Critical Properties of Binary MixturesBY COLIN P. HICKS*?Chemistry Department, University of the West Indies, Kingston 7, JamaicaCOLIN L. YOUNG$Department of Physical and Inorganic Chemistry, University of New England,Armidale, N.S.W. 2351, AustraliaANDReceived 25th March, 1974Gas-liquid critical pressures of mixtures of octamethylcyclotetrailoxane with neopentane, 2,3-dimethylbutane, cyclopentane and tetramethylsilane, critical temperatures of octamethylcyclotetra-siloxane + tetramethylsilane mixtures and the vapour pressure of tetramethylsilane near the criticalregion are reported. The critical temperature and pressure data for these and other mixtures arecompared with values calculated from the van der Waals one fluid theory assuming different combiningrules and equations of state.The limitations of the criticality condition solutions are discussedbriefly.We have previously lm3 reported measurements of the mixture critical temperaturesfor a wide range of quasi-spherical molecules of different sizes. In this paper wereport critical pressure measurements on mixtures of the large quasi-spherical moleculeoct amethylcy clo te trasiloxane (0 MCTS) with three quasi-spherical alkanes and tetra-methylsilane. A comparison of experimental critical temperatures and pressures ismade with values calculated from the van der Waals one fluid theory for these systems,and for values taken from the literature for hydrocarbon + n-alkane systems.The prediction of gas-liquid critical properties of hydrocarbon + n-alkane mixturehas been investigated by several Often a random mixing lo or relatedmodel has been used together with a modification of the approximate solution of thecriticality conditions given by Rowlinson.OSpear, Robinson and Chao used a more general solution to the criticality condi-tions based on earlier work by Kuenen,l together with the Redlich-Kwong equationof state.12 Kay and co-workers 5 * used an iterative solution to the criticalityconditions similar in concept to that used here, but with important differences. Theycompared the predictions of the Redlich-Kwong, Dieterici and Redlich-Ngo l4equations of state. Teja and Rowlinson studied the critical properties predicted whenthe Vennix and Kobayashi 37 equation of state for methane is used in an iterativesolution to the criticality conditions.In this work we have used a general solutionto the criticality conditions with two different equations of state.The theoretical interpretation of the critical properties of non-polar mixtures as afunction of composition is usually achieved by the assumption that there exists ahypothetical equivalent (pure) substance, which has the same configurational freeenergy as the mixture for specified conditions.present address : Division of Chemical Standards, National Physical Laboratory, Teddington,Middlesex TWl1 OLW$ present address : Department of Physical Chemistry, University of Melbourne, Parkville,Victoria 3052, Australia.12C. P . HICKS AND C. L. YOUNG 123The combinatorial contribution to the free energy is assumed to be separable andindependent of volume or pressure. A number of further assumptions are then madeconcerning the equivalent substance in order that its configurational free energy maybe evaluated for any specified conditions : (1) that the equivalent substance obeys aparticular reduced equation of state ; (2) that the reducing parameters for the equationof state may be obtained from a particular " recipe " or " prescription " which dependsonly upon composition and the energy and volume parameters characterising inter-actions between like and unlike molecules ; (3) that the parameters characterising theinteraction between unlike molecules may be obtained from combining rules appliedto the parameters for the interactions of like molecules.The equivalent substance, and associated assumptions, enable the criticalityconditions lo(a2G/i3x2),,, = 0 (1)(i33G/i3~3)p,, = 0 (2)to be solved for the critical temperature, volume and pressure.There is good evidence that the van der Waals one fluid prescriptionTisV& = C xixjTf,V,cii j( 3 4is a reasonable assumption for molecules of similar sizes lS* l6 (i.e., size ratios ofV2/V1 < 2).In this pair of equations Tts and VEs are the critical temperature andvolume of the equivalent substance, while if i = j the critical properties are those ofthe pure components, and if i # j the hypothetical critical properties are calculatedby means of the combining rules.The most used combining rules for the critical temperature and volume are theLorentz-Berthelot combining rules5 = 1The parameters 5 and q, or their equivalents, have been introduced by various workersto increase the usefulness of the combining rules.lo* 15-23 t is usually less than unity even for simple systems.15* 2o For more complicatedsystems is still usually less than unity 1-3* 6* ' but there are several cases where itis unity or slightly greater.2* 23The solution of the criticality conditions is not straightforward, but for not toodissimilar species a first-order approximation O gives a solution for gas-liquid criticalpoints which is satisfactory for most purposes.24 Previously we have used the van derWaals equation of state in this first-order approximation to calculate critical tempera-tures.l* 2* 23 Several other equations of state give broadly similar values.Howeverthis is not the case for critical pressures and volumes, which are sensitive to the reducedequation of state assumed for the equivalentIn this work we consider three different solutions of the criticality conditions. Oneof these solutions uses the approximate solution with van der Waals' equation of statefor the equivalent substanCe,'O and two use a general (rigorous) solution. One of thegeneral solutions uses a double Taylor expansion 25 about the critical point of theequivalent substance, and the other uses the equation of state proposed by Bjerreand B a l ~ . ~ 124 CRITICAL PROPERTIES OF BINARY MIXTURESEXPERIMENTALC H E MI C A LSOMCTS was prepared by the hydrolysis of dichlorodimethylsilane '' and purified on a1 m spinning band distillation column and then by fractional crystallisation.Gas chromato-graphy revealed only about 0.02 mol % impurity (probably decamethylcyclopentailoxane).Neopentane, 2,2-dimethylpropane, was puriss grade obtained from Fluka. The statedpurity was 99.92 mole %. 2,3-dimethylbutane was a National Physical Laboratory sampleof purity 99.89 mole %.Cyclopentane was a Phillips sample with a minimum purity 99.96 mole %.Tetramethylsilane was n.m.r. grade obtained from PCR Inc., Gainsville, Florida.APPARATUSThe sealed tube method was used to measure the critical temperatures of the OMCTS+tetramethylsilane mixtures.The apparatus for measuring critical pressures was a modification of that described else-where.28 In that publication, its operation for measuring the vapour pressure of pure sub-stances as a function of temperature was described.In the present work the sample wasconfined over mercury in a J-tube. The J-tube used in the present work had a demountableglass-to-metal-to-glass high pressure seal between the portion (A) which was heated in thefurnace and the portion kept at room temperature (B) (see fig. 1).28 The two componentswere injected into the part (A) using syringes with long Teflon needles. The total samplelength was about 15 mm. The seal was assembled and the J-tube attached to the vacuumline. The sample was frozen in liquid air, the J-tube evacuated, the sample melted and re-frozen and the J-tube evacuated again.Mercury was then poured on the sample as in thecase of determining the vapour pressure of pure substances. The J-tube was topped up withhydraulic fluid coupled to the dead weight tester and aligned in the furnace (while cold).The furnace was then switched on and the temperature rapidly raised to about 20 K belowthat of the lowest temperature studied. The stirrer was started. It consisted of a small ballbearing with about 0.1 mm clearance between the capillary tube walls. The ball was movedup and down through the sample by a pair of magnets at the rate of 30 strokes per minute.The temperature was raised slowly at the rate of 0.1 K min-I and the pressure adjusted sothat there was a minute vapour phase. The pressure was then increased slightly by addingsmall weights to the dead weight tester until the vapour phase disappeared.The temperaturewas measured when the vapour phase reappeared with the dead weight tester weights balancedby the vapour pressure of sample and hydrostatic heads of mercury and hydraulic fluid. Thisprocedure was repeated about 10 times. Dew points were observed by measuring the tem-perature at which liquid phase first appeared on lowerihg the temperature at a similar rateto that used on the heating cycle, the pressure being constant. At temperatures in excessof the critical temperature two dew points were observed at the same temperature, i.e., retro-grade condensation occurred [see ref. (lo), (29) and (30) for a discussion of this phenomenon].RESULTSThe critical properties of the pure components are given in table 1, and the criticaltemperatures of OMCTS + tetramethylsilane mixtures in table 2.The composition of the mixture was not accurately known in the work on criticalpressures because it was difficult to degas the sample effectively and to be sure thatnegligible sample was lost, the total sample size being of the order of mol.Thecomposition could, however, be estimated satisfactorily from the known compositiondependence of Ti.2* (We assume that Tk is that of a binary system and not affectedby having mercury in the heated zone.)The measured pressures were corrected for the difference between the earth'sgravitational field at Armidale (9.791 11 m s - ~ ) and that for which the dead weighC.P. HICKS AND C . L. YOUNG 125TABLE l.-cRITICAL PROPERTIES OF THE PURE COMPONENTSsubstana Tc/K 10-5 m-2 Vc/cmJ mol- 1OMCTS 586.5 (28) 13.40 (28) 984 (28)c yclopent ane 511.6 (38) 45.08 (38) 260 (38)neopen t ane 433.75 (38) 31.99 (38) 303 (38)2,3-dimet hyl but ane 499.93 (38) 31.27 (38) 358 (38)tetramet hylsilane 450.4" 28.14* 357 -fReference numbers are given in brackets.* this work ; estimated assuming ZC = 0.269.TABLE 2.-cRITICAL TEMPERATURES OF OMCTS( 1) + TETRAMETHYLSILANE(2) MIXTURESx1*0.092 90.198 20.233 50.402 9TcIK t475.8498.0506.9535.1X10.445 00.543 00.693 40.829 9T a l e537.8548.6568.0575.9* estimated error k 0.001 ; t estimated error t- 0.3 K.TABLE 3 .-CRITICAL TEMPERATURES AND PRESSURES OF OMCTS MIXTURES0.8650.6450.4900.2550.1 150.9550.6750.3200.2450.2100.1980.1 700.9250.41 50.3250.21 50.1600.9500.5550.3950.3420.2820.145Tc/KOMCTS(l)+ cyclopentane(2)583.7 19.09573.8 25.90565.8 30.16547.4 38.77OMCTS(l)+ neopentane(2)585.3 15.03565.6 23.40526.5 33.29509.7 35.93501.8 37.01497.7 37.21490.5 37.35529.8 43.8618.7325.6029.9138.6243.7614.6623.1633.1935.8736.9637.1637.310 MCTS( 1) + 2,3-dimet hyl but ane(2)583.6 16.45 16.10555.0 26.73 26.54545.9 28.69 28.53532.2 31.14 3 1.02524.6 32.51 32.42OMCTS( 1) + tetramethylsilane(2)585.0 15.13551.2 25.16533.7 28.69527.5 29.76518.1 31.18488.1 33.4914.7624.9928.5729.6731.1133.46(i) without mercury vapour correction ; (ii) with mercury vapour correction given by eqn ( 5 )aestimated from independent x, Tc measurements ; b estimated error k0.5 K ; =estimated errorIf: 7 kN m-2126 CRITICAL PROPERTIES OF BINARY MIXTUREStester was calibrated (9.806 65 m s-~).It was also necessary to make some estimateof the effect of mercury on the measured pressures. This effect has been investigatedby Pak and Kay,6 who used gallium in place of mercury as the pressure transferringmedium in the heated zone. The vapour pressure of gallium is negligible at thecritical temperatures of organic compounds. They found that the partial pressureexerted by the mercury in the hydrocarbon was about 10-14 % less than the vapourpressure of pure mercury at the same temperature, and proposed that the partialpressure of mercury (PHg) be calculated from the equation(5) log,,(P,,/N m-2) = 9.765 72 - 3037.6 (K/T).TABLE 4.-vAPOUR PRESSURE, p , OF TETRAMETHYLSILANET/K lO-sp/N m-2 TIK 10-5plN m-2450.4448.1444.3442.0439.2437.728.14 434.5 22.2527.16 43 1.7 21.2726.18 429.0 20.2925.20 425.3 19.3124.20 422.7 18.3323.22 419.1 17.35We have used this equation, although it is doubtful whether it gives PHg to withinbetter than about +25 % over the temperature and pressure range studied here.The critical temperatures and pressures of the four mixtures studied are given intable 3, together with the compositions estimated from the known compositiondependence of T'L.The vapour pressure of tetrainethylsilane was also measured in this work, and isgiven in table 4.THEORYTo solve the criticality conditions we assume that the free energy of the mixturecan be divided into a combinatorial part given by the Flory theory,31 Acb, and aninteractional part, AT,, which is the configurational free energy of an assumed equi-valent substance.A = (6)(7) = A:,+RT(x, In 4' +x2 In &)where#, = x,/(x, +rx,) = 1-42.If we make the corresponding states assumption thatA2 = A:S(T/fes VIhes) (9)where fes and he, are temperature and volume reducing parameters respectively, thenwe can proceed to solve the criticality conditions in terms of the properties of theequivalent substance.APPROXIMATE SOLUTIONRowlinson lo has presented an approximate solution to the criticality conditionswhich we have discussed and used previously.'.2 * 2 3 For the mixture critical tem-perature, Tk, and critical pressure, p;, the solutions arC. P. HICKS AND C. L. YOUNG 127where the equation of state properties on the right hand side of these equations arefor the equivalent substance at its critical point; TZs, & and VSs are the criticaltemperature, pressure and volume respectively of the equivalent substance, andf:s = (afeslax2) (12)h6s = (aheslax2). (13)Although eqn (10) and (1 1) are only approximate, they have been derived withoutreference to any particular equation of state. To make use of them it is necessary toassume an equation of state for the equivalent substance.GENERAL SOLUTIONOnly a brief outline of the major points of the general solution is given here.Eqn (1) and (2) can be rewritten asDetails of the treatment will be given in a separate p~blication.~~(a2&/aX2)v,T (a2A:s/ax2)v,T ( d P / a X ) Y , d 2 = 0 (14)(a3&/aX3)v,27 4- (a3A,*,/ax3)v,7- 3(a2P/ax2)v,~Q -3(a2p/ax av),~2 + ( a 2 p / a v 2 ) , , , ~ 3 = o (15)Q = (ap/ax)V,T/(ap/av)T,x (16)whereand where Acb and AZs are the separate contributions to the Helmholtz free energy ofthe mixture which were defined earlier.Rewriting eqn (14) and (15) in reduced formenables us to defineso that the critical points of the mixture are now the solutions of the simultaneousequationsa(Gi,T) = 0 (19)/?(RT) = 0. (20)In eqn (17)-(20) = V/V& = T/Zs, P" = P/&128 CRITICAL PROPERTIES OF BINARY MIXTURESo! and /? may be calculated from eqn (17) and (18) if' we know the composition,the mixture prescription, the size parameter fr in eqn (8)j and the reduced equation ofstate for the equivalent substance.and by using a Newton-Raphson iterative procedure,33 starting from the approximate solution given by eqn(10) and (1 1).The iteration failed to converge to a solution in a few cases when theapproximate solugon was inadequate. In these cases we subtracted 0.05 from theinitial value for V . If the iteration still broke down a further 0.05 was subtractedfrom runti1 a solution was found. This technique was based upon the observationthat the initial value for Twas always very near the final solution, but the initial valuefor was_always greater than the final solution if the two differed significantly.Once V and Thad been obtained, the corresponding value of 3 was derived fromthe reduced equation of state.The actual mixture critical properties were thenobtained from the reduced critical properties by use of T:,, VEs and &. pzs is notgiven directly by the prescription for the equivalent substance, and a value for 2" mustbe adopted before it may be calculated from the prescription. During the solutionof the criticality conditions for r a n d T, the value of Z,", assumed was that for therelevant equation of state. However to generate the actual mixture critical propertiesfrom the reduced values, a value of 2" was chosen which would make the calculatedcritical pressures run smoothly from one pure substance critical pressure to the other.This will not happen if the equation of state value is used, and so the inole fractioncombination of the pure substance values for Z",proposed by Pitzer and Hultgren 34 was employed.This practice in which the equation of state is used to generate reduced values ofthe critical properties, after which an experimentally realistic value of 2" is used inthe calculation of the actual critical properties from the reduced critical properties,differs from the practice of previous worker^.^'^In this work we have solved these equations forzc = x,z;1+x22;2 (21)EQUATIONS OF STATE(1) BJERRE A N D BAK EQUATION OF STATEWe have used the two parameter equation of state proposed by Bjerre and Bak.26They tested a number of two parameter equations of state and found that the mostsatisfactory equation for 0.56 < T/T" < 0.95 was10 4; 75 1 p- --3 (P-1/6) 16 T*(v+1/4)22" = 3/10 (23)where, as before, properties with a tilde are reduced with respect to the equivalentsubstance critical properties.(2) DAVIS A N D RICE TAYLOR SERIESWe have also used the double Taylor series expansion about the critical point ofa pure substance proposed by Davis and Rice.25 These workers wrote the reducedpressure of a one component system in the vicinity of its critical point as a Taylorseries expansion in reduced temperature and density, and evaluated the coefficientsof the leading terms in the expansion for argon.The work of Davis and Rice does not provide a theory of the critical point, anddoes not reproduce the known singularities which appear as the critical point iC.P . HICKS AND C. L. YOUNG 129approached. It does, however, give us an accurate reduced equation of state for thenear critical region for the equivalent substance, and is valid as long as there is a pointwithin 6 on each side of the critical point about which such analytic expansions canbe made.The values of the coefficients of the Taylor series expansion obtained by Davisand Rice were used where available, and higher order coefficients were taken fromwork by Hicks 24 where necessary. The Taylor series was taken to the sixth orderof the derivatives of pressure with respect to density, and the temperature derivativeof pressure was expanded to the fourth order with respect to density.The restrictedrange of validity of the Taylor series expansion meant that the configurational energy,which requires an integration from V = GO down to Y = V, could not be calculated.Instead it was calculated using the Beattie-Bridgman equation with the generalcoefficients determined by Su and Chang.36 Calculations showed that small changesin the configurational energy used did not affect the predicted critical propertiessignificantly. The restricted range of validity also meant that solutions to thecriticality conditions could not be found for all mixtures studied here.COMPARISON WITH EXPERIMENTOMCTS MIXTURESThe three methods already outlined have been used to predict the critical propertiesof the OMCTS mixtures which were studied experimentally.In each case the van derWaals one fluid prescription, eqn (3), was used, and Y for eqn (8) was taken to be theratio of liquid molar volumes at 293.15 K.Van der Waals' equation was used to give the equation of state properties requiredin the approximate solution to the criticality conditions.Different pairs of combining rules were used for the critical temperature, T;2, andcritical volume, Y;2, and those used are summarised in table 5.TABLE 5 .-COMBINING RULESno. name(1) Berthelotf Lorentz< and 7 for use in eqn (4)5 = 1 p I = 12(1112)+ 26 ViI v,;(11 +Id w:p+ VC,j)"(2) Hudson-McCoubrey 17+ Lorentz =- = 1(3) Berthelot+ Good-Hope * = I2(Z112)* 26 v;l vg2(11 + 1 2 ) ( q+ Vc,tY(4) Hudson-McCoubrey + Good-Hope = -I I and 1, are the first ionisation potentials of each species.The standard deviations between predicted and experimental critical properties aregiven in table 6.For these mixtures all the methods predict critical pressures whichare below those observed ; a value of < greater than unity would be needed to improveagreement, whereas a value of < less than unity is needed to fit critical temperatures atequimole fraction (assuming q = 1). In spite of this the Hudson and McCoubreycombining rrrle for T;.,, which gives values smaller than the Berthelot or geometricinem rule, gives larger standard deviations in every case.I-130 CRITICAL PROPERTIES OF BINARY MIXTURESN-ALKANE MIXTURESWe now consider the application of the general solution together with the Bjerreand Bak equation of state to n-alkane + n-alkane, n-alkane + cycloalkane and n-alkane + benzene mixtures.The experimental data are taken from a recent compilation.The Davis and Rice Taylor series was not used in this comparison because the criticalpoints of many of these mixtures lie outside its range of validity.TABLE 6.-STANDARD DEVIATIONS OF CALCULATED AND OBSERVED CRITICAL PROPERTIES FOKTHE O M n S MIXTUREScritical temperature TC/KmixturesapproximatesolutionVan der Waalsequation(1) (2)OMCTS-5 cyclopentane 7.7 24.9OMCTS+ iieopentane 3.2 27.8OMCTS+ 2,3-dimethylbutane 6.1 16.3OMCTS+ tetramethylsilane 5.0 16.0general solutionDavis and Rice Taylor series Bjerre and Bak equation(1) (2) (3) (4) (1) (2) (3) (4)t3.8 23.5 4.7 20.4 4.8 26.3 5.9 23.84.7 * * * 2.6 29.0 1.9 26.43.1 15.4 3.4 14.0 4.3 17.3 4.8 16.9 * * * * 3.6 16.2 4.6 14.4critical pressure pC/N m-2OMCTS+ cyclopentane 5.6 7.5 5.9 6.5 4.8 5.0 5.6 7.1 4.5 5.9OMCTS+ neopentane 6.4 8.8 6.0 * * * 5.6 7.7 4.1 6.2OMCTS+2,3-dimethylbutane 4.1 5.4 4.0 4.5 3.4 6.3 3.7 5.0 3.1 4.2OMCTS+tetramethylsilane 5.7 6.8 * * * * 4.9 5.7 4.1 4.9* Solution outside the range of the Taylor series.7 The pairs of combining rules used for eachcolumn are shown by the numbers in parenthesis. See table 5 for details of combining rules.The calculations were made using, as before, the van der Waals one fluid prescrip-tion, and Y as the ratio of molar volumes (extrapolated values were used if necessary)at 293.15 K.(1) n- A LK ANE + n- AL K ANE MIXTURESThe standard deviations between predicted and experimental critical propertiesusing each pair of combining rules are given in table 7.When the molecules are ofsimilar size critical temperatures and pressures are predicted reasonably well, but asthe chain length difference increases the predictions worsen.Overall pair 3 is best and pair 4 is worse for these mixtures.(2) n-ALKANE-kCYCLOALKANE MIXTURESThese mixtures do not form a good test of the combining rules. The standarddeviations between predicted and experimental critical properties using each pair ofcombining rules are given in table 8. Both pairs 1 and 3 predict the critical propertieswith fair accuracy, pair 3 being marginally superior, as in the case of the n-alkane+n-a1 kane mixtures.(3) n-ALKANE+BENZENE MIXTURESStandard deviations between predicted and experimental values are given in table 9.Again the pairs 1 and 3 predict the observed critical properties with the best overallaccuracyC .P. HICKS AND C . L. YOUNG 131CONCLUSIONSOf the pairs of combining rules considered the combination of the Good and Hoperule for VC,, and the Berthelot rule for Ti2 is the best for these mixtures. However,when the size difference of the two molecules is large none of the combining rulespredict the observed critical properties well. This is probably due as much to theinadequacy of the van der Waals one fluid prescription as to the inadequacy of theequation of state used.TABLE 7.-sTANDARD DEVIATION BETWEEN EXPERIMENTAL CRITICAL PROPERTIES AND THOSEPREDICTED BY THE VAN DER WAALS ONE FLUID MODEL, THE BJERRE AND BAK EQUATION OFSTATE AND VARIOUS COMBINING RULES FOR THE n-ALKANE+ n-ALKANE MIXTURES(1)0.83.67.813.611.326.70.73.22.95.75.011.50.20.92.63.07.30.51.42.92.10.92.72.64.87.90.30.32.11.61 .o0.21 .o0.4(2)1.70.83.23.35.618.30.70.83.44.18 .O10.00.51.31.73.58.60.71 .o1.36.26.19.59.811.111.90.31.42.49.011.80.53.20.5TC/K(3)0.83.67.613.215.420.10.70.83.44.18.010.00.21 .o2.73.26.50.51.42.92.10.92.72.54.78.40.30.32.22.10.50.21 .o0.5(4)1.60.73.13 .O2.719.30.70.93.04.48.48.60.51.32.03.49.30.71 .o1.16.26.29.29.311.012.20.31.42.48.811.40.53.20.5(1)2.01.64.77.96.012.40.81.21.83.24.26.20.30.71.52.13.90.8---1.5----0.30.61.73 .O4.10.30.90.7(2)2.12.04.56.79.827.40.81.52.43.94.89.80.30.81.72.35.90.8---1.7----0.30.71.83.34.40.31 .o0.7(3)1.91.33.96.24.98.60.81.11.52.53.45.10.20.61.41.83.5----1.3----0.30.51.52.73.80.30.80.74)2.01.73.65.37.027.40.81.32.13.44.110.00.30.71.72.05.5----1.6----0.30.61.73 .O4.00.30.90.7The two different entries in this table for Cs+ C9 correspond to measurements by differentworkersI32 CRITICAL PROPERTIES OF BINARY MIXTURESThese results lend support to the hypothesis of Good and Hope l 8 that " ageometric mean distance rule " is superior to the arithmetic mean rule.However,in view of the uncertainty in applying any equation of state and mixture prescriptionto the criticality conditions, too much emphasis must not be placed on this observation.TABLE 8.-STANDARD DEVIATION BETWEEN EXPERIMENTAL CRITICAL PROPERTIES AND THOSEPREDICTED BY THE VAN DER WAALS ONE FLUID MODEL, THE BJERRE AND BAK EQUATION OFSTATE AND VARIOUS COMBINING RULES FOR THE n-ALKANE+ CYCLOALKANE MIXTURESmixture TC/K l O - 5 p C I N m-2(1) (2) (3) (4) (1) (2) (3) (4)c-C5 + n-CS 0.9 1.4 0.9 1.4 0.3 0.3 0.3 0.3c-C5 + n-C6 0.4 1.9 0.4 1.9 0.2 0.3 0.2 0.3c-C5 + n-C7 0.8 3.8 0.8 3.8 0.2 0.5 0.1 0.3c-C5+n-Cs 1.4 5.9 1.4 5.7 0.6 1.0 0.3 0.8c-C5+n-Cg 2.2 7.7 2.3 7.5 0.9 1.5 0.5 1.211.7 6.6 12.0 7.1 6.3 6.6 5.1 5.63.5 3.5 3.5 3.50.4 0.9 0.4 0.9 0.3 0.3 0.2 0.30.3 2.3 0.3 2.3 0.1 0.2 0.1 0.20.3 3.9 0.4 3.9 0.3 0.5 0.2 0.41.1 5.2 1.1 5.1 0.3 0.7 0.1 0.51.0 7.6 1.0 7.4 0.7 1.2 0.5 1 .o1.4 14.3 1.6 13.9 1.7 2.5 1.3 2.1- - - -The two different entries in this table for C'C6+n-C8 correspond to measurements bydifferent workers.TABLE ST STANDARD DEVIATION BETWEEN EXPERIMENTAL CRITICAL PROPERTIES AND THOSEPREDICTED BY THE VAN DER WAALS ONE FLUID MODEL, THE BJERRE AND BAK EQUATION OFSTATE AND VARIOUS COMBINING RULES FOK BENZENE(BZ) + n-ALKANESmixture TC/K(1) (2) (3) (4)59.0 56.5 60.8 56.916.4 15.2 16.5 15.35.2 4.6 5.2 4.65.4 2.6 5.4 2.63.6 0.7 3.2 0.74.0 1.0 4.0 1 .o4.4 3.4 4.5 3.26.4 3.4 6.4 3.24.8 5.8 5.0 5.58.6 6.7 8.6 6.55.9 7.4 6.0 7.16.4 15.5 6.3 14.87.5 26.6 8.5 26.3lO-spc/N m-2(1) ( 2 ) (3) (4)18.3 15.7 17.4 15.51.2 1.5 1.2 1.5- - - -0.3 0.2 0.4 0.30.5 0.2 0.7 0.30.5 0.2 0.7 0.20.4 0.5 0.6 0.20.7 1 .o 0.6 0.61.4 2.5 0.8 2.13.1 3.9 2.2 3.3- - - -- - - -The two different entries in this table for some mixtures correspond to measurements bydifferent workers.Financial support from the Queen Elizabeth I1 Fellowship Committee to C.L. 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