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Van der Waals-like global phase diagrams

Robert L. Scott

Department of Chemistry and Biochemistry, University of California, L os Angeles, California

90095, USA

Received 28th April 1999, Accepted 2nd July 1999

Since the rst global phase diagram (1970) based upon the van der Waals equation of state for binary

mixtures, there have been many newer ones, most recently that obtained by Yelash and Kraska ( Ber.

Bunsen-Ges. Phys. Chem., 1998, 102 213) by combining the Carnahan and Starling hard-sphere equation of

state (J. Chem. Phys., 1969, 51, 635) with the original van der Waals attraction. Of the various such equations

the results for the original van der Waals equation seem to be the least satisfactory in comparison with the

experimental one-component uid, and probably also for binary mixtures. However the Carnahan and

Starling equation diverges only at a density much higher than close-packed. Sanchez (J. Chem. Phys., 1994,

101, 7003) has developed a new 4,3 Pade approximant that ts the virial coefficients of the hard sphere uid

and which diverges almost exactly at the close-packed density. If this expression is used in a van der Waals-like

equation of state, it produces the same kind of closed-loop diagrams that Yelash and Kraska found, althoughover a substantially smaller area of the global phase diagram. Extensions of global phase diagrams to mixtures

of molecules of dierent size (even to polymer systems) seem promising. In particular a simple equation of state

with the Flory mixing equation and the Prigogine entropy parameter (T he Molecular T heory of Solutions,

North Holland, Amsterdam, 1957, ch. XVI) has led to a satisfactory account of tricritical points in

hydrocarbon systems and should be investigated further.

I Introduction

In the early years of this century van Laar 1 used the van derWaals equation to calculate a number of binary-mixturephase diagrams, ones we now distinguish as being of threedierent kinds, depending upon the way critical lines appear

in the pressuretemperature (p,T) projection of the three-dimensional pressuretemperaturecomposition diagram.However, he restricted himself to the geometric mean for theunlike attraction parameter and, presum-[a

12\ (a

11a

22)1@2]

ably because of the difficulty of hand calculations, apparentlynever determined the boundaries between the various types.

For decades the van der Waals equation fell into disrepute,and it was not until 1968 that van Konynenburg and Scott2h4used the equation to determine the rst modern global phasediagram (shown later as Fig. 5). They found that the equationcould account qualitatively for ve of the six major types ofbinary phase diagrams shown in Fig. 1, all but the one thatshows a low-temperature closed loop (later called Type VI5).Since that time, many others have extended the concept,

notably Griffiths and coworkers, who examined the three-component lattice gas6 and improved our understanding ofthe two-component van der Waals mixture7 by nding theshield region.

However, all the known experimental systems of the closed-loop type involve hydrogen bonding between the unlike com-ponents, and the standard explanation of the closed loop hasinvolved the introduction of special orientational forces.Barker and Fock8 used the quasi-chemical model to calculateclosed-loop diagrams, an approach extended recently byHueda, Costas and Scott.9 Similar results have been obtainedfrom other models in a number of studies.10h13 Alternativelyone may obtain the same result by introducing temperature-

Presented at the 1st Workshop on Global Phase Diagrams,(Bunsen-Kolloquium 77), Walberberg, Germany, 21st24th March1999.

dependent equilibrium constants for complex formation,14 oreven more simply by making the parameter in the van dera

12Waals equation temperature-dependent ; however these lastcan hardly be called models.

We restrict the term van der Waals-like equations ofstate to those formed by adding the usual van der Waals

attraction term to an equation of state for hard([

a/Vm)spheres. Earlier work on non-van der Waals-like equations ofstate, e.g., the Lennard-Jones calculations of Boshkov,15 had

Fig. 1 Pressuretemperature projections of p, T, x phase diagramsfor the six major types of binary mixture behavior. The solid lines arethe vapor pressures of the two pure components, the dashed lines arethree-phase lines, and the dotted lines are critical lines.(L

1L

2G)

[Reprinted by permission from ref. 22, Copyright, 1987, AmericanChemical Society.]

Phys. Chem. Chem. Phys., 1999, 1, 42254231 4225

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led to closed-loop behavior without explicit directional inter-actions, but all of these involved some temperature depen-dence of the attraction term. Very recently Yelash andKraska16 have found that, if a van der Waals-like equation ofstate is formed with the Carnahan and Starling equation ofstate17 for hard spheres, the resulting global phase diagramshows a region of low-temperature closed-loop behavior(Types VI and VII).

We propose in what follows to review some of the van derWaals-like equations of state, including a new one, and to

consider the resulting global phase diagrams.

II van der Waals-like equations of state

We consider as van der Waals-like any equation of statethat has the following form:

Z \pVm/RT\f(y) [ y/T* (1)

where p is the pressure, is the molar volume, R is the molarVm

gas constant, and T is the thermodynamic temperature. Thevariable is a dimensionless density, whiley \ b/(4V

m)

T* \ RT b/(4a) is a dimensionless temperature (not to be con-fused with where is the critical temperature); theT/T

cT

cparameters a and b are the usual van der Waals constants.The function f(y) is just the hard-sphere compression factor.

From eqn. (1) we can obtain the molar Helmholtz free energyA

m:

Am

(T, y)/RT\P

f(y) dy/y [ y/T* (2)

A simplifying feature of these equations is that contributionsfrom energy and entropy are completely separate. The termson the right are either temperature-independent (entropy) orinversely proportional to T (energy).

The hard-sphere compression factor Z can be expanded in avirial series, in which the rst eight coefficients have beendetermined, the last several by van Rensburg:18

f(y) \ 1 ] 4y ] 10y2 ] 18.36477y3 ] 28.2245y4

] 39.74y5 ] 53.54y6 ] (3)

We now consider four of the many possible closed-form hard-sphere functions f(y) :

van der W aals

f(y) \1

1 [ 4y\ 1 ] 4y ] 16y2 ] 64y3 ] 256y4

] 1024y5 ] 4096y6 ] (4)

Scott

f(y) \1 ] 2y

1 [ 2y\ 1 ] 4y ] 8y2 ] 16y3 ] 32y4

] 64y5 ] 128y6 ] (5)

Carnahan and Starling

f(y) \1 ] y ] y2 [ y3

(1 [ y)3\ 1 ] 4y ] 10y2 ] 18y3 ] 28y4

] 40y5 ] 54y6 ] (6)

Sanchezvan Rensburg

f(y) \ Z4,3

\1 ] 1.024385y ] 1.104537y2 [ 0.4611472y3 [ 0.743038y4

1 [ 2.975615y ] 3.007000y2 [ 1.097758y3

(7)

The second (Scott) was rst proposed in a reviewchapter19 as an improvement on the original van der Waals

form ; it has no theoretical signicance, but has been usedoccasionally in dealing with mixtures and with global phasediagrams.

The third (C&S) is a weighted average17 of the PercusYevick pressure equation and the scaled particle equation(equivalent in three dimensions to the PercusYevick com-pressibility equation) and gives a remarkably close t of inte-gers to the correct hard-sphere virial coefficients.

The last (SvR) is a Pade approximant determined bySanchez20 to yield on expansion the rst eight virial coeffi-

cients exactly. Sanchez has shown that the approximantZ4,3is fairly robust (distinctly better than his Z3,4

).Of these, the original van der Waals form is clearly the

poorest representation of the real hard-sphere uid.Each of these expressions diverges when the denominator

becomes zero. The rst two diverge at y \ 1/4 and y \ 1/2respectively, well before the density of the close-packed crystalat y \p/(3 \J2) \ 0.740 480. Conversely, the Carnahan andStarling denominator has three degenerate roots at y \ 1, wellbeyond the close-packed density. The Sanchezvan Rensburgdenominator looks very similar to that of Carnahan and Star-ling, but it has only one real root at y \ 0.739 514, sur-prisingly close to that of the close-packed crystal.

Since it is not obvious that a simple combination of a hard-sphere f(y) with the van der Waals attractive energy should bea good representation of a real uid, it is useful to see whatthese combination equations yield for the properties of theone-component uid. Table 1 shows the critical constantsderived from eqn. (1). (For convenience and simplicity we labelthese van der Waals-like equations by the names of theirhard-sphere contributors.) It should be noted how similar arethe critical constants for the last three; only the van der Waalsvalues are substantially dierent. This is surely a result of thefact that, at these relatively low densities, the rst few virialcoefficients are not very dierent.

More serious is the failure of any of these van der Waals-like equations to yield a satisfactory value of the compressionfactor at the critical point of a simple uid ( e.g., argon). Indeedit can be shown that no van der Waals-like equation (e.g., a

hard-sphere set of all-positive virial coefficients combined withthe van der Waals energy) can yield a value of lower thanZ

c1/3.

Figs. 2 and 3 show for each of these equations the reducedvapor pressure and the reduced liquid densityp/p

co/o

c\ y/y

cas a function of reduced temperature Also shown areT/T

c.

the values for the RedlichKwong equation (a favorite ofchemical engineers, but not a van der Waals-like equation)and the experimental values for argon. (It should be notedthat no classical equation such as these can reproduce theexperimental densities near the critical point; experimentallythe curve is nearly cubic, while analytic theories necessarilyyield quadratic curves.)

What is interesting is that, while the Carnahan and Starling

equation yields vapor pressures close to the experimental (thevan der Waals vapor pressure is way o), the situation isreversed for the liquid density; the Scott equation gives betteragreement than either the van der Waals or Carnahan andStarling equations, whose values are equally far from those forargon (in opposite directions).

The values obtained using the Carnahan and Starling equa-tion and those using the Sanchezvan Rensburg equation arevirtually indistinguishable down to reduced temperatures T/T

cof about 0.6, where the computed densities are about threetimes the critical density (i.e., y \ 0.4). Below this point,however, the liquid densities dier increasingly, as shown inFig. 4. The dierences produced by the two dierent denomi-nators appear most strikingly as one approaches zero tem-

perature. The Carnahan and Starling uid not only ends at amuch higher density (corresponding to y \ 1), but the degen-eracy of the roots produces a zero slope (i.e., an innite coeffi-

4226 Phys. Chem. Chem. Phys., 1999, 1, 42254231

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Table 1 Critical constants for various f(y)

yc

\ b/(4Vc) RT

cb/(4a) p

cb2/(16a) Z

c\ (pV/RT)

c

van der Waals 0.083 333 33 0.074 074 07 0.002 314 81 0.375 00Scott 0.133 974 59 0.098 076 21 0.004 809 47 0.366 03Carnahan and Starling 0.130 443 88 0.094 328 70 0.004 416 81 0.358 96Sanchezvan Rensburg 0.130 100 45 0.094 209 84 0.004 400 42 0.359 02Experimental (argon) 0.291

cient of thermal expansion!) Whether any of this will interesta pragmatic experimentalist, who knows that uids do notreach these temperatures and densities, is doubtful.

III Global phase diagrams

Fig. 5 shows the original (1968) global phase diagram deducedfrom the van der Waals equation for mixtures of equal-sized

Fig. 2 The reduced vapor pressure as a function of reduced tem-p/pc

perature for various equations of state: VdWvan der Waals;T/Tc

SScott; C&SCarnahan and Starling; RKRedlichKwong.

Fig. 3 The reduced density of the coexisting liquid as a functiono/ocof reduced temperature for various equations of state: identica-T/T

ctions as in Fig. 2.

spheres slightly revised to show the Griffiths(b

11

\ b

22

),shield region in the center. The usual coordinates for thesediagrams are the following dimensionless parameters:

m \ (b11

[ b22

)/(b11

] b22

) (8)

Fig. 4 The reduced density of the coexisting liquid shown overo/oc

the complete temperature range to S-vRSanchezvanT/Tc

\ 0 :Rensburg; other identications as in Fig. 2.

Fig. 5 The global phase diagram for equal-sized molecules (m\ 0)from the van der Waals equation of state. Except for the addition of

the Griffiths shield region in the upper center, this is the diagramoriginally shown in refs. 2 and 3. [Reprinted by permission from ref.22, Copyright, 1987, American Chemical Society.]


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f\Aa

22b

222

[a

11b

112

BNAa22

b222

]a

11b

112

B(9)

j\Aa

22b

222

[2a

12b

11b

22

]a

11b

112

BNAa22

b222

]a

11b

112

B(10)

The parameter m measures the dierence in size (molarvolume) between the two species ; here we assume that theunlike parameter otherwise one needs ab

12\ (b

11] b

22)/2;

fourth parameter. The parameter f measures the dierence in

cohesive energy (or internal pressure) between the twospecies; for it is simply the ratio of the dierenceb11

\ b22

,between the (low-temperature) energies of vaporization andtheir sum. The third parameter introduces a property of themixture, and j is a measure of the (low-temperature)a

12,

energy (of mixing). The scaling of the parameters is chosen sothat even extreme dierences in the a s and b s can beaccommodated on one graph; moreover, it is only the relativevalues of the a s and b s that determine the parameters.

If i.e., if the unlike attraction is the geo-a12

\ (a11

a22

)1@2,metric mean of the like attractions, then a simple relationexists between the parameters j and f :

(1 [ j)2 ] f2 \ 1 (11)

This describes the half-circle shown in Fig. 5.To use eqns. (1) and (2) for binary mixtures, one adopts a

model in which the mixture is considered as a single uidwith average values anda

mb

m:

am

\ x12a

11] 2x

1x

2a

12] x

22a

22(12)

bm

\ x1

b11

] x2

b22

\ b11M1 ] x

2[2m/(1 [ m)]N (13)

Eqn. (1) can be used unchanged, but to eqn. (2) must be addeda term for the entropy of mixing, traditionally the entropy ofideal mixing:

Am

(T, y, x)/RT\P

f(y) dy/y [ y/T* ] x1

ln x1

] x2

ln x2

(14)

(Note that these simple forms obscure the fact that, for mix-tures, T* contains which is a function of composition, aa

m,

feature that must be recognized when taking derivatives withrespect to mole fraction. Moreover, unless isb

11\ b

22, b

malso a function of composition, and appears in both T*b

mand y.)

Fig. 5 shows the dierent types of phase diagrams found inthe original studies. The labels II-A and III-HA refer to thepresence of azeotropes and heteroazeotropes, respectively, inType II and Type III diagrams. Not distinguished in thisgure, although found by van Konynenburg and Scott, isType a subdivision of Type III with a pressure minimumIII

m,

in the critical line. Of special interest is the small regionlabelled Type IV, in which, near the critical point of com-ponent 1, there is a small three-phase region resulting from akink in the critical line; while no examples of the type areknown for molecules of equal size, the more general globalphase diagram accounts for the Type IV behavior of thesystem methane ] n-hexane.21 Fig. 5 is a revision22 of theoriginal diagram3,4 and includes the shield region aroundf\ 0, j\ 0.45, rst found theoretically by Griffiths andcoworkers6,7 but never observed experimentally.

Other choices for f(y), e.g., the Scott equation, yield a globalphase diagram (Fig. 6) in which the boundaries between typesare shifted quantitatively, but, until recently, no new qualit-ative features were found with any van der Waals-like equa-

tions of state. Thus the results reported by Yelash andKraska16 for the Carnahan and Starling f(y) were more than alittle surprising; that they found Types VI and VII in a region

Fig. 6 Comparison of the global phase diagrams (m \ 0) obtainedfrom the van der Waals equation (dashed line) and from the Scottequation (solid line). The diagrams are symmetrical around f\ 0, soonly the positive f region is shown. The Griffiths shield region hasbeen omitted (see Fig. 5). (Unpublished work.)

just below the f\ 0 line contradicted some of the usual ideasabout low-temperature closed loops. However, given theeccentric properties of the Carnahan and Starling equation athigh densities, it seemed useful to repeat the calculations witha well-behaved hard-sphere f(y), i.e., the previously outlinedSanchezvan Rensburg Pade approximant.

The resulting phase boundaries for it are compared withthose for the Carnahan and Starling equation in Fig. 7. One

still nds Type VI and VII behavior, although in a distinctlysmaller region. Thus it is clear that the unusual behavior thatYelash and Kraska found is not just a result of going to den-sities higher than the close-packed or of the degeneracy of theCarnahan and Starling denominator. However, it should beemphasized that, in these regions of the global phase diagram,where the densities are very high (close to the point of diver-gence of the equation of state), the results are extremely sensi-tive to the exact form of the denominator of f(y).

[After this manuscript was prepared for the WalberbergWorkshop, a paper by Yelash, Kraska and Deiters23 pro-posed two new hard-sphere equations which, although simpler

Fig. 7 A portion of the global phase diagram showing closed-loop

regions obtained using the Carnahan and Starling f(y) (dashed lines)and the Sanchezvan Rensburg f(y) (solid lines). Note the exaggeratedscale of the ordinate j.


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and less accurate than the SvR form, have a factor (3 [ 4y)in the denominator that produces a divergence at y \ 0.75,close to that of the close-packed crystal. When combined withthe van der Waals attraction term, both produce closed loops,and the better of the two (a biquadratic equation rather thana cubic) yields a global phase diagram very similar to thatshown in Fig. 7.]

The closed loops are found mainly at low temperatures andhigh densities, so their signicance for experiments on realuid mixtures is still debatable. For example, no experimental

evidence for closed loops in nonpolar systems is likely to befound until one can get a uid system to a temperature belowthe maximum pressure of the Type VI critical line. In our cal-culations using the Sanchezvan Rensburg f(y), we neverfound the maximum above a reduced temperature ofT/T

c0.25 in the Type VI region; however, in the Type VII region itis found at higher reduced temperatures, ending above

at the cusp between the boundary and theT/Tc

\ 0.5 VmVII

VIIV boundary, what Yelash and Kraska call the doublecritical end cusp (DCEC). If there are any real systems thatfall in this very narrow region of f and j, and if this van derWaals-like equation is appropriate, this temperature is highenough that experimentalists might nd a closed loop in abinary mixture of nonpolar components.

Technical details about some of the calculations will befound in the Appendix.

IV Dierences in size (b22

> b11

)

There have been only a few examples of van der Waals-likeglobal phase diagrams for systems of molecules of dierentsize (m[ 0), most of them coming from our laboratory.3,4,24,25(Some studies of the eect of size dierences have been carriedout for non-van der Waals-like equations of state, e.g., withthe RedlichKwong equation of state by Deiters and Pegg.26)In order to extend global phase diagrams to such systems, inparticular to systems of nonpolar polymers, additionalassumptions or restrictions are advisable:

1. Most molecules are more linear than globular, so the

assumption that is simply the arithmetic mean of andb12

b11

[eqn. (13)] seems reasonable.b22

2. For chain molecules, the Flory mixing equation is prob-ably preferable to the ideal mixing equation. Ideal entropy ofmixing is, of course, correct for the dilute gas mixture, whilethe Flory entropy is surely preferable for the dense liquidmixture. Which is preferable at densities corresponding to thecritical density of a one-component uid is unknown and islikely to remain unknown unless someone can carry out acareful simulation. Most of the critical lines we have calcu-lated are at densities greater than this, so we prefer to use theFlory equation throughout.

3. A corresponding states treatment of molecules of dier-ent length requires the introduction of a third parameter c

(rst introduced by Prigogine27) that allows for the entropydierences produced by a chain molecule with internal degreesof freedom.

A full global phase diagram with three independent param-eters f, j, and m would be three-dimensional, so one usuallyassumes the geometric mean for i.e., eqn. (10). With thisa

12,

restriction, an early calculation3,22 produced the j, m diagramshown in Fig. 8. Although this gure was based upon idealentropy and the van der Waals f(y), it nevertheless accountedqualitatively for the distribution of mixtures of n-alkanesbetween Types II, IV, and III. A later calculation(unpublished), using the Flory entropy of mixing, shifts theboundaries slightly, but leaves the general picture unchanged.

Wei and Scott24 examined the question of whether large

dierences in size, i.e., large values of m, could shift the Grif-ths shield region over to the geometric-mean line. They usedmainly the ideal entropy and found that, at best, the shield

Fig. 8 The van der Waals global phase diagram for systemssatisfying the geometric mean eqn. (11). The solid line is the locus oftricritical points. The dashed line is the boundary between Types IVand III. The dotted lines show boundaries of azeotropic subtypes.Shown on the diagram are the f,m coordinates of various n-alkanesystems. Diagram corrected from the original diagram in ref. 3.[Reproduced by permission from ref. 24.]

region could only occur with very large j values associatedwith large m values of opposite sign, describing this situationas a mixture of Teon and mercury. A few inadequate cal-culations were made with the Flory mixing entropy, but thisresearch needs to be pursued with it as well as with better f(y)functions.

Further research on mixtures of chain molecules (or indeedof any nonglobular molecules) should probably replace eqns.(1) and (2) with the more general ones that Pegg et al.25 used:

Z \pVm/RT\ 1 ] c[ f(y) [ 1] [ y/T* (15)

Am

(T, y, x)/RT\ ln y ] cP

[ f(y) dy [ 1]/y [ y/T*

] x1

ln(x1

b11

/bm

) ] x2

ln(x2

b22

/bm

) (16)

Here c appears in a form suggested by Beret and Prausnitz.28They evaluated the parameter to t experiment, but, at least

for n-alkanes, it can be expressed simply in terms of thenumber of carbon atoms n :

c(n \ 1) \ 1 ; c(n [ 1) \ 1 ] n/3 (17)

Fig. 9 Upper and lower critical-end-point lines in the tricriticalregion of mixtures of ethane with n-alkanes. (Here is the number ofn

2carbon atoms in the higher hydrocarbon or the average value in theternary quasi-binary mixture.) The two lines converge to the tri-critical point and the region between the two is where three phases

can occur. The solid lines are calculated from eqn. (16). The dashedlines show a selection of experimental points for binary and quasi-binary mixtures.29,30 [Reproduced by permission from ref. 25.]


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Then for the mixture is simply the mole fraction average ofcm

andc1

c2

cm

\ x1

c1

] x2

c2

(18)

Pegg et al.25 used this equation [with the van der Waalsf(y) and no adjustable parameters] with remarkable success tointerpret the behavior of mixtures of n-alkanes in the vicinityof tricritical points. Fig. 9 shows the experimental29,30 andtheoretical upper and lower critical-end-point lines for binaryand quasi-binary mixtures of ethane with mixtures of higher

hydrocarbons from to Eqn. (16) predicts the tri-C16 C20 .critical point to be at the higher hydrocarbon whilen2

\ 16.48the experimental value occurs at 17.61. With the Scott equa-tion, one gets (unpublished). What one would getn

2\ 17.3

with other (better ?) choices of f(y) is unknown, and suchgames should probably not be encouraged.

How successfully this approach can be generalized to treatdierences of size in other kinds of chain molecules remains tobe seen.

V Some general comments

Anyone working in the eld of global phase diagrams is surelyaware of two serious problems:

1. No simple equation of state can ever be expected to yielda quantitative t to experimental results. We are lucky whenwe get a reasonably semi-quantitative understanding.

2. The known experimental results for binary mixtures ofnonpolar nonelectrolytes cover only a small area of the globalphase diagram (at or above the geometric mean for the unlikeattraction). Unfortunately this probably means that some ofthe interesting phase diagrams deduced theoretically willnever be found experimentally, at least not for nonpolarsystems.

Indeed it is likely that most, if not all, of the nonpolarsystems classied as I or V are really II or IV, with the low-temperature upper critical solution temperature (UCST)hidden below the solid line. This should be the case wheneverthe low-temperature enthalpy of mixing is known to be posi-tive. Dickinson et al.31 used the van der Waals and Scottequations of state to show that the atness of the freezingcurve of the system methane ] n-hexane can easily be inter-preted in terms of metastable liquidliquid phase separation 5to 15 K below.

Acknowledgements

wish to thank all of my collaborators over the years whoIhave contributed to our work on global phase diagrams, inparticular (but not exclusively) Peter van Konynenburg, IanL. Pegg, and my colleague Charles M. Knobler.

AppendixA general picture of the character of a global phase diagramcan be obtained by examining phase diagrams along the linej \ 0. This line necessarily separates Types II, IV, andIII

m,

III (occurring only with positive j values) from Types I andVand maybe VI, VII, and only with nega-V

m(occurring

tive j values). The tricritical line necessarily crosses the j\ 0line, but others may not.

An important feature of the j\ 0 line is the termination ofthe critical line at the maximum (i.e., at the divergence ofy

0the denominator). This occurs necessarily at T\ 0, but thepressure at this limit depends upon the parameter f and thep

0choice of the equation of state. In particular one can derive ananalytic expression for the limiting mole fraction for anyx

0f(y) : If one writes thenw0

\ 1 [ 2x0

,

F0

w02 ] y

0G

0@ (1 [ w

02) \ 0 (19)

where is the numerator of the selected f(y), and is theF0

G0@

rst derivative (with respect to y) of the denominator, bothevaluated at Eqn. (19) is linear in (quadratic in andy

0. w

02 x

0)

is easily solved; the resultant (or is the same for allw0

x0

)values of f. This can then be used to obtain for anyw

0p

0particular value off.

p0/p

c\

y02(f[ w

0)

pc*w

0(1 [ f)

(zero j only) (20)

where is the critical pressure of component 1 and ispc

pc*

taken from Table 1. Evidently changes sign atpcb2/(16a) p0and the character of the phase diagramf\ w

0\ (1 [ 2x

0),

will change there.For the Carnahan and Starling f(y) with its degenerate

denominator, is zero, so and turns outG0@ w

0\ 0, x

0\ 1/2, p

0to be positive innity for all values of f. For all the other f(y),with a nondegenerate denominator, can be positive orp

0negative, but is normally nite.

This termination is important in recognizing dierent typesof phase diagrams, because for any positive value of j, thecritical line pulls away from the T\ 0 axis and goes top

0positive innity at a corresponding to the critical solutionT

0temperature of a close-packed simple mixture at (i.e.,x

0\ 1

2one gets only Type II, III, or IV phase diagrams). Conversely,

for any negative value of j, the critical line ends at T\ 0,and a negative pressure of nite magnitude that isx

0\ 0 p

0independent of the values of f and j (as long as j is negative)and a function only of the choice of equation of state:

p0/p

c\ y

02/p

c* (negative j only) (21)

[It may be noted that eqn. (21) is the limiting value of eqn.(20) for f\ 0 and is necessarily at a more negative pressurethan that given by eqn. (20) for any nonzero f.]

With the van der Waals or Scott f(y), the andIVIIIm

boundaries end asymptotically on the j\ 0 line atIIImIII

T\ 0, From eqns. (19) and (20) one nds that thep0

\ 0.boundary ends atIVIII

mf\ w

0\ 1 [ 2x

0\ 1/J2 \ 0.7071

with the van der Waals f(y) and at f\ w0

\ 1 [ 2x0

\

with the Scott f(y). The asymptotic termina-1/J3 \ 0.5774tion of these lines makes Type impossible and, in addition,Vm

precludes Types VI and VII for these equations of state.With the Sanchezvan Rensburg f(y) and j\ 0, changesp

0sign at f\ 0.332 934 and the phase diagramx

0\ 0.333 533,

changes from Type I to Type II; moreover, in contrast to thevan der Waals or Scott diagrams, this is the asymptotic end ofa line that extends into the region of negative j values,marking the boundary between Types I and VI.

It is interesting to note that this line separating Types I andII, III and V, etc., which for all van der Waals-like equationsof state coincides with the j\ 0 line, can be dierent for otherequations of state. Thus Deiters and Pegg,26 who call this thezero-Kelvin endpoint curve, found that for the Redlich

Kwong equation, it did not coincide with the j\

0 line, butwas found below that line (except at f\ 0).

References

1 J. J. van Laar, Proc. Sect. Sci., K. Ned. Acad. W et., 190405, 7,517, 636, 646; 190506, 8, 33; 190607, 9, 226.

2 P. H. van Konynenburg, PhD Dissertation, UCLA, 1968.3 R. L. Scott and P. H. van Konynenburg, Discuss. Faraday Soc.,

1970, 49, 87.4 P. H. van Konynenburg and R. L. Scott, Philos. T rans. R. Soc.

L ondon, 1990, 298, 495 (includes a major fraction of ref. 2).5 P. Clancy, K. E. Gubbins and C. G. Gray, Faraday Discuss.

Chem. Soc., 1978, 66, 116.6 D. Furman, S. Dattagupta and R. B. Griffiths, Phys. Rev. B: Solid

State, 1977, 15, 441.

7 D. Furman and R. B. Griffiths, Phys. Rev. B: Solid State, 1978,17, 1139.

8 J. A. Barker and W. Fock, Discuss. Faraday Soc., 1953, 15, 188.


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9 Y. Hueda, M. E. Costas and R. L. Scott, J. Phys. Chem. B, 1997,101, 8676.

10 G. R. Anderson and J. C. Wheeler, J. Chem. Phys., 1978, 69, 2082.11 J. S. Walker and C. A. Vause, J. Chem. Phys., 1983, 79, 2660.12 L. R. Corrales and J. C. Wheeler, J. Chem. Phys., 1989, 91, 7097.13 G. Jackson, Mol. Phys., 1991, 72, 1365.14 V. Talanquer, J. Chem. Phys., 1992, 96, 5408.15 L. Z. Boshkov, Dokl. Akad. Nauk SSSR, 1987, 294, 901.16 L. V. Yelash and T. Kraska, Ber. Bunsen-Ges. Phys. Chem., 1998,

102, 213.17 N. F. Carnahan and K. E. Starling, J. Chem. Phys., 1969, 51, 635.18 J. van Rensburg, J. Phys. A, 1993, 26, 4805.

19 R. L. Scott, in Physical Chemistry, An Advanced T reatise, ed. D.Henderson, Academic Press, New York, 1971, vol. 8A, ch. 1, p. 1.

20 I. Sanchez, J. Chem. Phys., 1994, 101, 7003.21 A. J. Davenport and J. S. Rowlinson, T rans. Faraday Soc., 1963,

59, 78.22 R. L. Scott, Acc. Chem. Res., 1987, 20, 97.

23 L. V. Yelash, T. Kraska and U. K. Deiters, J. Chem. Phys., 1999,110, 3079.

24 I. C. Wei and R. L. Scott, J. Stat. Phys., 1988, 52, 1315.25 I. L. Pegg, C. M. Knobler and R. L. Scott, J. Chem. Phys., 1990,

92, 5442.26 U. K. Deiters and I. L. Pegg, J. Chem. Phys., 1989, 90, 6632.27 I. Prigogine, T he Molecular T heory of Solutions, North Holland,

Amsterdam, 1957, ch. XVI.28 S. Beret and J. M. Prausnitz, AIChE J., 1975, 21, 1123.29 M. C. Goh, J. Specovius, R. L. Scott and C. M. Knobler, J. Chem.

Phys., 1987, 86, 4120.30 M. C. Goh, R. L. Scott and C. M. Knobler, J. Chem. Phys., 1988,

89, 2281.31 E. Dickinson, C. M. Knobler and R. L. Scott, J. Chem. Soc.,

Faraday T rans. 1, 1973, 69, 2179.

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