589 (1942).
Christensen, C., J. Gmehling, P. Rasmussen and U. Weidlich,
“Heats of Mixing Data Collection,” Chemistry Data Series,
Vol. III, Parts 1 and 2, DECHEMA, Frankfurt am Main, FRG
(1984).
Christensen, J. J., R. W. Hanks and R. M. Izatt, “Handbook of
Heats of Mixing,” Wiley, New York, NY (1982).
Christensen, J. J., R. L. Rowley and R. M. Izatt, “Handbook of
Heats of Mixing, Supplementary Volume,” Wiley, New York,
NY (1988).
Dominguez, M., A. M. Mainar, H. Artigas, F. M. Royo and J. S.
Vrieta, “Isobaric VLE Data of the Ternary System (1-Butanol
+ n-Hexane + 1-Butylamine) and the Three Constituent
Binary Mixtures at 101.3 kPa,” J. Chem. Eng. Japan, 30,
484– 490 (1997).
Dominguez, M., S. Martin, H. Artigas, M. C. López and F. M.
Royo, “Isobaric Vapor-Liquid Equilibrium for the Binary
Mixtures (2-Butanol + n-Hexane) and 2-Butanol +
1-Butylamine) and for the Ternary System (2-Butanol +
n-Hexane + 1-Butylamine) at 101.3 kPa,” J. Chem. Eng.
Data, 47, 405– 410 (2002).
Eng, R. and S. I. Sandler, “Vapor-Liquid Equilibria for Three
Aldehyde/Hydrocarbon Mixtures,” J. Chem. Eng. Data, 29,
156– 161 (1984).
Fredenslund, Aa., R. L. Jones and J. M. Prausnitz, “Group-
Contribution Estimation of Activity Coefficients in Nonideal
Liquid Mixtures,” AIChE J., 21, 1086– 1099 (1975).
Gaw, W. J. and F. L. Swinton, “Thermodynamic Properties of
Binary Systems Containing Hexafluorobenzene,” Trans.
Faraday Soc., 64, 2023– 2034 (1968).
Gmehling, J. and T. Holderbaum, “Heats of Mixing Data
Collection,” Chemistry Data Series, Vol. III, Part 3 (1989),
Part 4 (1991), DECHEMA, Frankfurt am Main, FRG.
674 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005
Gmehling, J., J. Menke and M. Schiller, “Activity Coefficients at
Infinite Dilution,” Part 3, DECHEMA, Frankfurt am Main,
FRG (1994), pp. XIV– XXIII.
Hildebrand, J. H., J. M. Prausnitz and R. L. Scott, “Regular and
Related Solutions,” Prentice-Hall, Englewood Cliffs, NJ
(1970).
King, M. B., “Phase Equilibrium in Mixtures,” Pergamon,
Oxford (1969), p. 373.
Malanowski, S. and A. Anderko, “Modelling Phase Equilibria,”
Wiley, New York, NY (1992), pp. 133– 136.
Nakanishi, K., H. Shirai and T. Minamiyama, “Vapor-Liquid
Equilibria of Binary Systems Containing Alcohols,” J. Chem.
Eng. Data, 12, 591– 594 (1967).
Prigogine, I. and R. Defay, “Chemical Thermodynamics,” transl.
D. H. Everett, Longmans, London (1954), pp. 464– 465.
Redlich, O. and A. T. Kister, “Algebraic Representation of
Thermodynamic Properties and the Classification of
Solutions,” Ind. Eng. Chem., 40, 345– 348 (1948).
Renon, H. and J. M. Prausnitz, “Local Compositions in
Thermodynamic Excess Functions for Liquid Mixtures,”
AIChE J., 14, 135– 144 (1968).
Sandler, S. I., “Chemical and Engineering Thermodynamics,”
3rd ed., Wiley, New York, NY (1999), p. 291.
Senra, A. M. P., I. M. A. Fonseca, A. G. M. Ferreira and L. Q.
Lobo, “Vapour-liquid equilibria of {xCH3F + (1 – x)HCl} at
temperatures of 159.01 K and 182.33 K,” J. Chem.
Thermodynamics, 34, 1557– 1566 (2002).
Smith, W. R. and R. W. Missen, “Chemical Reaction
Equilibrium Analysis,” Wiley-Interscience, New York, NY
(1982); Krieger, Malabar, FL (1991), pp. 43, 155– 156.
Wilson, G. M., “Vapor-Liquid Equilibrium. XI. A New
Expression for the Excess Free Energy of Mixing,” J. Am.
Chem. Soc., 86, 127– 130 (1964).
Zielkiewicz, J., “(Vapour + liquid) equilibrium in (N,N-
dimethylacetamide + methanol + water) at the temperature
313.15 K,” J. Chem.Thermodynamics, 35, 1993– 2001
(2003).
Manuscript received February 17, 2005; revised manuscript received
June 22, 2005; accepted for publication July 27, 2005.

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On criteria for occurence of azeotropes in isothermal and isobraric binary systems

  • 1. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 667 F rom the point of view of separation processes involving vapour-liquid systems (e.g. fractional distillation), it is important to know if an azeotrope exists in a particular system. If complete experimental vapour-liquid equilibrium data are available, this information is known explicitly. If, however, the data are embedded in a correlation for liquid-phase activity coefficients (γi) (or, equivalently, for the excess molar Gibbs function (gE)), or if the data come from a model for γi or gE, this information is only known implicitly. Models may include a solution model such as a regular solution (Hildebrand et al., 1970), or a group-contribution method based on molecular structure, such as UNIFAC (Fredenslund et al., 1975), for use when no experimental or model parameters are available. The possible existence of an azeotrope may be considered for a system either at constant temperature (T) or at constant pressure (P). The former may be more important from a theoretical On Criteria for Occurrence of Azeotropes in Isothermal and Isobaric Binary Systems Ronald W. Missen Department of Chemical Engineering and Applied Chemistry, University of Toronto, 200 College Street, Toronto, ON, Canada M5S 3E5 point of view, but since separation processes involving vapour- liquid systems operate essentially at constant pressure, this situation is more important from this practical point of view. Attention has been paid in the literature to the isothermal case, but we address both situations here. These considerations lead to the question of obtaining criteria for the occurrence of azeotropes in general. Some guidance in the literature for isothermal systems comes from the “Bancroft rule” (King, 1969) and from criteria developed by Prigogine and Defay (1954) for regular solutions. The former, which states, in effect, that if two species have the same saturation vapour pressure p* (at T′, say), then there is an azeotrope at T′, is self-evident, since any departure from ideality in such a case leads to an azeotrope. Comprehensive criteria for isothermal systems are given by Brandini (1974) in terms of activity coeffi- cients at infinite dilution (γ∞). We include here a different Criteria are developed for the occurrence of azeotropes in binary nonelectrolyte systems for both isothermal and isobaric situations in terms of liquid-phase activity coefficients at infinite dilution (γ∞). In the case of isothermal systems at temperature T, for a positive azeotrope, γ∞ lvc ≥ p*mvc/ p*lvc, where lvc refers to the less volatile component, mvc to the more volatile component, and p* to saturation vapour pressure at T; for a negative azeotrope, γ∞ mvc ≤ p*lvc/p*mvc. In the case of isobaric systems at pressure P, for a minimum-boiling azeotrope, γ∞ lvc(TBmvc) ≥ P/p*lvc(TBmvc), where TB refers to the boiling point at P; for a maximum-boiling azeotrope, γ∞ mvc(TBlvc) ≤ P/p*mvc(TBlvc). The criteria are also given in terms of the parameters of selected correlations for the excess molar Gibbs function (gE). Examples of the use of the criteria are provided. Various methods that generate values of γ∞ can be used in conjunction with the criteria, for example, in screening procedures. Des critères basés sur les coefficients d’activité en phase liquide pour une dilution infinie (γ∞) ont été mis au point pour prédire l’apparition des azéotropes dans les systèmes non électrolytiques binaires dans des conditions isothermes et isobares. Dans le cas des systèmes isothermes à la température T, pour une azéotrope positive, γ∞ lvc ≥ p*mvc/p*lvc , où lvc réfère au composant moins volatil, mvc au composant plus volatil et p* à la pression de vapeur saturante à T; pour une azéotrope négative, γ∞ mvc ≤ p*lvc/p*mvc . Dans le cas de systèmes isobares à la pression P, pour une azéotrope d’ébullition minimum, γ∞ lvc(TBmvc) ≥ P/p*lvc(TBmvc) , où TB réfère au point d’ébullition à P; pour une azéotrope d’ébullition maximum, γ∞ mvc(TBlvc) ≤ P/p*mvc(TBlvc). On donne également les critères pour les paramètres de corrélations sélectionnées pour la fonction molaire d’excès de Gibbs (gE). Des exemples d’utilisation des critères sont donnés. Diverses méthodes donnant des valeurs de γ∞ peuvent être utilisées avec ces critères, par exemple dans les méthodes de tamisage. Keywords: isothermal azeotrope criteria; isobaric azeotrope criteria; binary systems
  • 2. 668 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 development of these together with a graphical interpretation. We also give a corresponding new treatment for isobaric systems. Examples to illustrate application of the criteria developed are provided. ISOTHERMAL CRITERIA FOR OCCURRENCE OF AZEOTROPES Qualitative Considerations Figure 1 illustrates schematically P(x1) relationships for a fixed- T plane intersecting a P-T-x1 saturated-liquid surface, where x1is mole fraction of component 1. It shows P(x1) for various degrees of departure from Raoult’s law (A, for an ideal system) in both positive and negative directions. Curves B, C, and D show increasing degrees of positive deviation: B for no azeotrope and D for a positive azeotrope (at the maximum, where x1 = y1 (not shown, the mole fraction in the coexisting vapour phase, whether actual or virtual)); C represents the boundary between these two types of behaviour, as the indication of incipient occurrence of a positive azeotrope with respect to increasing extent of nonideality. (For still greater positive deviation than in D, single liquid-phase stability eventually breaks down, and increasing degrees of partial miscibility set in, leading ultimately to complete immiscibility, the greatest degree of positive deviation from Raoult’s law.). Correspondingly, curves E and G (negative azeotrope at the minimum) illustrate increasing degrees of negative deviation, with F representing incipient negative azeotrope formation with respect to increasing nonide- ality. (There is no corresponding single-phase stability breakdown in this direction.) Assumptions We assume the following: (A1) The system is a binary nonelectrolyte system of components 1 and 2 that are completely miscible in the liquid phase. (A2) Information for γi or gE is for a saturated liquid phase, actually or potentially in equilibrium with a vapour phase. (A3) The γi or gE data refer to Raoult’s law ideality; that is, γi → 1 as xi → 1. (A4) The γi or gE data refer either to isothermal or to isobaric (case to follow) conditions. (A5) The dependence of γi or gE on P is negligible, but the dependence on T must be allowed for. (A6) The pressure is sufficiently low that the Poynting correc- tion (Sandler, 1999) is equal to one. Criteria in Terms of Activity Coefficients at Infinite Dilution The activity coefficients at infinite dilution are designated γ1 ∞ (at x1 = 0) and γ2 ∞ (at x2 = 0) for components 1 and 2, respectively. Our goal is to obtain criteria in terms of these quantities. In the absence of any fugacity corrections at low pressure (assumption (A6)), the total pressure at T is given by P = γ1(T,x1)x1p1* (T) + γ2(T,x1)x2p2* (T) (1) From Equation (1), on differentiation at constant T, we obtain ∂ ∂ = + ∂ ∂ − + ∂ ∂ P x p x p x p x p x1 1 1 1 1 1 1 2 2 2 2 2 1 * * * *γ γ γ γ (2) = 0 (at an azeotrope) (3) Consider incipient positive azeotrope formation with component 1 as the more volatile component (p1* > p2* at T). This occurs as x1 → 1 at C in Figure 1. In such a case, x2 → 0, γ1 → 1, γ2 → γ2 ∞, x1p1* ∂γ1/∂x1 = x1p1*γ1 ∂lnγ1/∂x1 → p1* ∂lnγ1/∂x1 → 0, and x2p2* ∂γ2/∂x1 → 0. From Equations (2) and (3), we conclude that a criterion for incipient formation of a positive azeotrope is γ2 ∞ = p1*/p2* ( > 1). Instead of component 1 being the more volatile component, an arbitrary designation may have component 2 in this role. The situation depicted in Figure 1 is then reversed (although not shown). Incipient azeotrope formation occurs as x1 → 0 or x2 → 1, in which case, γ1 → γ1 ∞ and γ2 → 1. From Equations (2) and (3), the criterion for an incipient azeotrope is γ1 ∞ = p2*/p1* ( > 1). In either case, if the component designa- tions are changed to more volatile component (mvc) and less volatile component (lvc), instead of components 1 and 2, the criterion for an incipient positive azeotrope can be written as one equation: p*2 P X1 F T CONSTANTT CONSTANTT CONST p*1 0 1 G E A B D C Figure 1. Schematic representation of P(x1) at fixed T showing positive and negative deviations from Raoult’s law. A Raoult’s law B Positive deviation, no azeotrope C Positive deviation, incipient positive azeotrope D Positive deviation, positive azeotrope at maximum E Negative deviation, no azeotrope F Negative deviation, incipient negative azeotrope G Negative deviation, negative azeotrope at minimum
  • 3. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 669 γlvc mvc lvc p p ∞ = * * criterion for incipient positive azeotrope formaation           (4) As the degree of positive departure from ideality increases, in the sense of curve D in Figure 1, azeotrope formation occurs. In every case, γ∞ lvc increases relative to the value in Equation (4). Thus, a criterion for the existence of a positive azeotrope is γlvc mvc lvc p p ∞ ≥       * * criterion for positive azeotrope (5) If γ∞ lvc < p*mvc/p*lvc, no azeotrope exists. Similarly, consider incipient negative azeotrope formation as x1 → 0 at F in Figure 1, with component 1 as the mvc (or the reverse as x1 → 1, if component 2 is the mvc). Considerations based on Equations (2) and (3) corresponding to those above lead to the following criterion for an incipient negative azeotrope: γmvc lvc mvc p p ∞ = * * criterion for incipient negative azeotrope formaation           (6) As the degree of negative departure from ideality increases in the sense of curve G in Figure 1, azeotrope formation occurs. In every case, γ∞ mvc decreases relative to the value in Equation (6). Table 1. Expressions for γ1 ∞ and γ2 ∞ from selected gE equations gE equation γ1 ∞ γ2 ∞ Redlich-Kister, Equation (8) Van Laar Wilson NRTL exp ( ) exp exp exp 1 1 1 1 0 0 1 RT g RT g A B k K k k k K k = = ∑ ∑−                 Λ 22 21 21 12 21 12 12 12 1 1 1 1exp( ) exp( ) exp exp - exp − − + ( )  Λ Λ Λ τ τ α τ τ 2 2 21 12 2 1+ ( ) τ α τexp - Table 2. Examples of use of criteria (5) and (7) for positive and negative azeotropes System T/ºC mvc dev. gE criterion (5) criterion (7) ref. corr.a γ∞ lvc P*mvc pos. γ∞ lvc P*lvc neg. P*lvc azeo. P*mvc azeo. n-C5H12 (1) 40 (1) + VL 4.07 1.52 Y ← N/A → (a) + 40 (1) + W 4.02 1.52 Y ← N/A → (a) propionaldehyde (2) 40 (1) + NRTL 3.85 1.52 Y ← N/A → (a) n-butyraldehyde (1) 45 (1) + VL 3.05 2.30 Y ← N/A → (a) + 45 (1) + W 3.12 2.30 Y ← N/A → (a) n-C7H16 (2) 45 (1) + NRTL 3.04 2.30 Y ← N/A → (a) isobutyraldehyde (1) 45 (1) + VL 2.72 3.41 N ← N/A → (a) + 45 (1) + W 2.76 3.41 N ← N/A → (a) n-C7H16 (2) 45 (1) + NRTL 2.73 3.41 N ← N/A → (a) DMA (1) + CH3OH (2) 40 (2) – RK ← N/A → 0.406 0.019 N (b) CH3F (1) + HCl (2) –90.8 (2) – RK ← N/A → 0.267 0.668 Y (c) C6F6(1) + C6H6(2) 60 (2) ± RK 1.20 1.04 Y 0.898 0.979 Y (d) (a) Eng and Sandler (1984); (b) Zielkiewicz (2003); (c) Senra et al. (2002); (d) Gaw and Swinton (1968) aRK (Redlich-Kister); VL (van Laar); W (Wilson) Figure 2. Regions of occurrence of positive and negative azeotropes in binary systems at fixed T according to criteria (5) and (7). (note change of variables in lower left and upper right fields)
  • 4. 670 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 a gE expression is available, but is outside the scope of this paper. As noted above, Brandini (1974) presents essentially the same criteria as (5) and (7) for isothermal systems, but expresses them in terms of components 1 and 2, and provides no graphical interpretation corresponding to Figure 2. The designation as components 1 and 2 requires four statements rather than the two in criteria (5) and (7). Criteria in Terms of Parameters of gE Correlations As noted above, the information required to apply the criteria for existence of azeotropes may be embedded in equations for the excess molar Gibbs function (gE). In Table 1, expressions are provided for γ1 ∞ and γ2 ∞ from four such equations: Redlich-Kister (1948), van Laar (Carlson and Colburn, 1942), Wilson (1964), and NRTL (Renon and Prausnitz, 1968). The van Laar results are readily apparent from the Carlson-Colburn forms of the activity coefficients. The Redlich-Kister equation is an expansion in terms of composition that can be represented by: g x x g x xE k K k k = − = ∑1 2 0 1 2( ) (8) The results in Table 1 can be derived from Equation (8). Examples of Use of Criteria Table 2 shows randomly selected examples of the use of criteria (5) and (7) to indicate presence or absence of an azeotrope. The first nine entries come from the work of Eng and Sandler (1984) on three aldehyde systems (column 1) at the T given in column 2. In each of these cases, component (1) is the more volatile component (mvc), and the deviation from ideality is positive. For each system, they compared the fitting of several gE correla- tions (corr.), including van Laar (VL), Wilson (W) and NRTL equations. The use of criterion (5) is applicable here, and columns 6 to 8 give the results – azeotropes in the first two systems, but no azeotrope in the third, as consistently shown by each gE correlation. The next two entries (Zielkiewicz, 2003; Senra et al., 2002) refer to negative deviation and the use of criterion (7) in conjunction with the RK equation. The last entry in Table 2 is for the rare case of a double azeotrope, in the hexafluorobenzene + benzene system at 60°C (Gaw and Swinton, 1968). This is the situation in which both a positive and a negative azeotrope occur at a particular T. In this case, both criteria (5) and (7) are required, and they indicate the existence of the two azeotropes observed. In all 12 cases in Table 2, the results of application of criterion (5) or (7) agree with experimental results. ISOBARIC CRITERIA FOR OCCURRENCE OF AZEOTROPES Qualitative Considerations Figure 3 illustrates schematically T(x1) relationships for fixed-P planes intersecting a P-T-x1 saturated-liquid surface. The behaviour labelled by C′, D′, F′, and G′ corresponds to that labelled by C, D, F, and G in Figure 1. Thus, D′ represents a minimum-boiling azeotrope at fixed P, which corresponds to a positive azeotrope at fixed T. Similarly, G′ represents a maximum- boiling azeotrope, which corresponds to a negative azeotrope. The behaviour indicated at C′ and F′ represents incipient formation (with respect to increasing extent of nonideality) of Thus, a criterion for the existence of a negative azeotrope is γmvc lvc mvc p p ∞ ≤       * * criterion for negative azeotrope (7) If γ∞ mvc > p*lvc/p*mvc, no azeotrope exists. Figure 2 shows the results expressed by the criteria of (5) and (7) in graphical form. In the lower left (closed) field, the diagonal line representing incipient azeotrope formation divides the regions of negative azeotrope occurrence and no azeotrope according to (7). Similarly, the continuation of the diagonal line through the upper right (open-ended) field divides the regions of positive azeotrope occurrence and no azeotrope according to (5). The horizontal axis at γ = 1 represents ideal (Raoult’s law) behaviour, and forms one border of each “no azeotrope” field. The vertical axis at a pressure ratio = 1 represents the Bancroft rule, and forms one border of each azeotrope field. In a model sense, Figure 2 indicates the extent of nonideality, in terms of γ∞ lvc or γ∞ mvc that can be tolerated to avoid the occurrence of an azeotrope for a given ratio of vapour pressures. In Figure 2, the indefinite extension of criterion (5) for a positive azeotrope can be deceiving. Assumption (A1) notwith- standing, increasing nonideality, represented by increasing γ∞ lvc, leads eventually to a miscibility gap in the liquid phase. This cannot be predicted by criterion (5), and, if suspected, must be tested independently by a stability analysis. This is facilitated if TB1 C' D' X1 G' F' T P CONSTANTP CONSTANTP CONST TB2 0 1 Figure 3. Schematic representation of T(x1) at fixed P for various situations: D′ minimum-boiling azeotrope C′ incipient formation of minimum-boiling azeotrope G′ maximum-boiling azeotrope F′ incipient formation of maximum-boiling azeotrope
  • 5. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 671 minimum-boiling and maximum-boiling azeotropes, respec- tively. TB1 and TB2 are the boiling points of components 1 and 2, respectively, at P. The more volatile component (mvc) has the lower boiling point: TB1 < TB2. Criteria in Terms of Activity Coefficients at Infinite Dilution The activity coefficient intercepts at x1 = 0 and x1 = 1, γ1 ∞ and γ2 ∞, respectively, again represent the activity coefficients at infinite dilution. Our goal is to obtain criteria for the occurrence of azeotropes in terms of these quantities. From Equation (1), which represents P(T,x1), with dP = 0, we form the derivative ∂ ∂ = − + ∂ ∂ − + ∂ ∂ ∂ T x p x p x p x p x x p1 1 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 γ γ γ γ γ γ γ * * ln * * ln * lln * * ln *γ γ γ γ γ1 1 1 1 2 2 2 2 2 2 2 ∂ + + ∂ ∂ + T x dp dT x p T x dp dT (9) = 0 (at an azeotrope) (10) where (Smith and Missen, 1991) ∂ ∂ = − l T h RT i i E 2 nγ (11) where – hi E is the excess partial molar enthalpy of species i, and R is the gas constant. Corresponding to the isothermal case, consider first incipient formation of a minimum-boiling azeotrope as x1 → 1 at C′ in Figure 3. Then, x2 → 0, T → TB1, γ1 → 1, γ2 → γ2 ∞ (TB1), – hi E → 0, and Equations (9) and (10) reduce to − − = ∞ p T T p T dp dT B1 B1 2 B1 2 1 1 *( ) ( ) *( ) * γ 0 (12) From Equation (12), since dp1*/dT ≠ 0, we obtain the following criterion for incipient formation of a minimum-boiling azeotrope with component 1 as the mvc: γ2 1 1 1 2 1 2 1 ∞ = =( ) *( ) *( ) *( ) T p T p T P p T B B B B (13) If component 2 is arbitrarily designated as the mvc, then the criterion for incipient formation becomes γ1 2 2 B2 1 B2 1 B2 ∞ = =( ) *( ) *( ) *( ) T p T p T P p T B (14) Equations (13) and (14) can be combined into one criterion by using designations mvc and lvc instead of 1 and 2: Table 4. Examples of use of criterion (18) for maximum-boiling azeotropes System P/ TB1 (P)/ TB2 (P)/ mvc gE γ∞ mvc P kPa ºC ºC corr.a (TBlvc) P*mvc(TBlvc) azeo ref. 1-butanol (1) 101.3 117.7 77.1 (2) VL 0.44 0.31 N (a) + 101.3 117.7 77.1 (2) W 0.40 0.31 N (a) 1-butylamine (2) 101.3 117.7 77.1 (2) NRTL 0.43 0.31 N (a) 2-butanol (1) 101.3 99.6 77.1 (2) VL 0.41 0.51 Y (b) + 101.3 99.6 77.1 (2) W 0.35 0.51 Y (b) 1-butylamine (2) 101.3 99.6 77.1 (2) NRTL 0.42 0.51 Y (b) CH3OH (1) + diethylamine (2) 97.3 63.5 53.8 (2) M3 0.37 0.72 Y (c) CH3OH (1) + 1-butylamine (2) 97.3 63.5 76.6 (1) M2 0.79 0.61 N (c) (a) Dominguez et al. (1997); (b) Dominguez et al. (2002); (c) Nakanishi et al. (1967) aVL (van Laar); W (Wilson); M2, M3 (two-constant, three-constant Margules) Table 3. Examples of use of criterion (16) for minimum-boiling azeotropes System P/ TB1 (P)/ TB2 (P)/ mvc gE γ∞ lvc P azeo ref. kPa ºC ºC corr.a (TBmvc) P*lvc(TBmvc) 1-butanol (1) 101.3 117.7 68.8 (2) VL 8.6 7.9 Y (a) + 101.3 117.7 68.8 (2) W 12.4 7.9 Y (a) n-hexane (2) 101.3 117.7 68.8 (2) NRTL 6.7 7.9 Nb (a) 2-butanol (1) 101.3 99.6 68.8 (2) VL 6.8 3.6 Y (b) + 101.3 99.6 68.8 (2) W 8.3 3.6 Y (b) n-hexane (2) 101.3 99.6 68.8 (2) NRTL 5.8 3.6 Y (b) n-hexane (1) 101.3 68.8 77.1 (1) VL 1.83 1.36 Y (a) + 101.3 68.8 77.1 (1) W 1.83 1.36 Y (a) 1-butylamine (2) 101.3 68.8 77.1 (1) NRTL 1.82 1.36 Y (a) CH3OH (1) + triethylamine (2) 97.3 63.5 88.3 (1) M2 2.09 2.37 N (c) C3H7OH (1) + H2O (2) 101.3 87.8 100.0 (1) VL 3.10 1.57 Y (d) (a) Dominguez et al. (1997); (b) Dominguez et al. (2002); (c) Nakanishi et al. (1967); (d) Carlson and Colburn (1942) aVL (van Laar); W (Wilson); M2 (two-constant Margules) bsee text
  • 6. 672 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 γlvc B mvc lvc B mvc T P p T ∞ =( ) ( )* criterion for incipient formation of mminimum boiling azeotrope−           (15) Since the actual occurrence of a minimum-boiling azeotrope, as in curve D′ in Figure 3, results from a greater positive departure from ideality (that is, a greater value of γ∞ lvc) than represented in Equation (15), a criterion for this is γlvc B mvc lvc B mvc T P p T ∞ ≥( ) ( )* criterion for minimum - boiling azeotrrope           (16) If γ∞ lvc(TBmvc) < P/p*lvc(TBmvc), there is no azeotrope. Next, consider incipient formation of a maximum-boiling azeotrope as x1 → 0 at F′ in Figure 3. Or, conversely, if component 2 is arbitrarily designated as the mvc, incipient formation occurs as x2 → 0 (not shown in Figure 3). In either case, it occurs as xmvc → 0. Arguments similar to those above for a minimum- boiling azeotrope lead to the following criterion for incipient formation: γ∞ mvc B lvc mvc B lvc T P p T ( ) ( )* = criterion for incipient formation off maximum - boiling azeotrope           (17) Since the actual occurrence of a maximum-boiling azeotrope, as in curve G′ in Figure 3, results from a greater negative departure from ideality (that is, a smaller value of γ∞ mvc)than represented in Equation (17), a criterion for this is γmvc B lvc mvc B lvc T P p T ∞ ≤( ) ( )* criterion for maximum - boiling azeotrrope           (18) If γ∞ mvc(TBlvc) > P/p*mvc(TBlvc), there is no azeotrope. Figure 4 shows the results expressed by criteria (16) and (18) in graphical form. In the lower left (closed) field, the diagonal line representing incipient azeotrope formation divides the regions of maximum-boiling azeotrope occurrence and no azeotrope according to (18). Similarly, the continuation of the diagonal line through the upper right (open-ended) field divides the regions of minimum-boiling azeotrope occurrence and no azeotrope according to (16). Figure 4 is similar to Figure 2, but the variables and azeotrope fields have different significance. In the use of Figure 4 or criterion (16) for a minimum-boiling azeotrope, the possible occurrence of a liquid-phase miscibility gap for sufficiently large values of γ∞ lvc(TBmvc) must be consid- ered, as discussed above for a positive azeotrope. Criterion (16) cannot predict this. In the use of criteria (16) and (18), values of γ∞ lvc(TBmvc) and γ∞ lvc(TBlvc) are the ones naturally obtained from experimental vapour-liquid equilibrium data at P. If, however, γ∞ lvc and γ∞ mvc are obtained otherwise, for example, from gE(T′), their values, γ∞ lvc(T′) and γ∞ lvc(T′) must be adjusted to give the values required for criteria (16) and (18). This temperature adjustment is done by means of Equation (11). Thus, for γ∞ lvc ln γ γ lvc B mvc lvc T T lvc ET T h T T dT B mvc ∞ ∞ ′ ∞ ′         = − ∫ ( ) ( ) ( )1 R 2 (19) and for γ∞ mvc ln γmvc B lvc mvc T T mvc ET T h T T dT B lvc ∞ ∞ ′ ∞ ′         = − ∫ ( ) ( ) ( ) ³ 1 R 2 (20) where – hlvc E∞ and – hE∞ mvc are the excess partial molar enthalpies at infinite dilution of lvc and mvc, respectively. The excess enthalpy (heat of mixing) compilations of Christensen et al. (1982, 1988) and of Christensen et al. (1984) and Gmehling and Holderbaum (1989,1991) are useful for this purpose. Criteria in Terms of Parameters of gE Correlations As noted for isothermal systems, since γi ∞ can be obtained from gE correlations, the criteria for the occurrence of azeotropes in isobaric systems can be applied using the parameters of the gE correlations, as given in Table 1. The last three of these are used in the examples of applications of the criteria in the following section. Examples of Use of Criteria Tables 3 and 4 show randomly selected examples of the use of criteria (16) and (18), respectively, to indicate presence or absence of azeotropes. In these tables, vapour pressures required were calculated from Antoine constants provided by Boublík et al. (1984) for the butanols, and by Nakanishi et al. (1967) for methanol and the amines. In Table 3, the first nine entries for three systems come from the work of Dominguez et al. (1997, 2002) for positive deviations from ideality for systems involving butanols, n-hexane, and 1-butylamine (column 1) at 101.3 kPa (column 2). The boiling points (TB) of the components are listed in columns 3 and 4. The MINIMUM - BOILING AZEOTROPE NO AZEOTROPE P CONSTANT γlvcγlvcγ∞(TBmvc) P/plvc*(TBmvc) MAXIMUM - BOILING AZEOTROPE NO AZEOTROPE γmvγmvγc∞(TBlvc) P/pmvc (TBlvc)0 1P/p0 1P/p *0 1*(T0 1(TBlvc 0 1 Blvc)0 1) 1 Figure 4. Regions of occurrence of minimum-boiling and maximum- boiling azeotropes in binary systems at fixed P, according to criteria (16) and (18). (note change of variables in lower left and upper right fields)
  • 7. VOLUME 83, AUGUST 2005 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 673 mvc is indicated next, followed, in order, by the gE (or γ) correla- tion (corr.) used to fit experimental equilibrium data, the value of γ∞ lvc at TBmvc, the value of the ratio P/p*lvc(TBmvc), and the conclusion as to whether an azeotrope occurs. For each of these three systems (nine entries), the authors fitted the van Laar (VL), Wilson (W), and NRTL equations, among others, to the experimental data, and provided resulting values of γ∞ lvc(TBmvc) In each case except one, the existence of a minimum-boiling azeotrope is indicated, in agreement with experimental results. The one apparent exception is for the third entry. In this case, however, the NRTL estimate of γ∞ lvc(TBmvc) (together with two others not listed here) is low in comparison with the estimates from the van Laar and Wilson equations (graphical extrapolation indicates a value ≥ 9, in line with these last two). (This example shows that the experimental data from VLE measurements across the composition range do not always provide for a good statistical fit at the extremes for γ1 ∞ and γ2 ∞ for a given correla- tion.) The last two entries (Nakanishi et al., 1967; Carlson and Colburn, 1942) similarly illustrate situations in which no azeotrope and an azeotrope, respectively, occur. In Table 4, the first six entries for two systems come from the work of Dominguez et al. (1997, 2002) for negative deviations from ideality. For the system 1-butanol (1) + 1-butylamine (2), the conclusion is that there is no azeotrope. For the system 2- butanol (1) + 1-butylamine (2), the conclusion is the opposite – there is a maximum-boiling azeotrope. Both of these conclu- sions agree with the experimental result. The last two entries, from the work of Nakanishi et al. (1967), also provide one example in which an azeotrope occurs and one in which there is no azeotrope. CONCLUSION Any method that generates values of activity coefficients at infinite dilution, γ1 ∞ and γ2 ∞, in a binary system can be used to determine whether an azeotrope exists in the system at a specified (constant) T, in accordance with Figure 2 (based on criteria (5) and (7)); or in the system at a specified (constant) P in accordance with Figure 4 (based on criteria (16) and (18)). (For isothermal systems, the criteria are essentially the same as those given by Brandini (1974).) In a model sense, Figure 2 or Figure 4 shows the extent of nonideality that can be tolerated to avoid an azeotrope for a given pressure ratio; the greater the ratio, the greater the extent of nonideality allowable, consistent with qualitative considerations. Gmehling et al. (1994) have described various experimental methods for determining γ∞, including the use of gas-liquid chromatography and ebulliometry (as well as from VLE measure- ments). Malanowski and Anderko (1992) note these and describe two estimation methods. Applications should prove useful for screening for various purposes. ACKNOWLEDGEMENT Financial assistance has been received from the Natural Sciences and Engineering Research Council of Canada. NOMENCATURE A,B parameters in Van Laar equation gE excess Gibbs function (J mol-1) gk parameter in Redlich-Kister equation (8) – hi E excess partial molar enthalpy of component i (J mol-1) p* saturation vapour pressure (kPa) P pressure (kPa) R gas constant, 8.3145 (J mol-1 K-1) T temperature (K or °C) y mole fraction in vapour phase x mole fraction in liquid phase Greek Symbols γ activity coefficient Λ12, Λ21 parameters in Wilson equation τ12, τ21 parameters in NRTL equation Superscripts ∞ at infinite dilution Subscripts B boiling point i component i lvc less volatile component mvc more volatile component 1, 2 component 1, 2 REFERENCES Boublík, T., V. Fried and E. Hála, “The Vapour Pressures of Pure Substances,” 2nd ed., Elsevier, Amsterdam, 1984. Brandini, V., “Use of Infinite-Dilution Activity Coefficients for Predicting Azeotrope Formation at Constant Temperature and Partial Miscibility in Binary Liquid Mixtures,” Ind. Eng. Chem. Fundam., 13, 154– 156 (1974). Carlson, H. C. and A. P. Colburn, “Vapor-Liquid Equilibria of Nonideal Solutions,” Ind. Eng. Chem., 34, 581– 589 (1942). Christensen, C., J. Gmehling, P. Rasmussen and U. Weidlich, “Heats of Mixing Data Collection,” Chemistry Data Series, Vol. III, Parts 1 and 2, DECHEMA, Frankfurt am Main, FRG (1984). Christensen, J. J., R. W. Hanks and R. M. Izatt, “Handbook of Heats of Mixing,” Wiley, New York, NY (1982). Christensen, J. J., R. L. Rowley and R. M. Izatt, “Handbook of Heats of Mixing, Supplementary Volume,” Wiley, New York, NY (1988). Dominguez, M., A. M. Mainar, H. Artigas, F. M. Royo and J. S. Vrieta, “Isobaric VLE Data of the Ternary System (1-Butanol + n-Hexane + 1-Butylamine) and the Three Constituent Binary Mixtures at 101.3 kPa,” J. Chem. Eng. Japan, 30, 484– 490 (1997). Dominguez, M., S. Martin, H. Artigas, M. C. López and F. M. Royo, “Isobaric Vapor-Liquid Equilibrium for the Binary Mixtures (2-Butanol + n-Hexane) and 2-Butanol + 1-Butylamine) and for the Ternary System (2-Butanol + n-Hexane + 1-Butylamine) at 101.3 kPa,” J. Chem. Eng. Data, 47, 405– 410 (2002). Eng, R. and S. I. Sandler, “Vapor-Liquid Equilibria for Three Aldehyde/Hydrocarbon Mixtures,” J. Chem. Eng. Data, 29, 156– 161 (1984). Fredenslund, Aa., R. L. Jones and J. M. Prausnitz, “Group- Contribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures,” AIChE J., 21, 1086– 1099 (1975). Gaw, W. J. and F. L. Swinton, “Thermodynamic Properties of Binary Systems Containing Hexafluorobenzene,” Trans. Faraday Soc., 64, 2023– 2034 (1968). Gmehling, J. and T. Holderbaum, “Heats of Mixing Data Collection,” Chemistry Data Series, Vol. III, Part 3 (1989), Part 4 (1991), DECHEMA, Frankfurt am Main, FRG.
  • 8. 674 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 83, AUGUST 2005 Gmehling, J., J. Menke and M. Schiller, “Activity Coefficients at Infinite Dilution,” Part 3, DECHEMA, Frankfurt am Main, FRG (1994), pp. XIV– XXIII. Hildebrand, J. H., J. M. Prausnitz and R. L. Scott, “Regular and Related Solutions,” Prentice-Hall, Englewood Cliffs, NJ (1970). King, M. B., “Phase Equilibrium in Mixtures,” Pergamon, Oxford (1969), p. 373. Malanowski, S. and A. Anderko, “Modelling Phase Equilibria,” Wiley, New York, NY (1992), pp. 133– 136. Nakanishi, K., H. Shirai and T. Minamiyama, “Vapor-Liquid Equilibria of Binary Systems Containing Alcohols,” J. Chem. Eng. Data, 12, 591– 594 (1967). Prigogine, I. and R. Defay, “Chemical Thermodynamics,” transl. D. H. Everett, Longmans, London (1954), pp. 464– 465. Redlich, O. and A. T. Kister, “Algebraic Representation of Thermodynamic Properties and the Classification of Solutions,” Ind. Eng. Chem., 40, 345– 348 (1948). Renon, H. and J. M. Prausnitz, “Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures,” AIChE J., 14, 135– 144 (1968). Sandler, S. I., “Chemical and Engineering Thermodynamics,” 3rd ed., Wiley, New York, NY (1999), p. 291. Senra, A. M. P., I. M. A. Fonseca, A. G. M. Ferreira and L. Q. Lobo, “Vapour-liquid equilibria of {xCH3F + (1 – x)HCl} at temperatures of 159.01 K and 182.33 K,” J. Chem. Thermodynamics, 34, 1557– 1566 (2002). Smith, W. R. and R. W. Missen, “Chemical Reaction Equilibrium Analysis,” Wiley-Interscience, New York, NY (1982); Krieger, Malabar, FL (1991), pp. 43, 155– 156. Wilson, G. M., “Vapor-Liquid Equilibrium. XI. A New Expression for the Excess Free Energy of Mixing,” J. Am. Chem. Soc., 86, 127– 130 (1964). Zielkiewicz, J., “(Vapour + liquid) equilibrium in (N,N- dimethylacetamide + methanol + water) at the temperature 313.15 K,” J. Chem.Thermodynamics, 35, 1993– 2001 (2003). Manuscript received February 17, 2005; revised manuscript received June 22, 2005; accepted for publication July 27, 2005.