J. Chem. Thermodynamics 1998, 30, 1253]1261Article No. ct980394

The measurement of coexistence curves{ ( ) ( )for x CHCOOC H H 1 I x2 5 2( ) }CH CH CH in the critical region3 2 7 3

Xueqin An, Jiong Yang, and Weigou Shena

Department of Chemistry, Lanzhou Uni ersity, Lanzhou, Gansu 730000,P. R. China

� Ž . ŽCoexistence curves for binary liquid mixtures of x CHCOOC H q 1 y2 5 2. Ž . 4x CH CH CH have been determined by measurement of the refractive index. The3 2 7 3

experimental results have been used in the determination of the critical exponent b and thecritical amplitude B, and the study of the diameters of the coexistence curves. The values ofthe critical exponent b are found to be consistent with theoretical predictions. The coexistencecurves have been successfully described by a combination of the Wegner equation and the

Ž . yrexpression for the diameter. The exponents r and b in the relations 1 y f rf A M andc cy1 .865 yb Ž .B f A M , where f is the critical volume fraction of CHCOOC H , have beenf c c 2 5 2Ž . Ž .found to be 0.42 " 0.01 and 0.31 " 0.04 , respectively, consistent with observations from

experimental studies for other chain-molecule solution systems, and support theLandau]Ginsburg]Wilson type model we proposed recently. q 1998 Academic Press

KEYWORDS: critical phenomena; coexistence curve; refractive index; diethyl maleate;nonane

1. Introduction

Recently, we proposed a Landau]Ginsburg]Wilson type model to describe themolar mass dependence of the critical amplitudes for chain-molecule solutions ofboth small molecules and polymers.Ž1. From this model we derived a general formfor the difference of the volume fractions of two coexisting phases as:

B fy1 .865 A Myb , 1Ž .f c

where b s 0.29 and f is the critical volume fraction of the non-chain-moleculecŽcomponent i.e. the critical volume fraction of a polar liquid for the small

.chain-molecule solutions we studied , M is molar mass of the chain molecule, andB is the amplitude associated with the coexistence curve of temperature againstf

volume fraction. In addition, f was experimentally found to be dependent on thecmolar mass as follows:

1 y f rf A Myr , 2Ž . Ž .c c

with the universal exponent r s 0.41Ž2. for the chain-molecule solutions for bothsmall molecules and polymers. To test the validity of the universality of equations

a To whom correspondence should be addressed.

0021]9614r98r101253 q 09 $30.00r0 q 1998 Academic Press

X. An, J. Yang, and W. Shen1254

Ž . Ž .1 and 2 , we have measured the coexistence curves near the critical points forŽ . Ž3 ] 6.diethylmaleate q n-alkane with alkane carbon numbers 6, 7, 8, and 10.In this paper we report the measurements of the coexistence curves for� Ž . Ž . Ž . 4x CHCOOC H q 1 y x CH CH CH . The critical amplitude B and the2 5 2 3 2 7 3

Ž . Ž . Ž .critical exponent b have been deduced from T , n , T , x , and T , f curvesŽwhere n, x, and f are the refractive index, the mole fraction and the volume

.fraction, respectively , and the behaviour of the diameters r of the coexistencedcurves have been examined. The coexistence curves have been successfullydescribed by the combination of the Wegner equation and the expression for the

Ž . Ž .diameter. The values the exponents b and r in equations 1 and 2 have beendetermined and compared with the values predicted by theory, or found in otherexperimental studies.

2. Experimental

Ž .Diethyl maleate CHCOOC H obtained from Beijing Chemicals Factory was2 5 2Žpurified by fractional distillation under vacuum. The n-nonane mass fraction s

.0.99 was supplied by Aldrich Chemical Company Inc. and was stored over. y104 10 m molecular sieves.The coexistence curves were determined by measurement of the refractive

indices using the method of minimum deviation. The apparatus and theexperimental procedure for the measurement of the refractive index, and thetechniques for the determination of the critical temperature and the criticalconcentration have been described previously.Ž7. During the measurements, the

. y3temperature was constant to within "2 10 K. The accuracy and precision in the. y2 . y3measurement of temperature were better than "1 10 K and "1 10 K,

. y3respectively. The accuracy of measurement was "3 10 K for the temperature. y4Ž .difference T y T , "2 10 for the refractive index in each coexisting phase, andc

. y3"1 10 for the critical mole fraction x .c

3. Results and discussion

� Ž .The critical mole fractions and the critical temperature of x CHCOOC H q2 5 2Ž . Ž . 4 Ž . Ž1 y x CH CH CH were determined to be 0.473 " 0.001 and 310.1 "3 2 7 3

.0.1 K, respectively. The refractive indices for each coexisting phase were measuredat various temperatures. The results are listed in columns 2 and 3 of table 1, and

Ž .shown in figure 1 a .Ž .In order to obtain the T , x coexistence curve, the refractive indices of pure

diethyl maleate and nonane at various temperatures, and the refractive indices of aseries of binary mixtures with known mole fractions in the single phase region atT s 310.29 K were also measured. The results are listed in tables 2 and 3. Therefractive index may be expressed as a linear function of temperature in a certaintemperature range by:

. .n T , x s n T8, x q ­ nr­T T y T8 , 3Ž . Ž . Ž . Ž . Ž .x

. .­ nr­T s x ­ n r­ T q 1 y x ­ n r­ T , 4Ž . Ž . Ž . Ž . Ž .x A B

Ž .The coexistence curves of diethyl maleate q nonane 1255

Ž . Ž . Ž . � Ž . ŽTABLE 1. Coexistence curves of T , n , T , x , and T , f for x CHCOOC H q 1 y2 5 2. y7. Ž . 4x CH CH CH . Refractive indices n were measured at wavelength l s 6.328 10 m, and T s3 2 7 3 c

310.064 K. Volume fraction is denoted by f . Subscripts 1 and 2 relate to upper and lower phases,0respectively

Ž .T y T rK n n x x f fc 1 2 1 2 1 2

0.002 1.4093 1.4106 0.457 0.493 0.431 0.4670.004 1.4091 1.4108 0.451 0.499 0.426 0.4730.008 1.4089 1.4110 0.445 0.504 0.420 0.4780.013 1.4087 1.4112 0.439 0.510 0.414 0.4840.020 1.4084 1.4115 0.431 0.518 0.406 0.4920.032 1.4082 1.4117 0.425 0.523 0.400 0.4970.041 1.4081 1.4119 0.422 0.529 0.397 0.5030.051 1.4080 1.4119 0.419 0.528 0.394 0.5030.064 1.4079 1.4121 0.416 0.534 0.391 0.5080.081 1.4077 1.4123 0.410 0.539 0.385 0.5130.104 1.4075 1.4125 0.403 0.544 0.379 0.5190.124 1.4074 1.4127 0.400 0.549 0.376 0.5240.151 1.4073 1.4128 0.397 0.552 0.373 0.5260.187 1.4071 1.4131 0.391 0.559 0.366 0.5340.230 1.4070 1.4134 0.387 0.567 0.363 0.5420.277 1.4068 1.4136 0.381 0.572 0.357 0.5460.320 1.4067 1.4137 0.377 0.574 0.353 0.5490.368 1.4066 1.4139 0.373 0.579 0.350 0.5530.421 1.4065 1.4141 0.370 0.583 0.346 0.5580.481 1.4064 1.4143 0.366 0.588 0.342 0.5630.552 1.4062 1.4146 0.359 0.595 0.336 0.5700.631 1.4060 1.4148 0.352 0.599 0.329 0.5750.720 1.4058 1.4150 0.344 0.604 0.322 0.5790.810 1.4057 1.4152 0.340 0.608 0.317 0.5830.926 1.4055 1.4155 0.332 0.614 0.310 0.5901.061 1.4054 1.4157 0.327 0.618 0.305 0.5931.213 1.4053 1.4161 0.322 0.627 0.300 0.6021.370 1.4052 1.4164 0.317 0.633 0.295 0.6081.550 1.4051 1.4168 0.311 0.641 0.289 0.6171.749 1.4050 1.4171 0.305 0.646 0.284 0.6231.982 1.4049 1.4175 0.298 0.654 0.277 0.6302.356 1.4047 1.4180 0.287 0.663 0.266 0.6392.742 1.4045 1.4185 0.274 0.671 0.254 0.6483.263 1.4044 1.4192 0.263 0.683 0.244 0.6604.064 1.4044 1.4202 0.251 0.699 0.231 0.6774.902 1.4042 1.4212 0.231 0.714 0.213 0.6936.006 1.4044 1.4222 0.221 0.727 0.203 0.7067.189 1.4046 1.4234 0.208 0.743 0.192 0.7238.605 1.4047 1.4246 0.189 0.757 0.173 0.737

10.235 1.4051 1.4259 0.175 0.771 0.161 0.752

Ž .where ­ nr­T is the derivative of n with respect to T for a particularxŽ . Ž . Ž .composition x, and ­ n r­ T and ­ n r­ T are the values of ­ nr­T forA B x

Ž . Ž .x s 1 and x s 0, respectively. The validity of equations 3 and 4 have beenconfirmed in previous work.Ž3,7. The results listed in table 2 were fitted to equation

. y4 . y4Ž . Ž . Ž .3 to obtain ­ n r­ T s y4.56 10 and ­ n r­ T s y4.70 10 with aA B. y4standard deviation of 1 10 in n. With a chosen value of T8 s 304.95 K,

corresponding to about the middle temperature for the coexistence curveŽ . Ž .determined in this work, the values of n T8, x were calculated from equations 3

X. An, J. Yang, and W. Shen1256

Ž . Ž . Ž . � Ž . ŽFIGURE 1. Coexistence curves of: a, T , n ; b, T , x ; and c, T , f for x CHCOOC H q 1 y2 5 2. Ž . 4x CH CH CH where T s temperature, n s refractive index, x s mole fraction, and f s3 2 7 3

volume fraction. v, experimental values of the concentration variables r of the coexisting phases;', experimental values of the diameter r of the coexisting phases; }}}, calculated concentrationdvariables and diameter of coexisting phases.

Ž . Ž . Ž .and 4 with the n T , x data listed in table 3. A polynomial form of n T8, x as afunction of x at T8 was obtained:

. . 2 . 3 . 4n T8, x s 1.3987 q 0.0177 x q 0.0353 x y 0.0324 x q 0.0144 x , 5Ž . Ž .

. y4with a standard deviation of 1 10 . The values of the refractive indices for theŽ .coexisting phases were then converted to mole fractions by calculating n T8 , x

Ž .The coexistence curves of diethyl maleate q nonane 1257

TABLE 2. Refractive index n at wavelengthy7.l s 6.328 10 m for

Ž . Ž .CHCOOC H and CH CH CH2 5 2 3 2 7 3at various temperature T

Ž . Ž .CHCOOC H CH CH CH2 5 2 3 2 7 3

TrK n TrK n

299.059 1.4365 310.255 1.3962300.807 1.4356 309.591 1.3965302.239 1.4350 308.793 1.3969303.149 1.4345 307.696 1.3974304.157 1.4340 306.741 1.3978306.079 1.4331 305.606 1.3984307.770 1.4324 304.383 1.3989309.287 1.4318 300.943 1.4006310.457 1.4313

Ž . Ž . Ž .through equations 3 and 4 , and iteratively solving equation 5 . The results areŽ .listed in columns 4 and 5 of table 1 and shown in figure 1 b .

The mole fraction was used to calculate the volume fraction through:

1rf s 1 y K q Krx , 6Ž . Ž .K s d M r d M , 7Ž . Ž . Ž .A B B A

where M is the molar mass, d is the mass density, and subscript A and B refer todiethyl maleate and n-nonane, respectively. The values of d and d wereA Bobtained from references 8 and 9. The values of f of the coexisting phases for thissystem at various temperatures are listed in columns 6 and 7 of table 1 and shown

Ž .in figure 1 c .Ž .The differences r y r of the ‘‘density’’ variables of the coexisting phases may2 1

be expressed by the Wegner expression:Ž10.

r y r s Bt b q B t bqD q . . . , 8Ž .2 1 1

where r and r are the ‘‘density’’ variables in the upper and lower coexisting1 2Ž . Ž .phases; t s T y T rT , T is the critical temperature; and b 0.3265 andc c c

Ž .Ž11,12.D 0.50 are universal critical exponents which describe the shape of thecoexistence curve. Values of B and B depend on the system and the choice of the1

. y7 � Ž . ŽTABLE 3. Refractive indices n at wavelength l s 6.328 10 m for x CHCOOC H q 1 y2 5 2. Ž . 4x CH CH CH at T s 310.29 K3 2 7 3

x n x n x n

0.000 1.3962 0.399 1.4071 0.800 1.42260.100 1.3982 0.500 1.4108 0.900 1.42660.200 1.4009 0.599 1.4144 1.000 1.43130.300 1.4040 0.701 1.4180

X. An, J. Yang, and W. Shen1258

� Ž . ŽTABLE 4. Values of the critical amplitudes B and critical exponents b for x CHCOOC H q 12 5 2. Ž . 4 Ž . Ž . Ž . Ž .y x CH CH CH coexistence curves of T , n , T , x , and T , f in equation 93 2 7 3

OrderŽ . Ž .parameter T y T - 1 K T y T - 10.3 Kc c

B b B b

n 0.066 " 0.002 0.324 " 0.003 0.064 " 0.001 0.320 " 0.002x 1.86 " 0.05 0.324 " 0.004 1.81 " 0.02 0.322 " 0.002f 1.85 " 0.04 0.325 " 0.004 1.81 " 0.02 0.322 " 0.002

order parameter. In a region sufficiently close to the critical temperature, equationŽ .8 becomes:

r y r s Bt b. 9Ž .2 1

Ž .The differences r y r of the ‘‘density’’ variables of the coexisting phases were2 1Ž .calculated from the results listed in table 1, and were fitted to equation 9 to

obtain the critical exponent b and the critical amplitude B. The results are shownŽ .in table 4. The values of b and B depend on the cutoff values of T y T , but forc

Ž .T y T - 1 K, the values of b for three choices of variables are all in goodcŽ .Ž11.agreement with the theoretical prediction of 0.3265 " 0.001 within the

experimental uncertainties.The diameter r of the coexistence curve may be expressed by:d

r s r q r r2 s r q Dt Z q . . . , 10Ž . Ž .d 2 1 c

where r is the value of r at the critical point, D is the system-dependentcparameter, and Z is the apparent exponent. For a binary mixture there is no apriori reason for choosing one ‘‘density’’ variable rather than another, but thevariables may be tested by examining the symmetry of the coexistence curve and by

Ž . Ž . Ž13,14.comparing the goodness of fits of equation 10 with 1 y a and 2b , wherea s 0.11Ž12. characterizes the divergence, as the critical point is approached, of theheat capacity at constant volume for the pure fluid. The experimental diameter

Ž .data were fitted to equation 10 . The results of the fits are compared in table 5,where the experimental value of n was obtained by extrapolating the refractivec, exptindices against temperatures in the single phase region up to the criticaltemperature; the value of x was determined by the technique of ‘‘equalc, exptvolume’’, and the value of f was then calculated from x throughc, expt c, expt

Ž . Ž .equations 6 and 7 . The uncertainties of the optimal parameters reported intable 5 include no systematic errors arising from converting n to x, and x to f.Such uncertainties in x and f were estimated to be about "0.005. Therefore the

Ž .values of n , x , and f obtained from the extrapolation of equation 10 arec c cconsistent with those observed. This constitutes evidence that no significant criticalanomaly was present in the refractive indices, and that the refractive indices wereproperly converted to mole fractions and volume fractions in our treatment. The

Ž .quality of the fit of equation 10 may be indicated by the values of the standard

Ž .The coexistence curves of diethyl maleate q nonane 1259

Ž .TABLE 5. Parameters of equation 10 and standard deviation s in r for the diameters of thedŽ . Ž . Ž . � Ž . Ž . Ž . 4coexistence curves of T , n , T , x , and T , f for x CHCOOC H q 1 y x CH CH CH .2 5 2 3 2 7 3

r is the critical value of the order parameter determined by the techniques described in the textc, expt

Ž . Ž . Ž .T , n T , x T , f

r 1.4099 " 0.0002 0.473 " 0.005 0.447 " 0.005c, expt

Z s 0.89

r 1.4099 " 0.0001 0.475 " 0.001 0.450 " 0.001cD 0.1141 " 0.0008 y0.04 " 0.02 0.15 " 0.02

y5 y3 y3. . .s 5.2 10 1.2 10 1.3 10

Z s 0.653

r 1.4097 " 0.0001 0.475 " 0.001 0.450 " 0.001cD 0.048 " 0.002 y0.016 " 0.007 0.068 " 0.007

y4 y3 y3. . .s 2.3 10 1.2 10 1.2 10

deviation s listed in table 5. We have found no significant difference between fitswith t 1ya and with t 2 b for both x and f . Therefore we conclude that f and xare almost equally good variables for the construction of order parameters for thesystem under study. This is consistent with what the symmetry indicates: no

Ž . Ž .significant symmetry difference between T , x and T , f is observed in figure 1.

Ž . Ž .FIGURE 2. A ln]ln plot of 1 y f rf against molar mass M for diethyl maleate q n-alkane :c cŽ .v, experimental values; }}}, calculated values from equation 2 with r s 0.42.

X. An, J. Yang, and W. Shen1260

Ž . Ž . Ž . Ž .TABLE 6. Parameters of equation 8 for the coexistence curves of T , n , T , x , and T , f for� Ž . Ž . Ž . 4x CHCOOC H q 1 y x CH CH CH2 5 2 3 2 7 3

Order parameter B B1

n 0.0653 " 0.00020.0675 " 0.0002 y0.021 " 0.002

x 1.850 " 0.0041.890 " 0.006 y0.38 " 0.05

f 1.836 " 0.0041.875 " 0.006 y0.37 " 0.05

When the critical exponents b and D are fixed at the theoretical valuesŽ . Ž11,12. Ž .b s 0.3265, D s 0.50 , and equation 8 is used to fit the phase separationdata, the parameters B and B can be obtained. The results are listed in table 6.1

Ž . Ž .Combination of equations 8 and 10 yields:

r s r q Dt Z y 1r2 Bt b y 1r2 B t bqD , 11Ž . Ž . Ž .1 c 1

r s r q Dt Z q 1r2 Bt b q 1r2 B t bqD . 12Ž . Ž . Ž .2 c 1

When Z, b , D, and T are fixed at 0.89, 0.3265, 0.5, and 310.064 K, respectively,cand the values of D, r , B, and B are taken from tables 5 and 6, the values of r ,c 1 1

y1 .865 Ž .FIGURE 3. A ln]ln plot of B f against molar mass M for diethyl maleate q n-alkane :f cŽ .v, experimental values; }}}, calculated values from equation 1 with b s 0.31.

Ž .The coexistence curves of diethyl maleate q nonane 1261

�Ž . 4TABLE 7. Critical amplitudes B and critical volume fractions f for CHCOOC H q n-alkane .f c 2 5 2M is the molar mass of n-alkane

y1.Ž .n-alkane Mr g mol f Bc f

Hexane 86.178 0.406 1.749Heptane 100.206 0.421 1.824Octane 114.233 0.436 1.816Nonane 128.260 0.447 1.836Decane 142.287 0.458 1.904

Ž . Ž . Ž .r , and r can be calculated from equations 11 , 12 , and 10 . The results are2 dshown as lines in figure 1. The calculated values are in good agreement with theexperimental results.

Table 7 summarizes the values of B and f for five experimentally investigatedf cŽ . Ž . Ž .systems of diethyl maleate q n-alkane . According to equations 1 and 2 , ln]ln

Ž . y1.865plots of 1 y f rf and B f against the molar mass of n-alkane M willc c f cyield two straight lines. These two plots are shown in figures 2 and 3. A least-squares

Ž . Ž .fit results in the values of 0.42 " 0.01 and 0.31 " 0.04 for the exponents r andŽ . Ž1.b, which are in excellent agreement with calculated values b s 0.29 , and

observed values from experimental studies on chain-molecule solutions of bothsmall moleculesŽ2,15. and polymers.Ž2. This supports the universality of the exponentsof M and the Landau]Ginsburg]Wilson type model we proposed recently forchain-molecule solutions.

ŽThis work was supported by the National Natural Science Foundation Project.29673019 , the State Education Committee, and the Natural Science Foundation of

Ž .Gansu Province No: ZR-96-009 , P. R. China.

REFERENCES

1. An, X.; Jiang, F.; Chen, C; Shen, W. Chem. Phys. Letters 1998, 282, 403]408.2. An, X., Jiang, F.; Chen, C.; Shen, W. Pure and Applied Chem. 1998, 70, 609]614.3. An, X.; Lui, X.; Shen, W. J. Chem. Thermodynamics 1997, 29, 669]675.4. An, X.; Yang, J.; Shen, W. J. Chem. Thermodynamics 1998, 30, 13]19.5. An, X.; Mao, C.; Sun, G.; Shen, W. J. Chem. Thermodynamics 1998, 30, 689]695.6. An, X.; Chi, X.; Wang, T.; Shen, W. J. Chem. Thermodynamics 1998, 30, 1199]1205.7. An, X.; Shen, W.; Wang, H.; Zheng, G. J. Chem. Thermodynamics 1993, 25, 1373]1383.8. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Sol ents: 4th edition. Techniques of Chemistry,

Vol. II. Wiley-Interscience: New York. 1986.9. Thermodynamics Research Centre, API 44 Tables, Selected Values of Properties of Hydrocarbons and

Related Components, Vol. I. 1972.10. Wegner, F. J. Phys. Re¨ . 1972, B5, 4529]4536.11. Alpert, D. Z. Phys. Re¨. 1982, B25, 4810]4814.12. Le Guillou, J. C.; Zinn-Justin, J. Phys. Re¨ . 1980, B21, 3976]3998.13. Ewing, M. B.; Johnson, K. A.; McGlashan, M. L. J. Chem. Thermodynamics, 1988, 20, 49]62.14. Greer, S. C.; Das, B. K.; Kumar, A.; Gopal, E. S. R. J. Chem. Phys. 1983, 79, 4545]4552.15. An, X.; Li, P.; Zhao, H.; Shen, W. J. Chem. Thermodynamics 1998, 30, 1049]1059.

( )Recei ed 16 February 1998; in final form 20 May 1998

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