2. Static Properties of SolutionsVan der Waals and Related Models for HydrocarbonMixtures * -f-BY ROBERT L. SCOTT AND PETER H. VAN KONYNENBURG 1Received 6th January, 1970The van der Waals equation of state, in spite of its oversimplifications, gives useful qualitativeinformation about mixtures over a wide range of temperatures and pressures. When the Bronstedprinciple of congruence is used to evaluate the parameter a12 for the mixture, a wide range of prop-erties can be predicted : excess functions (including temperature and composition dependence) andphase eqiulibria (including lower critical solution phenomena at high temperatures), in good quali-tative agreement with experimental properties of mixtures of n-alkanes.Mixtures of n-alkanes serve, in a certain sense, as low molecular weight analoguesof polymer solutions and the extensive experimental data can be used to test varioustheoretical models. General qualitative features of alkane mixtures, which anysatisfactory model must produce, include : (a) the molar excess volume YE is negativeat all temperatures, increasingly so at higher temperatures, and with a minimum skewedin the direction of mixtures rich in the smaller component.(b) The molar excessenthalpy is small and positive at low temperatures and becomes negative at highertemperatures ; in a short intermediate range, the curve of RE against mol fraction isS-shaped. (c) For certain mixtures (e.g., CH4 + n-C6HI4), lower critical solutionphenomena occur near the gas-liquid critical point of the more volatile (smaller)component.Many of these properties can be interpreted in terms of the principle of congruencean extended theory of corresponding ~ t a t e s , ~ ' ~ or the new Flory equation of state.6*For several years we have been investigating the properties of van der Waals mixturesat elevated temperatures and pressures, particularly with respect to phase behaviour,8and have found that this model also reproduces-with a reasonable choice of para-meters-the qualitative behaviour of n-alkane mixtures.Because of the simplicityof the van der Waals equation and the physical reasonableness of its Q: and b parameters,it is useful to show how it leads to the observed behaviour of hydrocarbon mixtures.THE VAN DER WAALS EQUATIONThe equation of state of van der Waals was proposed in 1873 and was laterextended to binary mixtures lo in the formwhere p ,Allied Chemical Corporation.Angeles, California 90024, U.S.A.Pennsylvania 16802, U S A .p = RT/(P-b,,,)-a,/V2 (1)T, and R are the pressure, molar volume, thermodynamic temperature, and* This work has been supported by grants from the U.S.National Science Foundation and the=f Contribution No. 2522 from the Department of Chemistry, University of California, Los$ present address : Department of Chemistry, Pennsylvania State University, University Park,888 VAN DER WAALS MIXTURESmolar gas constant respectively. The parameters am and b, are usually written asquadratic averages for the binary mixture :a, = x~all+2xlx,a12+x,2a22, ( 2 )b, = xibl1+2~1x,b,z+xib,2, (3)where x1 and x2 are the mol fractions of the two components, and the parametersa l j and blj correspond to the appropriate pair interactions.In terms of modernpair potential energy functions u(r) for spherically symmetric molecules, a i j should beproportional to ~~~c~~ and b l j to clJ where E is the depth of the potential energy wellat its minimum and c the collision diameter. Recently, Leland, Rowlinson, andSather l 1 have used the Percus-Yevick approximation to support the argument thatthe prescriptions of eqn (2) and (3) are the most appropriate to use in a " one fluid "corresponding states treatment of mixtures.For mixtures of spherical molecules, for which o12 = (ell + 0 ~ ~ ) / 2 is a goodapproximation, the Berthelot combining rule b12 = (bt, + b$,)* would be appropriate,but for chain molecules the original van der Waals assumption that b, = (b, +bZ2)/2seems more reasonable.Then eqn (3) reduces tobm = Xlbll +xzbzz. (4)This leads to the conclusion that volumes are additive at the low temperaturelimit .As van der Waals showed, eqn (1) leads directly to an expression for the Helmholtzfree energy 2 for 1 mol of a binary mixture :~T~(T,Ex~)-.X,"(T,VO) = R T [ x ~ In x1 +xZ In xZ]-RT In [(T-bm)/Vo]-a,/E (5)where $(T,8") is thereference state of the unmixed components as ideal gases ofmolar volume vo. An equivalent development leads to an equation for the chemicalpotential p 1 of component 1 :p l ( ~ ~ x l ) - p ~ ( T , ~ o , l ) = RT In x,-RT In [ ( ~ - b m ) / ~ o ] +~ ~ [ b l 1 - ~ , 2 ( b 1 1 - 2 b l 2 + b2 2 ) I / ( r- bm) - 2(xla11+ x2a12)/ K (6)and a corresponding expression for component 2.The van der Waals equation is only an approximation at best and cannot beexpected to yield more than qualitative agreement with experiment, even for sphericalmolecules. It has the advantages, however, that (a) it is simple and its two parametershave a fairly clear physical meaning, (b) once the parameters are chosen, eqn (5) (or(6)) leads to explicit and unambiguous mathematical predictions, and (c) it yieldsqualitatively reasonable results over the entire range of temperature and pressure,including the coordinates of critical points (although not the correct form for functionsin the critical region).For mixtures of chain molecules, eqn (1-6) have the disadvantages that (a) withoutthe addition of a third parameter (e.g, the Prigogine-Hijmans-Flory c), they predictthat the critical compression factor ( p V/RT)" is invariably Q in disagreement with theexperimental evidence that for n-alkanes it decreases regularly with increasing chainlength, and (b) the special statistical effects of mixing chain molecules (e.g., the" Flory-Huggins " entropy) are not included, and there appears to be no entirely self-consistent way to add these.In the absence of correction terms, we may expectincreasing divergence between van der Waals predictions and experiment as the chain-length of the n-alkane increasesROBERT L . SCOTT A N D PETER H .VAN KONYNENBURG 89THE VAN DER WAALS PARAMETERSThe parameters a and b can be evaluated in various ways : from the properties ofthe gas at moderate pressures (e.g., the second virial coefficient B = b -a/RT) ; fromthe critical constants p", p, and T" : or from the properties of the liquid at low temp-erature and (essentially) zero pressure; e.g., the molar volume the coefficient ofthermal expansion a = (3 In r,aT),, the thermal pressure coefficient y = (aplaT),.Were the equation of state exact, all these methods would be equivalent and wouldlead to the same values for a and b. However, since the equations are only approx-imate the parameters selected will depend upon the method of evaluation. If one isinterested primarily in the dense fluid at low temperatures, it is preferable to use onlyproperties at high densities and low temperatures to determine a and b.Since we havebeen interested in the critical region as well we have elected to use a different method.Williamson and Scott l 2 have shown that the product of the liquid molar volume Pand the standard molar energy of vaporization AVOo, which should equal the van derWaals a, when evaluated at 25°C for n-alkanes from C5 to C16, closely fits the simpleequationwhere n is the number of carbon atoms in the n-alkane. An even better fit is obtainedby using the general quadratica = VAuDo = (81.065 kJ cm3 m0l-~)(n+0.879 5)2,a = (158.28+ 121.200 n+82.133 4 n2) kJ em3 mok2.(7)(8)Given these a and the experimental p a t 25"C, one may calculate the b from eqn(I), obtaining the simple linear fit,An even better fit is given by the quadratic equationb = (16.44 cm3 mol-l)(n+ 1.32) cm3 mol-l.b = (22.78+ 16.22 n+0.01 n2) cm3 mol-l.(9)(10)An alternative expression for b, V-R/y, evaluated at 25"C, yields an equationb = (16.5 cm3 mol-l)(n+ 1.33), with constants similar to those in eqn (9).Unfortun-ately, other procedures (e.g., using the critical constants) yield different results,showing more complex dependence of a and b upon n. For illustrative purpose weshall use the values of a and b given by eqn (7) and (9), but this uncertainty about the" best " values of a and b, reflecting the inexactness of the equation of state, meansthat detailed numerical agreement between theory and experiment is not to be expectedand, if found, may be at least partly fortuitous.If we adopt the principle of congruence, according to which the properties of ann-alkane mixture are set equal to the properties of the pure n-alkane of averagen = xlnl +x2n2, then the parameters a,n and b, are obtained by substituting ii into theappropriate equations for a and b.When this procedure is applied to eqn (7) and(9), and compared with the general eqn (2) and (3), one finds the geometric mean fora,, = (a11a2,)* and the arithmetic mean for b,, = (bll+b2,)/2 and thus eqn (4).The more nearly exact eqn (8), when combined with the principle of congruence, leadsto aI2 values slightly smaller than the geometric mean.-THE EXCESS FUNCTIONSFrom eqn (I) and (4) we can derive the various thermodynamic functions formixing at constant temperature and pressur90 VAN DER WAALS MIXTURESExcept for the entropy and the Gibbs free energy, where the ideal solution term mustbe subtracted, eqn (11) yields the excess functions themselves.While these are bestcalculated directly, it is useful for qualitative understanding to look at the variousexcess functions as power series expansions in temperature and pressure.cE = - A(a/b) - RTA In (pob2/a) - (RT)2A(b/a) -@T)'A(b2/a2) + . . .RE = - A(u/b) + (Rn2A(b/a) + 2(RT)3A(b2/a2) + . . .+p[(Ab+ RTA(b2/u) +2(RT)2A(b3/a2) + . . .] - . . . (12)(13)(14)+p[Ab - 2(RT)2A(b3/a2) - . . .] + . . .TSE = RTA In (pob2 la) +- 2(RT)2A(b/a) -k 3(RT)3A(b2/a2) -I- . ..-p[RTA(b2/a) + 4(RT)2A(b3/a2) + . . .] + . . . .PE = Ab+RTA(b2/a)+2(RT)2A(b3/a2)+ . . . -p[RTA(b4/a2)+ . . .I+. . . (15)where A( ), following eqn (1 l), means the difference between the quantity for themixture and the weighted average for the unmixed components, i.e. A(a/b) = (a,,,/b,)- xl(al 1) - ~ ~ ( a ~ ~ / b ~ ~ ) , etc. and where po is a reference pressure (here irrelevantsince it cancels out).The general behaviour of the excess functions will thus depend upon the propertiesof the A-functions, their signs, magnitudes, and concentration dependences. Formixtures of n-alkanes and probably for many other homologous series, all the A-functions are invariably negative and include a factor xIx2(nl -n2)2 which makes thefunction roughly parabolic.A further factor dependent upon x, n1 and n, skews thecurve (except for Ab, which if non-zero is simply parabolic) so that the minimumoccurs at a mixture richer in the component of smaller n. The " higher " A-functionsare somewhat more skewed than is A(a/b).If we consider the excess functions at low pressures where p-dependent terms arevirtually negligible, we conclude that : (a) the excess Gibbs free energy GE is invariablypositive for all x and increases with temperature. The function GE/RT decreaseswith temperature initially (from a hypothetical value of +co at T = 0) and thenincreases again. If we take GE/RT = 3 as a rough criterion for liquid-liquid phaseseparation,l this guarantees partial miscibility at sufficiently low temperatures (butprobably below the melting curve for many systems so not experimentally observed)and raises the possibility of two critical solution points in one system, an upper(" normal ") one at low temperatures and a lower one at higher temperatures, with aregion of complete miscibility between.(b) The excess entropy gE and the excessvolume VE are invariably negative for all x, and decrease (increase in magnitude) withincreasing temperature. The skewing of the curves becomes more marked as Tincreases. (c) The excess enthalpy is positive at low temperatures and decreases as Tincreases, be_coming negative at higher temperatures. In the low temperature limit(i,e., when V z b,) the leading term, - A(a/b) in eqn (13) is the only significant one.With the linear average of eqn (4) for b,, eqn (13) then reduces to the van Laar equa-tion :or, with eqn (7) and (9) for a and b,(17)xlb, 1 ~ 2 b 2 2 - (0.95 kJ mol-')(n, - n2)2x1~2(xlbll +x2bZ2) - (nl+1.32)(n2+1.32)(ii+1.32)'The right-hand side of eqn (17) must necessarily be positive, but the numerical valueis much too small to account for the magnitude of the positive RE observed for alROBERT L .SCOTT A N D PETER H . VAN KONYNENBURG 91n-alkane mixtures at low temperatures. Use of the more nearly accurate eqn (8)increases the small dimensional factor in eqn (17) to 8.6 kJ mol-l, but even this issomewhat too small to fit the heats of mixing at 25°C. This fact is in accord with theobservation that the principle of congruence leads to too small values of RE whenthe heats of vaporization of the pure n-alkanes are used.Of the A-functions, only Ab, which for eqn (9) is exactly zero but for eqn (10) isweaklynegative[Ab = - (0.1 cm3 mol-')(n, -n2)2x1x2], and A(a/b), whichisvery small,are extremely sensitive to the precise specification of a12 and b12.In view of this, itseems appropriate to regard the dimensional constant in eqn (17) for -A(a/b), andmaybe that in Ab also, as empirically adjustable to fit the low-temperature excessfunctions. The other A-functions, which determine the variation of the excessfunctions with temperature and pressure, are almost completely insensitive to smallvariations in a12 and b12 and can be evaluated from the simpler eqn (7) and (9).The second term in eqn (13) is proportional to ( R n 2 , there being no linear term, sothe excess enthalpy will decrease rapidly as it becomes negative.Moreover, sinceA(b/a) is more skewed than A(afb) there will be a narrow intermediate range oftemperature in which the curve of gE against x is S-shaped, with negative values formixtures rich in the component of smaller n and positive values for mixtures rich inthe other component.All of these predictions correspond qualitatively with the experimental data onhydrocarbon mixtures; the fact that the experimental excess properties change morerapidly with temperature than predicted is a measure of the inexactness of the vander Waals equation of state.PHASE EQUILIBRIAOur interest in the van der Waals equation applied to binary mixtures was initiallydirected almost entirely toward the problem of phase equilibrium at elevated temp-eratures and pressure.Since eqn (1)-(4) define analytic functions at all temperaturesand pressures, the usual thermodynamic conditions for equilibrium between phases ccand p [T" = Tb, pa = pB, pf = p?, and p; = pi] may be applied and the equationssolved explicitly. In particular, one may obtain complex equations defining thecritical lines in a three dimensional p,T,x diagram.Sixty years ago van Laar l4 investigated the phase behaviour of van der Waalsmixtures, but he was handicapped by the difficulties of hand-calculations and by hisself-imposed restriction to the geometric mean for a12. Now with the availability ofhigh speed computers it is possible to examine the full implications of the van derWaals equation, i.e., to determine phase equilibria for all possible values of the a and b.Van Konynenburg * has calculated a large number of p,T,x diagrams covering a widevariety of values of a, mostly for b, = b22, but some for b22 = 2bl ,.A brief r6sumCof his work must suffice here.We classify the various types of phase diagrams according to the nature of theirP,T projections,15 in particular by the presence or absence of three phase lines andazeotrope lines, and by the way critical lines connect with these. Critical lines mayend in various ways : at a one-component gas-liquid critical point C1(T;C,pE,x2 = 0)or C2(T&p&x2 -= l), at the limiting upper critical solution point C,(T,,p = CO,~,) in aclose-packed ( V = b,) system, at the end of a three-phase line (L1L2G) with an upperor lower critical end point (UCEP or LCEP).Azeotrope lines, if present, end tan-gentialiy on critical lines. In some cases a critical line is essentially gas-liquid(G-L, Vc M 3b,) or liquid-liquid (L-L, Vc = b,) over its entire course ; in other casesit changes continuously from one type to the other. Nine major types are distin-guished 92 VAN DER WAALS MIXTURES1, one critical line (G-L), C , to C2.I-A, same as I, with the addition of a negative azeotrope.11, two critical lines : CI to C2 (G-L) ; C, to UCEP (L-L).11-A, same as 11, with the addition of a positive azeotrope.111, two critical lines : C, to UCEP (G-L); C, to C2(L-L to G-L).111-HA, same as 111, except that the three phase line lies at lower pressures thaneither pure component, producing a “ heteroazeotrope ” type diagram.IV, three critical lines: C1 to UCEP (essentially G-L); LCEP to C,(L-L) tov, two critical lines : C, to UCEP (essentially G-L) ; LCEP to Cz (L-L toV-A, same as V with addition of a negative azeotrope.If we assume b,, = (bI1 + b2,)/2, but not the geometric mean for a, 2, any van derG-L; C , to UCEP (L-L).G-L).Waals mixture can be specified by three parameters, 5, c, and AWithin the limitations of the van der Waals equation, 5 is the difference of thecritical molar volumes divided by the sum, while 5 is the difference of the criticalpressures divided by the sum.The usefulness of the parameter A is that it is the onlyone which includes a12 and is proportional to the low-temperature limit for RE(i.e., containing the difference factor in - A(a/b)).Fig.1 summarizes van Konynenburg’s results for bl = b2, (5 = 0). Only positivevalues of c are shown, for the diagram is symmetrical with respect to reflection throughthe c = 0 axis. The dashed line represents those mixtures for which a,, = (al1aZ2)*,the geometric mean; only types I1 and I11 diagrams are found for such systems(when b , , = b22) ; one must relax the combining rule for a,, to find the other seven.The especially interesting type IV with both upper and lower critical solutionphenomena occurs only in a narrow region where (a, la22)* <al2 <(a, , + a2J/2.Systems for which b l l # b2, have not been investigated so thoroughly, but ourresearch programme is continuing.Results for 5 = 5. and for the special case ( = 0indicate that the general form of fig. 1 is topologically invariant, but that the positionsof the regions are shifted quantitatively. In particular, for > 1 and > 1, the wedge-shaped region for type IV moves across the locus of geometric means for a,,, so thatlower critical solutim phenomena are now possible for systems satisfying eqn (7).Since the n-alkane systems do not deviate markedly from the geometric mean fora12, we may use that relation to eliminate the parameter A from the set 5, c, A withrelationThen the behaviour of such systems can be represented by fig. 2. The coordinates ofseveral n-alkane systems are indicated on the figure.(All are calculated using eqn (7)and (9), although the experimental data on methane and ethane do not fit them verywell.)When the two alkanes do not differ too much in size (e.g., c6 + C , 6), the phasediagram will be type 11, but as increases, one passes through type IV, with its upperand lower critical solution temperatures, to type 111, i.e., from ‘‘ complete miscibility ”(1-A),+r2 == 1. (21ROBERT L . SCOTT AND PETER €3. VAN KONYNENBURG 93IIFIG. 1.-Types of phase equilibria for iiiixlures of molecules of equal size (E = 0). Nine majorregions of characteristic p,T,x phase diagrams (see text) are separated by the full lines. The dashedline is the locus of the geometric mean for a12 (eqn. (21)).Negative values of are not shown ; thefull diagram is symmetrical around the < = 0 axis.FIG. 2.-Types of phase equilibria for mixtures obeying the geometric mean for a12 (eqn. .(21)).Negative values of are not shown because the origin is a centre of symmetry. < and 4 coordinatesare shown for a few binary n-alkane mixtures94 VAN DER WAALS MIXTURESthrough " limited miscibility " to " absolute immiscibility " in the nomenclature ofDavenport and Rowlinson.16 The fact that the boundaries sketched in fig. 2 roughlyfit the methane systems (where C,+C, is type IV) is probably fortuitous, for theethane systems do not fit so well (experimentally 17* l8 CZ+Cl9 to C2+C22 are typeIV). For solutions of polymethylene in n-alkanes (e.g., C6 + Cm), the simple van derWaals theory predicts type I11 instead of the type IV 0b~erved.l~ This increasingdisagreement for very long chain lengths may be due to the disregard of correctionsfor chain entropy effects (e.g., both the parameter c and the configurational mixingterm).TI I1 I I I r I I02' 3:: 3.6 3 -7 3 8azzh 1FIG. 3.-The locus of critical end points for systems changing continuously from type 11 throughtype 111 to type IV.The points along the curve represent exact calculations of end points forsystems lying along the dotted line in fig. 1. The reduced temperature Tr = T/T,O.Fig. 3, calculated from the van der Waals model (for = 0), illustrates how theupper and lower critical end points coalesce as one crosses the type IV region fromtype I1 to type TIT.For most hydrocarbon mixtures the low temperature upper criticalend point is hidden below the melting curve, but it seems clear that the basic behaviouris type IV, not type V. Not only do the positive excess enthalpies at low temperaturerequire this, but upper critical end points have now been found in a number of methane+hydrocarbon systems 8 s 16* 2o ; fig. 4 shows how the two critical solution points movecloser together as the solubility parameter 6 (proportional to af/b) of the hydrocarbonincreases. There are different curves for C6, C,, and Cs because these systems havedifferent values of 5 ; since both alkanes and alkenes show the same general behaviourthe phenomenon is clearly not restricted to a special class of hydrocarbonsROBERT L.SCOTT AND PETER H. VAN KONYNENBURG 95Even more striking confirmation of this general form is provided by measurementson several " quasi-binary " mixtures of methane with a pair of CG-isomers. Fig. 5shows the variation of the critical end points with the proportions of 2-methylpentaneand 2-ethyl-1-butene in a ternary mixture with methane. (Similar results are obtainedfor the system methane + 3,3-dimethylpentane + 2-methylhexane.)200I90I eo170I60M L-150140I30I20I10~ - H E X A N E\ \ A I-HEXENE2,2-DIMETHYLPENTANE 4,2,4 - 0 I ME THY L PEN TAN E \\ \ \ o 2-METHYL-I-PENTENE12,2,4 - T RI M E THY L -\ \\ PENTANE 03,3-OIMETHYLPENTANE 0\FIG. 4.--Correlation of upper and lower critical end points in methanefhydrocarbon systems with thesolubility parameter 6 of the hydrocarbon.A Davenport et 2o ; 0, van Konynenburg.8 Thesolid lines are not the best fit of the data but represent experimental measurements okternary mix-tures such as that shown in fig. 5.OTHER MODELS AND POSSIBLE REFINEMENTSMost of the conclusions here developed from a van der Waals model have beenderived from other theoretical treatments. It is not our intention to argue that thevan der Waals equation of state is superior to others which have been used. Thereare better equations of state which, with reasonable choices of several adjustable par-ameters, should (and do) give better agreement between calculated and observedproperties. The importance of these van der Waals calculations is that with (a) avery simple equation of state, (b) a simple dependence of the two parameters a and bupon n, and (c) the principle of congruence, they lead to qualitatiw agreement withexperimental behaviour over virtually the whole range of temperature, pressure, andcomposition. From the convergence of the results of the various theoretical ap-proaches the phenomena observed are of quite general significance and the pre-dicted behaviour of alkane mixtures is closely related to the actual equation of stateof the pure hydrocarbons and should not be sensitive to the special parameters or thespecial forms of any particular theory96 VAN DER WAALS MIXTURESThis van der Waals model might be improved (in the sense of better fit with experi-ment and closer approach to other theories) by the following refinements.(a) Theintroduction of a special a,,'not in accord with the principle of congruence. Thisseems to be necessary in order to fit RE at low temperatures; Orwoll and Floryfound it necessary to introduce an empirically adjusted X , , for a similar purpose.FIG. 5.-Upper and lower critical end points in the ternary system methane+Zmethylpentane+2-ethyl-1-butene. f3 is the relative mol fraction x3/(x2 +x3). x indicates the gas-liquid criticaltemperature of methane.(b) The replacement of RT( r- b m ) in eqn (1) by c,RTf( v/bm), wheref( p/bm) is chosen 21to give a better fit to the known results for the equation of state of hard spheres, andc, is a function of fi which introduces a scaling parameter for the entropy of a chainmolecule.Unfortunately, Cm must in reality be a function of density, for it must havethe value unity in the perfect gas region, another value at the critical density, and stillanother value at normal liquid densities. (c) The replacement of the configurationalfree energy of ideal mixing RT[x, In xl +x2 In x,], in eqn ( 5 ) by the Flory expression,RT[xl In (xlbl /bm) +x2 In (x2b22/bm)]. Unfortunately this too, while an improvementat liquid densities, would be inappropriate for dilute gas mixtures.These refinements would effect marked improvements in the fit of theory andexperiment in certain regions of temperature, density, and composition. HoROBERT L. SCOTT AND PETER H . VAN KONYNENBURG 97seriously these would affect the predicted thermodynamic behaviour in other regions(e.g., the critical region) is unknown.At present we are surprised and gratified thatthe unrefined van der Waals model is so successful and informative.l J. N. Bronsted and J. Koefoed, Kgl. Danske Videnskab. Selskab, Fys. Med., 1946,22, no. 17.I. Prigogine, The MoZecuZar Theory of Solutions (North-Holland, Amsterdam, 1957), chap. 17.Th. Holleman and J. Hijmans, Physica, 1962, 28, 604 ; ibid., 1965, 31, 64.S. N. Bhattacharyya, D. Patterson and T. Somcynsky, Physica, 1964, 30, 1276.D. Patterson and G. Delmas, Trans. Furaday SOC., 1969,65,708. ' P. J. Flory, R A. Orwoll, and A. Vrij, J. Amer. Chem. SOC., 1964,86, 3507. ' R. A. Orwoll and P. J. Flory, J. Amer. Chem. Sac., 1967,89,6814.P. H. van Konynenburg, Critical Lines and Phase Equilibria in Binary Mixtures (Ph. D. Diss.,U.C.L.A.), 1968.J. D. van der Waals, On the Continuity of the Gaseous and Liquid States (Ph.D. Diss., Leiden,1873).lo J. D. van der Waals, Z. phys. Chem., 1890,5, 133.l1 T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans. Faraday Soc., 1968, 64, 1447.l2 A. G. Williamson and R. L. Scott, Trans. Faraday SOC., 1970, 66,335.l3 J. L. Copp and D. H. Everett, Disc. Faraday SOC., 1953, 15, 174.l4 J. J. van Laar, Proc. Akad. Wetenschappen (Proc. Section Sci.), 1906,9,226 ; ibid., 1907,10,34.J. S. Rowlinson, Liquids and Liquid Mixtures, 2nd ed. (Butterworth, London, 1969), chap. 6. '' A. J. Davenport and J. S. Rowlinson, Trans. Faraday SOC., 1963,59,78.l7 J. P. Kohn, Y. J. Kim, and Y. C. Pau, J. Chem. Eng. Data, 1966,11, 333.l 8 A. B. Rodrigues and J. P. Kohn, J. Chem. Eng. Data, 1967, 12,191.l9 P. I. Freeman and J. S. Rowlinson, Polymer, 1960,1,20; C. H . Baker, W. Byers Brown, G.Gee, J. S. Rowlinson, D. Stubley, and R. E. Yeadon, Polymer, 1962, 3, 215; G. Allen andC. H. Baker, Polymer, 1965, 6, 181.E. A. Guggenheim, Mol. Phys., 1965,9,43, 199.2o A.. J. Davenport, J. S. Rowlinson and G. Saville, Trans. Faraday SOC., 1966, 62, 322