Fluid Phase Equilibria 214 (2003) 67–78
Phase behaviour of binary fluid mixtures: a global phase diagram
solely in terms of pure component properties
Ji-Lin Wang, Richard J. Sadus∗
Centre for Molecular Simulation, Swinburne University of Technology, P.O. Box 218, Hawthorn, Vic. 3122, Australia
Received 28 March 2003; accepted 6 June 2003
Abstract
Calculations of the critical properties of binary mixtures are reported using the Carnahan–Starling–van der Waals
(CSvdW) equation of state in conjunction with the one-fluid model and the Lorentz–Berthelot combining rules for
unlike interactions. The calculations are used to determine the global phase diagram of binary mixtures. A feature of
the calculations is that no adjustable parameters were used in the combining rules. This means that the global phase
diagram can be constructed solely in terms of experimentally measurable quantities such as the ratio of the critical
temperatures and critical volumes of the component molecules. The global phase diagram accounts for almost
all of the known types of fluid phase behaviour, including closed-loop liquid–liquid immiscibility. Comparisons
with experimental data indicate that the global phase diagram is a good qualitative predictor of phase behaviour.
Therefore, the phase behaviour type of a binary mixture can be confidently predicted solely from the ratios of critical
temperatures and volumes of the pure components.
© 2003 Elsevier B.V. All rights reserved.
Keywords: Phase diagram; Critical property; Binary mixture; Fluid
1. Introduction
The fluid phase equilibria of many binary mixtures have been observed experimentally over a wide
range of physical conditions including very high pressures [1]. Classification schemes [1–6] have been
developed based on the critical equilibria behaviour of binary mixtures and increasingly accurate equations
of state [7] have been reported and applied successfully to the prediction of both binary [1,8–12] and
ternary [1,9] mixtures. In addition, the application of molecular simulation [13] to phase equilibria is
expanding continually.
The phase behaviour of the majority of commonly occurring binary mixtures can be broadly classified
by their critical properties in terms of a few basic types [1,2]. It has been demonstrated recently [5,14] that
∗
Corresponding author. Tel.: +61-3-9214-8773; fax: +61-3-9819-0823.
E-mail address: [email protected] (R.J. Sadus).
0378-3812/$ – see front matter © 2003 Elsevier B.V. All rights reserved.
doi:10.1016/S0378-3812(03)00318-2
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J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
almost all the known types of behaviour, including closed-loop immiscibility, can be calculated using the
Carnahan–Starling–van der Waals (CSvdW) [15] or Guggenheim [16] equations of state. The CSvdW and
Guggenheim equations of state model the intermolecular interactions of fluids by combining an accurate
hard-sphere repulsion term with simple van der Waals dispersion interactions in the spirit of the original
van der Waals equation of state. The success of these relatively simple “hardsphere + attractive term”
equations of state to predict most aspects of binary mixture critical phenomena means that they can be
used to generate a so-called global phase diagram [3–5,14].
A global phase diagram is a two-dimensional map, which identifies regions of different phase behaviour in accordance with the properties of the x and y coordinates. Critical transitions of different kinds
generate boundary lines between different types of phase behaviour. Usually, the properties used for the
coordinates of the global phase diagram involve unlike or cross terms resulting from the contribution
of interactions between dissimilar molecules in the mixture. For example, the co-volume of a binary
mixture (bm ) has contributions from the interactions of both like (b11 and b22 ) and unlike interactions
(b12 ). The contribution of like interactions is an experimentally accessible quantity because it is simply
a property of the pure components. In contrast, the contribution of unlike interactions to a property of
a mixture is not a directly observable quantity. Typically, the contributions of unlike interactions are
extracted by analyzing experimental binary mixture measurements relative to pure component data or
they are approximated using combining rules. Except for mixtures of simple molecules, adjustable parameters are required in the combining rules to optimize the agreement of calculations with experimental
data.
In contrast to previous work [2–5,14], our aim is to construct a global phase diagram without using adjustable combining rule parameters. The use of adjustable parameters to construct a global phase
diagrams severely limits the predictive value of the diagram. In most cases, the adjustable parameters can only be obtained by fitting theory to experimental data. Furthermore, global phase diagrams
are often restricted to highly idealized conditions such as hard spheres or components with a specified
size difference. This means that such global phase diagrams are inappropriate for most real mixtures.
We avoid these problems by determining the global phase diagram in terms of experimental properties of the pure components. The result is a global phase diagram that can be used for real binary
mixtures.
2. Theory
The overall phase behaviour of binary mixtures can be conveniently classified in terms of variations
in the critical properties of the binary mixtures. Details of the calculation of the critical properties of
mixtures and the underlying theory are given extensively elsewhere [1,17] and only a brief outline of the
main points is given here. The critical properties of a binary fluid mixture can be obtained by determining
the temperature (T), volume (V) and composition (x), which satisfy the critical conditions,
2 2 ∂ A
− ∂ A
−
2
∂V
∂x
∂V
1
T
T =0
W = 2 (1)
2 A
∂
∂
A
∂x1 ∂V T
∂x12 T,V ∂W
∂V T
X = 2 ∂ A
∂x1 ∂V T
∂X
∂V T
Y = 2 ∂ A
∂x1 ∂V T
J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
∂W
∂x1
T,V
=0
∂ A
∂x12 T,V
∂X
∂x1 T,V 2 > 0
∂ A
2
∂x1 T,V 2
69
(2)
(3)
where A denotes the Helmholtz function. The condition Y > 0 (Eq. (3)) guarantees the thermodynamic
stability of the calculated critical point. Consequently, the calculation of critical equilibria involves locating the temperature, volume and composition, which satisfy two simultaneous equations (Eqs. (1) and
(2)), and checking the thermodynamic stability of the solution. The solution of these critical conditions
was obtained by applying the Hicks–Young algorithm [18].
The Helmholtz function is obtained from conformal solution theory [19] using the one-fluid model.
V T
A = fes A∗0
(4)
,
− RT ln hes + RT xi ln xi
hes fes
i
where R is the universal gas constant, A∗0 the configurational contribution to the Helmholtz function and
fes and hes are the characteristic conformal parameters of the equivalent substance. It is customary to
estimate the conformal parameters of pure fluids from critical temperatures and critical volumes relative
c
c
c
c
to the properties of a reference substance (denoted by the subscript 0), i.e. f11 = T11
/T00
, h11 = V11
/V00
,
etc. The conformal parameters for the equivalent substance are obtained from the van der Waals one-fluid
prescriptions
fes hes =
xi xj fij hij
(5)
i
hes =
j
i
xi xj hij
(6)
j
where the contribution from unlike interactions is given by the Lorentz–Berthelot combining rules.
fij = fii fjj
(7)
1/3
1/3
hij = 18 (hii + hjj )3
(8)
It should be noted that no external adjustable parameters have been introduced into either Eq. (7) or Eq. (8).
The configurational Helmholtz function, A∗0 , can be evaluated by direct integration of any suitable equation
of state with respect to volume. In this work, we have used the CSvdW equation [15].
p=
RT(1 + η + η2 − η3 )
a
− 2
3
V(1 − η)
V
(9)
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J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
where η = b/4V , b is the co-volume occupied by hard-spheres and a the contribution from van der
Waals-like attraction between molecules. At the critical point of a pure fluid a = 1.38287RTc V c and b =
0.13044V c . Previous work [5,14] on global phase diagrams has shown that the CSvdW can reproduce
the known types of phase behaviour. The equation of state parameters can be related to the conformal
parameters (a ∝ fh, b ∝ h).
3. Results and discussion
The phase behaviour of binary mixtures can be classified [1,2] into several basic types depending on
the type of critical behaviour (Fig. 1). Type I mixtures have a continuous vapour–liquid critical locus
linking the critical points of the two components. Types II–VI behaviour is associated with the addition
of various kinds of liquid–liquid critical transitions.
I
III
II
C
C
C
1
2
UCEP
pr
C
C
2
Vm
C
C
UCEP
1
C
2
LCEP
UCEP
C
V
C
UCEP
1
2
IV
1
UCEP
C
C
1
UCEP
1
2
LCEP
2
LCEP
VI
VII
C
1
C
C
UCEP
LCEP
UCEP
VIII
2
C
UCEP
1
LCEP
UCEP
LCEP
UCEP
2
C
C
UCEP
1
LCEP
2
Tr
Fig. 1. Phase behaviour classification scheme for binary mixtures. Types I, II, III, IV, V and VI behaviour are observed experimentally whereas behaviour of types Vm, VII and VIII have only been reported in the literature from calculations. Critical
equilibria of the binary mixtures (—), the critical points of the pure components (C), the vapour pressure curves (- - -, 1 and 2)
and three-phase liquid–liquid–vapour (— - - —) equilibria are illustrated.
J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
71
In type II behaviour, in addition to a vapour–liquid critical line, a locus of upper critical solution
temperatures (UCSTs) commences at low temperatures from an upper critical end point (UCEP), which
terminates a liquid–liquid–vapour (LLV) line. Types III–V behaviour is also characterised by discontinuities in vapour–liquid equilibria. In type III behaviour, the vapour–liquid critical line commencing
from the critical point of the component with the highest critical temperature only extents partly to the
critical point of the other component before veering abruptly to very high pressures. This high-pressure
region represents liquid–liquid equilibria and the critical locus displays a continuous transition between
vapour–liquid and liquid–liquid equilibria. The vapour–liquid locus commencing from the critical point
of the component with the lowest critical temperature ends on an UCEP.
Type IV behaviour is characterised by three distinct critical lines. The vapour–liquid critical line from
the component with the lowest critical temperature terminates at an UCEP whereas the other vapour–liquid
critical line ends on a lower critical end point (LCEP). In addition, a line of UCSTs commences from
another UCEP. It is apparent from Fig. 1 that the effect of these phenomena is to split the LLV line into
two distinct parts. Type V behaviour is similar to type IV expect that there is no UCST line.
For type VI behaviour, there is a continuous vapour–liquid line between the critical points of the pure
components and a locus of UCSTs commencing from an UCEP. There is also a critical line which links a
LCEP and an UCEP at either end of a LLV line. This region is associated with closed-loop liquid–liquid
equilibria. In addition to these experimentally observed types, some workers [5,8,14,20,21] have reported
other possible types (Vm and VII) based on calculations of the phase behaviour of binary mixtures. In
addition, type VIII behaviour may be possible [8] for some hypothetical systems.
The phase behaviour of types I–V mixtures can be predicted qualitatively using the simple van der Waals
equation of state in which intermolecular interaction is modelled by combining a hard-sphere term for
repulsion with a dispersion term. The van der Waals equation does not predict type VI phenomena, which is
characterized by both LCST and UCST phenomena resulting in closed-loop liquid–liquid immiscibility.
For quantitative predictions, the van der Waals equation has been superseded by many more accurate
equations of state [22]. Type VI behaviour and closed loop liquid–liquid equilibria is associated typically
with aqueous mixtures [1,23]. Despite this, it and the remaining types of phase behaviour can be predicted
[5,14] by using either the simple Guggenheim [16] or CSvdW [15] equations of state.
We have determined the global phase diagram (Fig. 2) of binary mixtures using the CSvdW equation
of state. In the absence of any contribution from combining rule parameters, the global phase diagram
c
c
can be determined solely from the ratios of the critical temperatures (Tr = T22
/T11
) and critical volumes
c
c
(Vr = V22 /V11 ). Fig. 2 shows that in the absence of deviations from the Lorentz–Berthelot combining
rules, phase behaviour of types I–VII can be expected depending on the value of Tr and Vr . This means
that the Lorentz–Berthelot combining rules alone can be used to predict types I–VII behaviour. Type VIII
behaviour was also observed but it is not marked on the global phase diagram because it was confined
to physically unrealistic packing fractions (η > 1). When Vr = 1, only types I, II and III behaviour is
observed. To observe type V behaviour requires a significant difference in both the critical volume and
critical temperatures of the components. The greatest diversity of phase behaviour is concentrated in a
very small range of temperature and volume. This region of the global phase diagram is shown in greater
detail in Fig. 3.
Figs. 2 and 3 illustrate the boundary states between the various phase behaviour types. Commonly, the
transition between states occurs via a tricritical point (TCP), a double critical end point (DCEP) or an
azeotropic critical end point (ACEP) or van Laar point. The thermodynamic criteria for these phenomena
are discussed by Yelash and Kraska [5]. The transition from type I to type II behaviour occurs via a
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J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
Fig. 2. Global phase diagram calculated from the CSvdW equation of state. A comparison is given with experiment
for binary mixtures of ammonia + n-alkanes/simple gases (䊊), water + noble gases (䊐), methane + hydrocarbons (䊏),
perfluoro-octane+n-alkanes (䉱), carbon dioxide+hydrocarbons (+), benzene+n-alkanes (), perfluorobenzene+hydrocarbons
(×) and methanol + n-alkanes/xenon (䊉). The coordinates and phase types for these mixtures are summarized in Table 1.
DCEP at a temperature of zero Kelvin which is commonly referred to as the zero Kelvin point (ZKP).
The transition between types II and III behaviour occurs via a DCEP [24] whereas a critical pressure set
point (CPSP) [25] is involved in the transition between types III and IIIm behaviour. A line of tricritical
points separates types I and V behaviour. Fig. 2 identifies regions in which behaviour of types IV, V and
VI occur. As discussed in considerable detail elsewhere [5,14,20] the boundary states between these types
of behaviour involve a TCP, a CPSP, a critical pressure landing point (CPLP) and a degenerated critical
pressure maximum or minimum (dCPM).
The transition between the various types of phase behaviour can be examined in greater detail by taking
slices of the global phases diagrams at different values of Vr . Fig. 4 illustrates the transitions observed
when Vr = 1.1. The type I/II (Fig. 4a) and type II/IIIm (Fig. 4b) transitions occur via a DCEP whereas
a CPSP is involved in the type IIIm/III transformation (Fig. 4c). When Vr = 1.7 (Fig. 5), a considerable
diversity of phase behaviour is observed over a relatively small temperature range. Initially, there is a
J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
73
3.00
III
Vm
CPS
P
2.80
dCP
M
CPLP
V
2.60
Tr
TCP
VII
CP
2.40
SP
D
C
E
P
IV
VI
IIIm
2.20
I
II
2.00
1.00
1.25
1.50
1.75
2.00
2.25
2.50
Vr
Fig. 3. Enlargement of the global phase diagram showing regions of types IV, VI, VII, V and Vm behaviour.
type VI/VII transition via a DCEP (Fig. 5a). Type VII behaviour is in turn transformed to type IV via a
dCPM (Fig. 5b).
The observation of types VI and VII behaviour is significant. Most previous work that has reported
types VI and VII behaviour did so in conjunction with combining rule parameters that represented a
significant deviation from the Lorentz–Berthelot combining rules. The results presented here indicate
that simply changing the relative values of the critical temperatures and volumes if sufficient to observe
types VI and VII behaviour.
It is of considerable interest to determine whether this simplified global phase diagram can be used to
predict the phase behaviour types of real binary mixtures. To determine this we have identified several
binary mixtures of different type: ammonia + n-alkanes [26]/simple gases [26], water + noble gases
[27], methane + n-hexane [28], 1-hexene [29] or other alkenes [2], perfluoro-octane + n-alkanes [30],
carbon dioxide + hydrocarbon [31,32], benzene + n-alkanes [33] and perfluorobenzene + hydrocarbons
[34], methanol + n-alkanes [26,35,36], or xenon [35] and methane + hydrocarbons [2]. These mixtures
represent a reasonably diverse range of molecules ranging from inert gases, hydrocarbons, fluorocarbons
and dipolar molecules. Data for these mixtures are summarized in Table 1 and they are located in Fig. 2
based on the Tr –Vr coordinates obtained from pure critical properties [37]. We note that other workers
[38,39] have also compared experimental data for some of these mixtures to the predictions from a global
phase diagram.
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J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
Transition
I
UCEP
ZKP
P
II
(a)
Transition
II
IIIm
DCEP
UCEP
(b)
IIIm
Transition
III
CPSP
UCEP
UCEP
UCEP
(c)
T
Fig. 4. The transition pathway between (a) type I/II, (b) type II/IIIm and (c) type IIIm/III behaviour when Vr = 1.1. The transition
between types I and II behaviour is via a ZKP with the liquid–liquid critical curve appearing topologically below solidification
at 0 K. The pure critical points (䊏) are identified.
It is apparent from the comparison with experiment that Fig. 2 is a reasonably good predictor of
the phase behaviour type. The mis-assignments are mostly restricted to types I/II and IV/V. Instead of
predicting type IV behaviour, Fig. 2 predicts type V. Similarly, sometimes type I behaviour is predicted
whereas the designated phase type should be II. It should be noted that the sole distinction between types
I/II and IV/V is the existence of an UCST curve. In many cases, the experimental assignment of both
types II and IV behaviour is uncertain because solidification occurs in the region of the phase diagram
where the UCST curve would normally be expected to occur. In view of such experimental difficulties,
the mis-assignment of types I/II or IV/V mixtures on the global phase diagram is not a serious deficiency.
The results are quite reasonable in view of the fact that no attempt has been made to account for the subtle
intermolecular interactions that often affect the phase behaviour observed for real binary mixtures.
J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
Tr = 2.31
VI
Transition
75
Tr = 2.33
VII
UCEP
DCEP
P
LCEP
UCEP
UCEP
UCEP
LCEP
LCEP
LCEP
(a)
VII
Tr = 2.36
UCEP
LCEP
UCEP
dCPM
Transition
Tr = 2.37
IV
UCEP
UCEP
LCEP
UCEP
LCEP
UCEP
(b)
T
Fig. 5. The transition path between (a) type VI/VII and (b) type VII/IV when Vr = 1.7. The pure critical points (䊏) and
liquid–liquid–vapour lines (— - - —) are shown.
The success of the comparison with experiment means that in many cases, the phase behaviour type of
a binary mixture can be predicted based solely on the relative difference in the critical properties of the
components. It is unlikely that Fig. 2 could be used to identify type VI behaviour for real binary mixtures
because such phenomena occur over a very narrow range of physical properties. However, in most cases,
the critical properties of the pure components can be used in conjunction with Fig. 2 to reliably distinguish
between types I/II, III and IV/V phenomena.
Kolafa et al. [40] have also reported a global phase diagram which is independent of adjustable combining rule parameters. Their diagram was presented as a “λ–τ projection” which should be equivalent to
1/3
a Vr –Tr diagram. They used the Mansoori–Carnahan–Starling–Leland [41] (MCSL) equation of state
for mixtures of hard spheres of different diameters (σ) coupled with the van der Waals attractive term
(MCSLvdW). The use of the MCSL equation means that there is no contribution from σ 12 terms. The one
√
fluid mixing rule (Eq. (5)) was used for am with a12 = a11 a22 . Another difference in their calculation is
that the Helmholtz function was obtained by directly integrating the equation of state plus an ideal mixing
term whereas our calculations used conformal solution theory. Conformal solution theory provides a more
theoretically rigorous way of including the contributions of configurational and mixing properties to the
Helmholtz function of the fluid.
In contrast to our results, Kolafa et al. [40] did not observe either types IIIm or V behaviour. This means
that their global phase diagram does not have the III/IIIm, IIIm/III, I/II and I/V boundaries identified in
Fig. 2. Instead of a type I/V transition, Kolafa et al. [40] observed a type II/IV transition and they did not
compare their global phase diagram with experimental data. For a pure fluid, the MCSLvdW equation
of state is identical to the CSvdW equation of state. Therefore, both the qualitative and quantitative
differences in the global phase diagrams can be probably attributed to the handling of unlike interactions
for the distance parameters. The calculations with the CSvdW equation used the one-fluid model to
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J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
Table 1
The reduced-temperature and reduced-volume coordinates and phase behaviour type of binary mixtures located on the global
phase diagram (Fig. 2)
Mixture
Vr
Tr
Type from
experiment
Type predicted
from Fig. 2
Ammonia + methane
Ammonia + ethane
Ammonia + propane
Ammonia + n-butane
Ammonia + argon
Ammonia + carbon monoxide
Ammonia + sulfur hexafluoride
1.375
2.05
2.82
3.54
1.04
1.29
2.75
0.47
0.75
0.91
1.05
0.37
0.328
0.785
III
II
II
II
III
III
III
III
II
III
III
III
II
III
Water + helium
Water + argon
Water + krypton
Water + xenon
1.017
1.34
1.625
2.1
0.008
0.23
0.323
0.447
III
III
III
III
III
III
III
III
Methane + n-hexane
Methane + 3,3-dimethylpentane*
Methane + 2,3-dimethyl-1-butene
Methane + 2-methyl-1-pentene
Methane + 1-hexene
3.74
4.18
3.46
3.57
3.58
2.67
2.82
2.64
2.66
2.65
V
IV
IV
IV
IV
V
V
V
V
V
Perfluoro-octane + n-hexane
Perfluoro-octane + n-heptane
Perfluoro-octane + n-tridecane
Perfluoro-octane + n-tetradecane
1.35
1.16
1.46
1.52
0.99
0.93
1.347
1.38
II
II
II
II
II
II
I
I
Carbon dioxide + n-butane
Carbon dioxide + n-pentane
Carbon dioxide + n-hexane
Carbon dioxide + n-tridecanea
Carbon dioxide + cyclohexane
Carbon dioxide + methylcyclohexane
2.71
3.23
3.93
8.76
3.27
3.91
0.71
0.65
0.6
2.22
0.55
0.53
II
II
II
IV
II
II
II
II
II
IV
IIIm
IIIm
Benzene + n-pentane
Benzene + n-hexane
Benzene + n-heptane
Benzene + n-octane
Benzene + n-nonane
Benzene + n-decane
Benzene + n-tridecane
1.17
1.43
1.67
1.9
2.12
2.35
2.7
0.83
0.9
0.96
1.01
1.06
1.09
1.2
II
II
II
II
II
II
II
II
II
II
I
I
I
I
Perfluorobenzene + n-pentane
Perfluorobenzene + n-hexane
Perfluorobenzene + n-heptane
Perfluorobenzene + n-octane
Perfluorobenzene + n-nonane
Perfluorobenzene + n-decane
Perfluorobenzene + n-dodecane
Perfluorobenzene + cyclohexane
Perfluorobenzene + benzene
1.06
1.14
1.33
1.52
1.7
1.91
2.01
1.05
1.25
1.14
0.95
1.01
1.07
1.11
1.15
1.25
0.96
0.95
II
II
II
II
II
II
II
II
II
I
II
I
I
I
I
I
I
II
Methanol + xenon
Methanol + ethane
Methanol + propane
Methanol + n-hexane
Methanol + n-heptane
1
1.254
1.72
3.136
3.66
1.77
0.6
0.72
0.99
1.055
II
II
II
II
II
II
II
II
II
I
a
The Tr –Vr coordinates of this mixture occur at outside of the range used in Fig. 2.
J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
77
evaluate the contribution from h12 whereas the MCSLvdW calculations do not involve an equivalent
cross term, i.e. there is no σ 12 . This suggests that the unlike distance parameters may have a subtle but
nonetheless important influence on the global phase diagram. The evolution of closed-loop behaviour in
the phase behaviour of binary mixtures is discussed elsewhere [42–44].
4. Conclusions
A global phase diagram has been developed solely in terms of the critical properties of the pure
components. Phase behaviour of types I–VII is predicted without the need to invoke deviations from
the Lorentz–Berthelot combining rules. The use of only experimentally accessible quantities means that
the global phase diagram can be used as a qualitative indicator of the expected type of phase behaviour.
Comparison with experimental data indicates that the global phase diagram is a reasonably good predictor
of types I/II, III and IV/V behaviour. It should be emphasised that our global phase diagram provides
only a qualitative indicator of the phase behaviour and that deviations from the Lorentz–Berthelot rules
are usually necessary for quantitative calculations.
List of symbols
a
attractive equation of state parameter
A
Helmholtz function
b
co-volume equation of state parameter
p
pressure
R
universal gas constant
T
temperature
V
volume
W
determinant defined by Eq. (1)
x
mole fraction
X
determinant defined by Eq. (2)
y
packing fraction
Y
determinant defined by Eq. (3)
Subscripts and superscripts
∗
configurational property
c
critical property
i
property of component i
j
property of component j
m
property of the mixture
r
reduced with respect to the critical property
Acknowledgements
J.-L.W. thanks the Australian Government for an Australian Postgraduate Award. The Australian Partnership for Advanced Computing provided a generous allocation of computing time.
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J.-L. Wang, R.J. Sadus / Fluid Phase Equilibria 214 (2003) 67–78
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