azeotropic behavior of dieterici binary fluids
TRANSCRIPT
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Fluid Phase Equilibria 257 (2007) 18–26
Azeotropic behavior of Dieterici binary fluids
Ilya Polishuk a,∗, Juan H. Vera b, Hugo Segura c,∗∗a The Department of Chemical Engineering & Biotechnology, The College of Judea and Samaria, 44837 Ariel, Israel
b Department of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2B2c Department of Chemical Engineering, Universidad de Concepcion, Concepcion, Chile
Received 4 April 2007; received in revised form 28 April 2007; accepted 1 May 2007Available online 5 May 2007
bstract
The results of the present study point to the fact that the EOS of Dieterici is able to predict single azeotropy ending at zero temperature. Inddition, the EOS of Dieterici is able to predict polyazeotropy, as in the case of van der Waals-type models, and even three azeotropes for binaryystems.
2007 Elsevier B.V. All rights reserved.
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eywords: Phase equilibria; Equation of state; Global phase diagram; Azeotrop
. Introduction
In contrast to the theoretical basis of the van der Waalsodel, that considers an overall central potential to subtract
ohesion forces from the hard core contribution to pressure,he equation of state (EOS) of Dieterici [1] is based on ahysical approximation in which only molecules having ainimum value of the kinetic energy could escape from the
ttraction of the bulk fluid to hit the surface of a container.n spite of the differences concerning the underlying molecu-ar model for calculating the macroscopic pressure of a fluid,he relation between the van der Waals equation and Dieterici’s
odel can be clearly identified [2]. Each of these EOSs pre-ict fluid phase equilibrium and critical points of pure fluids
nd their mixtures and, additionally, present similar predic-ions for the properties of non ideal gases. Combination of bothodel EOS is also possible [3,4]. For almost a century, the
Abbreviations: Az, azeotropic state; 3Az, poliazeotropic region, exhibitinghree homogeneous azeotropes; 3AzBP, triazeotropic system at the Bancr-ot point; CAzEP, critical azeotropic endpoint; CBP, critical Bancroft point;CAzEP, double critical azeotropic endpoint; DTAz, double tangent azeotropy;OS, equation of state; GPD, global phase diagram; PTAz, pure tangentzeotropy; ZT, zero temperature; ZTBP, zero temperature Bancroft point∗ Corresponding author. Tel.: +972 542179200.
∗∗ Corresponding author.E-mail addresses: [email protected] (I. Polishuk),
[email protected] (H. Segura).
2
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w
a
b
378-3812/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2007.05.001
eterici fluids
OS of Dieterici was rarely analyzed or implemented in prac-ice until it was revisited [5–11]. His results have encouragedurther investigations using the global phase diagram (GPD)pproach [12,13], in order to establish the capability of the modelor predicting vapor–liquid, liquid–liquid, liquid–liquid–vapor,iquid–liquid–liquid and liquid–liquid–liquid–vapor equilibria.owever, although the fluid phase behavior predicted by theieterici model seems to be well characterized so far, minor
ttention has been given to the prediction of azeotropy. Thus,he present study continues our previous GPD investigation onhe phase diagrams predicted by the EOS of Dieterici for binary
ixtures, focusing now on a deeper analysis of azeotropic phe-omena.
. Theory
The Dieterici EOS has the form:
= RT
v − be−a/vRT (1)
here the constants a and b for pure fluids are expressed as
4 (RTc,i)2
i =e2 Pc,i
(2)
i = 1
e2
RTc,i
Pc,i(3)
se Eq
aw
a
b
wkoeh
ξ
ζ
λ
wIisza(
A
w
A
aEsbc
λ
wE
•
•
p
λ
DacTbIdtKflo
G(f
ζ
λ
mTm
ζ
λ
waFCact
I. Polishuk et al. / Fluid Pha
nd e is Euler’s number. In order to extend Eq. (1) to mixtures,e have considered the following conventional mixing rules:
= x21a1 + 2x1x2
√a1a2(1 − k12) + x2
2a2 (4)
= x1b1 + x2b2 (5)
here xi is the mole fraction of component i (x2 = 1 − x1) and12 is a cross interaction parameter. The general characteristicsf Eq. (1) and its extension to mixtures have been discussedlsewhere [2–8]. Similar to our previous study [13], we considerere the following dimensionless global parameters:
= b2 − b1
b2 + b1(6)
= a2/b22 − a1/b
21
a2/b22 + a1/b
21
(7)
= a2/b22 − 2a12/b
212 + a1/b
21
a2/b22 + a1/b
21
(8)
here ξ, ζ, λ are the coordinates of a GPD and b12 = (b1 + b2)/2.n order to perform a GPD investigation of the azeotropic behav-or of the Dieterici fluid, it is expedient to consider two referencetates namely: the gas–liquid critical azeotropy and the low (orero) temperature azeotropy. The necessary conditions of anzeotrope ending in a gas–liquid critical azeotropic endpointCAzEP) are given by [14]:
2v = A3v = Axv = 0 (9)
here A is the Helmholtz function of Eq. (1) defined as
=∫ ∞
v
(P − RT
v
)dv + RT
(ln
1
v+
2∑i=1
xi ln xi
)(10)
nd Anx mv = (∂n+mA)/(∂xn ∂vm). Applying Eqs. (9) and (10) toqs. (1)–(3) and, after some straightforward algebra, it is pos-ible to demonstrate that the critical azeotrope of a Dietericiinary mixture is expressed in terms of global coordinates andoncentration as follows:
= −
ξ(3 + ξ[4 − 8x1 + ξ − 6x1x2ξ]) + (1 + [1 − 2x1]ξ)
× (2 + ξ[3 − 6x1 + ξ])ζ
2(1 + ξ − x1[2 + 3x2ξ])(11)
here x1 = 1 − x2 is the mole fraction of the critical azeotrope.q. (11) implies that:
at a constant ξ value (i.e. molecules with fixed covolume orsize ratio), a critical azeotrope of a given mole fraction x1displays a straight line on the ζ–λ plane;at constant ξ, ζ, λ values, Eq. (11) reduces into a quadraticequation of the azeotropic mole fraction. Consequently, amaximum of two azeotropes ending at the gas–liquid criticalpoint could be predicted by the Dieterici EOS.
fflfl
atD
uilibria 257 (2007) 18–26 19
For the case of equally sized molecules (ξ = 0), Eq. (11) sim-lifies to the following expression:
= ζ
2x1 − 1(12)
Consequently, for mixtures of molecules of equal sizes,ieterici fluids exhibit single critical azeotropy. Critical
zeotropic boundaries in a GPD plot are given by the appli-ation of Eq. (11) to the limiting concentrations x1 = 0 and 1.hus, we can obtain partial GPDs that show the range whereinary mixtures exhibit critical azeotropy, as shown in Fig. 1.n general, the results shown in these figures are similar to pre-ictions of van der Waals-type EOS. However, in contrast tohe latter model (see, for example, the results reported by vanonynenburg and Scott [15]), the critical azeotropy of Dietericiuids is extremely sensitive to the difference of molecular sizesf pure components.
The intersection of the critical azeotropic boundaries on aPD yields the critical Bancroft point (CBP), whose coordinates
for the case of Dieterici fluids) are given analytically by theollowing expressions:
Bancrotf,crit = − 3ξ
2 + ξ2 (13)
Bancrotf,crit = ξ2
2
(7 − ξ2
2 + ξ2
)(14)
As mentioned above, the Dieterici EOS can predict a maxi-um of two azeotropes ending at the gas–liquid critical point.he condition for which these critical azeotropes have equivalentole fractions is given by the following relations:
DCAzEP = 3ξ(1 + [1 − 2x1]ξ)(ξ[ξ − 1 + 2x1]−2)
4 + ξ(6 + ξ + ξ3 + 6x21[ξ + ξ3]−6x1[2 + ξ+ξ3])
(15)
DCAzREP =
ξ2(7 + ξ[12 + ξ − 2ξ2{3 + ξ} − 6x21ξ{ξ2 − 3}
+ 6x1{1 + ξ}{ξ2 + ξ − 4}])4 + ξ(6 + ξ + ξ3 + 6x2
1[ξ + ξ3]
− 6x1[2 + ξ + ξ3])(16)
hich correspond to the coordinates of a double criticalzeotropic endpoint (DCaZEP). Fig. 1c shows the details ofig. 1b in the vicinity of the critical Bancroft point (B).urves A–C are the parametric representation of Eqs. (15)nd (16) for ξ = 1/3 and 0 ≤ x1 ≤ 1. The region A–B–C (alsoalled dish region) corresponds to a range of global parame-ers where two critical azeotropes take place. All the previousacts indicate that the behavior of critical azeotropes in Dietericiuids is similar to the patterns observed in van der Waalsuids.
Let us now consider the low temperature reference for thezeotropic behavior. In general, the characterization of lowemperature azeotropy is a challenging task when consideringieterici’s fluids. The reason is that low temperature isotherms
20 I. Polishuk et al. / Fluid Phase Equilibria 257 (2007) 18–26
critic
ollcindptvvtbbthivt
i
T
a
l
filt
Fig. 1. (a–c) Range of critical azeotropy for the Dieterici EOS. Solid lines:
f this EOS are always positive and, in the range of liquid-ike volume, they converge to an orthogonal frame of straightines at the P–v projection [13]. In addition, phase equilibriumalculations with EOSs at low temperatures are numericallyll-conditioned and, consequently, an alternative approach iseeded for obtaining numerical solutions of these states. Asemonstrated by Segura et al. [16], similarly to the case ofure components, the vapor pressure of an azeotrope satisfieshe Maxwell area criterion at the P–v diagram. Therefore, theapor pressure of an azeotrope can be solved by considering theapor pressure calculation of a hypothetical pure fluid, charac-erized by the concentration of the azeotropic mixture. It haseen demonstrated also that the vapor pressure of a pure fluidecomes asymptotic to the liquid phase fugacity at the sameemperature [14]. Consequently, the vapor pressure of a pure
ypothetical fluid at low temperature may be accurately approx-mated by the liquid phase fugacity of the mixture. Finally, theapor pressure of an azeotrope is stationary on mole fraction, sohat a basic condition for detecting low temperature azeotropyT
E
al azeotropic endpoints; dashed lines: double critical azeotropic endpoints.
s
lim→0
(∂ ln P
∂x1
)T
= 0 ≈ limT→0
(∂ ln f L
∂x1
)T
(17)
nd from the general definition of the fugacity, it follows that
n f L = ln(φP) = PvL
RT− 1 − ln
vL
RT+∫ ∞
vL
(P
RT− 1
v
)dv
(18)
In addition, according to Eq. (1), the liquid volume that satis-es the Maxwell area criterion of stable phase equilibria at very
ow temperature tends to the liquid spinodal volume which, inurn, is given by [13]:
√ √
lim→0vL → v∗,L = a − a a − 4bRT
2RT(19)
Consequently, integrating Eqs. (1), (4) and (5) by means ofq. (18) and, then, taking the limit of the result at zero temper-
se Equilibria 257 (2007) 18–26 21
a
T
ie
λ
wzeEpoteaatm
gc1
ζ
temg
pameataep
3
ubc(drs
Fig. 2. (a and b) Range of zero temperature and critical azeotropic behaviorfpB
ubpwsAcob
I. Polishuk et al. / Fluid Pha
ture with the aid of Eq. (19), yields
lim→0
(T
∂ ln f L,Az
∂x1
)T
= −1
a
(da
dx1
)⇔Az
da
dx1= 0 (20)
Finally, from Eqs. (17) and (20) we can deduce the follow-ng transitional limit for zero temperature azeotropic endpointsxpressed in global parameters (Eqs. (6)–8)):
= ζ + ξ(2 + ξ − 2x1ξ + (2 − 4x1 + ξ)ζ)
2x1 − 1(21)
here x1 is the mole fraction of an azeotrope ending now atero temperature. Clearly, Eq. (21) is a linear relation in x1 forvery coordinate of the GPD (ξ, ζ, λ). Therefore, in general, theOS of Dieterici predicts single azeotropy ending at zero tem-erature, contrary to the statement [8] concerning the absencef zero temperature limited azeotropic lines in the GPD. Forhe case of equally sized molecules (ξ = 0) Eq. (21) becomesquivalent to Eq. (12), indicating that critical and zero temper-ture azeotropes are characterized by the same mole fractionnd, therefore, the azeotropic concentration does not depend ofemperature or pressure (as occurs in van der Waals mixtures of
olecules of equal sizes).The intersection of zero temperature azeotropic boundaries
ives place to the zero temperature Bancroft point (ZTBP) whichan be calculated from Eq. (21) by solving λ(x1 = 0) = λ(x1 =) as
Bancrotf,ZT = − 2ξ
1 + ξ2 ; λBancrotf,ZT = 4 − ξ2 − 4
1 + ξ2
(22)
Replacement of Eq. (22) in Eq. (21) indicates that the zeroemperature Bancroft point is a geometrical focus of Eq. (21) forvery azeotropic mole fraction. Consequently, every physicalole fraction of a binary mixture with the global coordinates
iven in Eq. (22) is azeotropic at zero temperature.From the detailed inspection of Eqs. (13), (14) and (22) it is
ossible to conclude that the global coordinates of the criticalnd zero temperature Bancroft points coincide for Dieterici’sixtures composed by molecules of equal sizes (ξ = 0). How-
ver, this is not so as the molecular size difference increases,s it is shown in Fig. 2 for ξ = 1/3. Furthermore, and contraryo the previous results [8], the divergence of zero temperaturend critical Bancroft points in the GPD of molecules of differ-nt sizes indicates that the EOS of Dieterici is able to predictolyazeotropy, as in the case of van der Waals-type models.
. Results and discussion
Once we have established the basic reference conditions, lets consider the distinctive attributes of the azeotropes predictedy the Dieterici EOS for binary mixtures. Fig. 3a shows theomplete details of Fig. 2b in the range of the zero-temperature
ZTBP) and critical Bancroft (CBP) points. Fig. 3b, in turn,epicts the details of Fig. 3a around the microscopic [3 Az]egion, whose characteristics will be discussed below in thisection. In Fig. 3 we have included the region of polyazeotropy,[obt
or the Dieterici EOS. Solid lines: critical and zero temperature azeotropic end-oints; dashed lines: zero temperature azeotropic endpoints; [CBP] and [ZTBP]:ancroft points.
sually limited by the so-called horn region [17]. It shoulde pointed out that the most formal treatment of this latterarametric range has been presented by van Konynenburg [18]ho discussed a simplified linear theory based on a classical
caling of the vapor pressure curve for van der Waals fluids.lthough not clearly recognized, a limiting condition of sub-
ritical polyazeotropy (which corresponds to the boundary linesf the horn region) is the so-called tangent azeotropy discussedy Guminski [19] (see Wisniak et al. [20] and Segura et al.
21] for details). Briefly, a tangent azeotrope is a special casef physical azeotropy where isothermal (or isobaric) dew andubble-point curves exhibit a stationary point of inflection athe azeotropic concentration. Following ideas from the dis-
22 I. Polishuk et al. / Fluid Phase Eq
Fig. 3. (a) Details of Fig. 2b around Bancroft points. Dashed lines: zero tem-perature azeotropic endpoint; solid lines: critical azeotropic endpoint; dot lines:subcritical boundary of the horn region; dot-dot-dashed lines: critical boundaryof the horn region; dot-dashed lines: DCAzEP line. (b) Details of Fig. 3a aroundthe [3 Az] region. Dashed lines: zero temperature azeotropic endpoint; solidlines: critical azeotropic endpoint; dot lines: subcritical boundary of the hornrll
po
P
μ
G
y
wctpsirt<
pfD(s
•
••
ca(g(itgropiCaP
•
•
egion; dot-dot-dashed lines: critical boundary of the horn region; dot-dashedines: DCAzEP line; square-solid line: double tangent azeotropy; triangle-solidine: triazeotropic system at the Bacroft point.
lacement theory discussed by Malesinki [22], the conditionsf tangent azeotropy for a binary system are
(x1, T, vL) = P(y1, T, v
V) (23)
Li (x1, T, v
L) = μVi (y1, T, v
V) (i = 1, 2) (24)
L2x = (x1, T, v
V) = GV2x(y1, T, v
V) (GL,V2x ≥ 0) (25)
1 = x1 (26)
•
uilibria 257 (2007) 18–26
here μ is the chemical potential and y and x are theoncentrations of the vapor V and liquid L phases, respec-ively. Solving Eqs. (23)–(26) at x1 = 0 and x1 = 1 (formally,ure tangent azeotropic points), it is possible to obtain theub-critical boundaries of the horn region. In addition, crit-cal tangent azeotropy (i.e. the critical boundary of the hornegion) can be calculated by solving Eqs. (23)–(26) very closeo a critical condition (where vL → vV) in the range 0 < x11.Fig. 4a depict the main set of T–x projections involving the
rediction of homogeneous azeotropy with the EOS of Dietericior ξ = 1/3. From these T–x projections it can be seen thatieterici mixtures are characterized by a single line of azeotropy
dot-dot-dash lines), which can reach anyone of the followingtates:
the boiling points of pure components (systems A, B, D, E,F, H, I, J, K, L);the zero temperature axis (systems A, B, C, G);the gas–liquid critical line (systems C, G, H, I, J, K, L).
In the systems shown in Fig. 4a, tangent azeotropy isharacterized by a stationary point of the azeotropic linet the T–x projection, condition which follows from Eqs.23)–(26). Consequently, systems F and K exhibit two homo-eneous azeotropes at isothermal and/or isobaric conditionsi.e. they are polyazeotropic systems). Furthermore, as shownn Fig. 4b, Dieterici mixtures are characterized by three addi-ional classes of very interesting azeotropic behavior, whoselobal coordinates have been picked from the microscopicange shown in Fig. 3b. It should be pointed out that theriginal van der Waals EOS (and related models) neitherredicts inflections nor stationary points of negative concav-ty along the azeotropic line at the T–x projection [15,17].onsequently, the van der Waals EOS does not predict thezeotropic behavior of the systems I, M, N, O shown in Fig. 4.articularly:
System I belongs to the dish region. In the case of vander Waals mixtures, the dish region is characterized bymixtures exhibiting a convex azeotropic line at the T–xprojection which, in addition, does not connect the boil-ing points of pure components. In contrast, in the case ofthe EOS of Dieterici, the dish region is characterized bytwo branches of azeotropic lines starting from pure compo-nents and connecting the critical line in two points of criticalazeotropy.System M demonstrates that, in contrast to the van der WaalsEOS, the EOS of Dieterici has the feature of predicting con-vergent polyazeotropy [20] (i.e. a concave azeotropic line atthe T-projection), which is the behavior found in almost all
the experimental cases of polyazeotropy [23].System N shows clear evidence that the EOS of Dietericican predict, simultaneously, divergent and convergentpolyazeotropy.
I. Polishuk et al. / Fluid Phase Equilibria 257 (2007) 18–26 23
Fig. 4. (a) Azeotropic behavior predicted by the Dieterici EOS in the different regions indicated in Fig. 3a. Solid lines: critical liquid–gas lines. Dot-dot-dashed lines:azeotropic lines. Squares: tangent azeotropy. (b) Azeotropic behavior predicted by the Dieterici EOS in the different regions indicated in Fig. 3b. Solid lines: criticalliquid–gas lines; dot-dot-dashed lines: azeotropic lines; squares: tangent azeotropy.
24 I. Polishuk et al. / Fluid Phase Equilibria 257 (2007) 18–26
(Cont
•
prb
•
•
•
Fig. 4.
System O implies that the EOS of Dieterici can predictpolyazetropic systems characterized by three azeotropes.
The range of parameters where the EOS of Dieterici predictsolyazeotropic systems with three azeotropes (also called [3 Az]egion) corresponds to the shaded area shown in Fig. 3b, and theiroundary lines are characterized by the following mechanisms:
Tangent azeotropy at pure components (PTAz mechanism),which is the general mechanism that bounds the horn region(dotted lines in Fig. 3b). The effect of the PTAz mechanismon the T–x projection is clearly illustrated in Fig. 5a.Double point of tangent azeotropy (DTAz mechanism), where
the stationary points of opposite concavity observed alongthe azeotropic lines of systems N and M collapse in a sin-gle stationary point of inflection (square-solid-square line inFig. 3b). As follows from Boshkov’s [24] bifurcation analy-inued).
sis, besides Eqs. (23)–(26), the transitional mechanism thatcharacterizes the double point of tangent azeotropy is
GL3x(x1, T, v
L) = GV3x(y1, T, v
V) (27)
while the effect of this mechanism on the T–x projection isclearly illustrated in Fig. 5b.At the T–x projection, the azeotropic line connects the boilingpoints of pure components at the Bancroft point (triangle-solid-triangle line in Fig. 3b) and, additionally, it inflects inmidrange azeotropic concentrations (3AzBP mechanism). Inthis case and, as a necessary condition, the azeotropic lineshould exhibit concentration slopes of the same sign at theboiling points of pure components, as shown in Fig. 5. The
mathematical conditions that describe this transitional mech-anism are more complex than the previous mechanisms andare given by the simultaneous fulfillment of the followingrelations:
I. Polishuk et al. / Fluid Phase Equilibria 257 (2007) 18–26 25
Fig. 5. (a) Illustration of the effect of the PTAz mechanism on azeotropic phase diagrams. Solid lines: critical liquid–gas lines. Dot-dot-dashed lines: azeotropiclines; squares: tangent azeotropy. (b) Illustration of the effect of the DTAz mechanism on azeotropic phase diagrams; dot-dot-dashed lines: azeotropic lines; square:tangent azeotropy; diamond: inflecting point of azeotropy. (c) Illustration of the effect of the 3AzBP mechanism on azeotropic phase diagrams. Dot-dot-dashed lines:azeotropic lines; square: tangent azeotropy; diamond: inflecting point of azeotropy.
2 se Eq
a
b
c
ooudiaisg
4
cpttoootpe
LabGP
RTv
xx
y
Gμ
ζ
Scr
SLV
A
Cl
R
[[
[[
[[
[
[[
[[
[
6 I. Polishuk et al. / Fluid Pha
. Mathematical conditions for azeotropic behavior in purecomponents at the Bancroft point:
PL(0, T, vL2 ) − PV(0, T, vV
2 ) = 0;
μLi (0, T, vL
2 ) − μLi (0, T, vV
2 ) = 0 (i = 1, 2);
PL(1, T, vL1 ) − PV(1, T, vV
1 ) = 0;
μLi (1, T, vL
1 ) − μLi (1, T, vV
1 ) = 0 (i = 1, 2);
PL(0, T, vL2 ) − PL(1, T, vL
1 ) = 0 (28)
. Mathematical conditions for an inflecting point of azeotropy:
PL(y1, T, vL1 ) − PV(x1, T, v
V1 ) = 0;
μLi (y1, T, v
L1 ) − μV
i (x1, T, vV1 ) = 0 (i = 1, 2);
GV3x(y1, T, v
V) − GL3x(x1, T, v
L) = 0; y1 − x1 = 0
(29)
. Mathematical condition for an azeotropic line with equal signof slopes at pure components:
[GV2x(1, T, vV
1 ) − GL2x(1, T, vL
1 )] × [GV2x(0, T, vV
2 )
−GL2x(0, T, vL
2 )] ≥ 0 (30)
Fig. 5c clearly depicts the effect of the 3AzBP mechanismn the T–x projection. Notably the horn region of the EOSf Dieterici appears in a range of total miscibility for the liq-id phase in the sub-critical range. However, as it was alreadyiscussed [7], besides of a critical line that connects the crit-cal points of pure components, the EOS of Dieterici exhibitsn additional critical line that corresponds to a closed loop ofmmiscibility in the range of high pressure. Consequently, theystems shown in Figs. 4 and 5 cannot be classified directly asenuine azeotropic Type I systems.
. Conclusions
This study presents analytical relations for the coordinates ofritical and the low (zero) temperature azeotropes on the globalhase diagram of Dieterici’s fluids. It has been demonstratedhat the azeotropic behavior of the Dieterici’s fluid is similar tohe patterns observed in van der Waals-type fluids. The resultsf the present study contradict some previous observations ofther authors [8]. In particular, it is demonstrated that the EOSf Dieterici is able to predict a single azeotropy ending at zeroemperature. In addition, the EOS of Dieterici is able to predictolyazeotropy, as in the case of van der Waals-type models andven three azeotropes for binary systems.
ist of symbols
cohesion parametercovolumeGibbs energypressure[
[
[
uilibria 257 (2007) 18–26
universal gas constanttemperaturevolumeliquid phase mole fraction
¯ azeotropic mole fractionvapor phase mole fraction
reek letterschemical potencial
, ξ, λ global dimensionless parameters
ubscriptscritical statereduced property
uperscriptsliquidvapor
cknowledgements
This work was partially financed by FONDECYT, Santiago,hile (Project 1050157), and by NSERC, Canada. We would
ike to thank Dr. A. Mejıa for several helpful discussions.
eferences
[1] C. Dieterici, Ann. Phys. Chem. Wiedemanns Ann. 69 (1899) 685–705.[2] F.H. MacDougall, J. Am. Chem. Soc. 38 (1916) 528–555.[3] I. Polishuk, J.H. Vera, AIChE J. 51 (2005) 2077–2088.[4] I. Polishuk, J.H. Vera, J. Phys. Chem. (B) 109 (2005) 5977–5984.[5] R.J. Sadus, Phys. Chem. Chem. Phys. 4 (2002) 919–921.[6] R.J. Sadus, Fluid Phase Equilib. 212 (2003) 31–39.[7] R.J. Sadus, J. Chem. Phys. 115 (2001) 1460–1462.[8] J. Bumba, J. Kolafa, Phys. Chem. Chem. Phys. 6 (2004) 2301–2306.[9] I. Cachadina, A. Mulero, M. Silbert, J. Chem. Phys. 120 (2004) 8868–8869.10] R.J. Sadus, J. Chem. Phys. 120 (2004) 8870.11] F.L. Roman, A. Mulero, F. Cuadros, Phys. Chem. Chem. Phys. 6 (2004)
5402–5409.12] A. Mulero, I. Cachadina, Int. J. Thermophys. 28 (2007) 279–298.13] I. Polishuk, R. Gonzalez, J.H. Vera, H. Segura, Phys. Chem. Chem. Phys.
6 (2004) 5189–5194.14] H. Segura, J. Wisniak, Comput. Chem. Eng. 21 (1997) 1339–1347.15] P.H. van Konynenburg, R.L. Scott, Phil. Trans. R. Soc. Lond. A 298 (1980)
495–540.16] H. Segura, J. Wisniak, P.G. Toledo, A. Mejıa, Fluid Phase Equilib. 166
(1999) 141–162.17] J. Kolafa, Phys. Chem. Chem. Phys. 1 (1999) 5665–5670.18] P. H. van Konynenburg, Critical Lines and Phase Equilibria in Binary
Mixtures, PhD Thesis, University of California, Los Angeles, 1968.19] K.Z. Guminski, Rocz. Chem. 32 (1958) 569–581.20] J. Wisniak, H. Segura, R. Reich, Ind. Eng. Chem. Res. 35 (1996)
3742–3758.21] H. Segura, J. Wisniak, A. Aucejo, R. Munoz, J.B. Monton, Ind. Eng. Chem.
Res. 35 (1996) 4194–4202.
22] W. Malesinski, Azeotropy and other Theoretical Problems ofVapour–Liquid Equilibrium, Interscience Publishers, London, 1965.23] S. Loras, A. Aucejo, J.B. Monton, J. Wisniak, H. Segura, J. Chem. Eng.
Data 46 (2001) 1351–1356.24] L.Z. Boshkov, Ber. Bunsenges. Phys. Chem. 96 (1992) 940–943.