J. Chem. Thermodynamics 1998, 30, 1199]1205Article No. ct980386

The measurement of coexistence curves{ ( ) ( )for x CHCOOC H H 1 I x2 5 2( ) }CH CH CH in the critical region3 2 8 3

Xueqin An, Xianlin Cui, Tao Wang, 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 8 3

experimental results have been used in the determination of the critical exponent b andcritical 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. Theexperimental results have also been used in the assessment of the Wegner correction termsover a temperature range of about 11 K from the critical temperature. The coexistence curveshave been successfully described by a combination of the Wegner equation and the expressionfor the diameter. q 1998 Academic Press

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

1. Introduction

ŽIn previous work, the coexistence curves for binary solutions of diethyl maleate q. Ž . Ž .hexane , diethyl maleate q heptane , and diethyl maleate q octane near the

Ž . Ž1 ] 3.liquid q liquid critical points have been reported. As a continuation of thestudy of the molar mass dependence of the critical behaviour in binary liquidmixtures of chain molecules, we present in this paper the measurements of

Ž .coexistence curves for diethyl maleate q decane by measurement of the refractiveindices using the method of minimum deviation.Ž4. The critical amplitude B and

Ž . Ž . Ž .the 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 a combination of the Wegner equation and the expression for thediameter.

2. Experimental

Ž .Diethyl maleate CHCOOC H obtained from Beijing Chemicals Factory was2 5 2

a To whom correspondence should be addressed.

0021]9614r98r101199 q 07 $30.00r0 q 1998 Academic Press

X. An et al.1200

Ž .purified by fractional distillation under vacuum. n-Decane mass fraction s 0.99. y10was supplied by Aldrich Chemical Company Inc. and stored over 4 10 m

molecular sieves.The apparatus and experimental procedure have been described previously.Ž4.

The critical mole fraction was determined by the technique of ‘‘equal volume’’.Ž4. Asample of critical composition was prepared in a rectangular fluorimeter cellequipped with an Ace-thread connection and was used throughout for themeasurement of the whole coexistence curve.Ž4. The phase separation temperatureof the sample was determined and taken as the critical temperature. During the

. y3measurements, the temperature was constant to within "2 10 K. The accuracy. y2and precision in the measurement of the temperature were better than "1 10 K

. y3 . y3and "1 10 K, respectively. The accuracy of measurement was "3 10 K for. y4Ž .the temperature difference T y T , "2 10 for the refractive index in eachc

. y3coexisting phase, and "1 10 for the critical mole fraction x .c

3. Results and discussion

� Ž .The critical mole fractions and the critical temperatures of x CHCOOC H q2 5 2Ž . Ž . 4 Ž . Ž1 y x CH CH CH were determined to be 0.505 " 0.001 and 318.8 "3 2 8 3

.0.1 K, respectively. The refractive indices of the coexisting phases in the cell weremeasured at various temperatures. The results are listed in columns 2 and 3 of

Ž .table 1, and shown in figure 1 a .

Ž . Ž . Ž .TABLE 1. Coexistence curves of T , n , T , x , and T , f where n is the refractive index, x the mole� Ž . Žfraction, f is the volume fraction, and T is the temperature for x CHCOOC H q 1 y2 5 2

. y7. Ž . 4x CH CH CH . Refractive indices were measured at wavelength l s 6.328 10 m. Subscripts 13 2 8 3and 2 relate to upper and lower phases, respectively. T s 318.802 Kc

DTrK n n x x f f1 2 1 2 1 2

0.001 1.4106 1.4116 0.485 0.519 0.438 0.4720.004 1.4106 1.4121 0.485 0.536 0.438 0.4880.005 1.4106 1.4119 0.486 0.530 0.439 0.4830.011 1.4104 1.4121 0.478 0.538 0.431 0.4910.018 1.4102 1.4124 0.469 0.545 0.423 0.4980.019 1.4101 1.4123 0.465 0.543 0.419 0.4960.038 1.4099 1.4127 0.458 0.557 0.411 0.5100.088 1.4095 1.4132 0.442 0.573 0.396 0.5260.180 1.4090 1.4138 0.422 0.588 0.376 0.5410.364 1.4084 1.4146 0.397 0.612 0.352 0.5660.609 1.4080 1.4152 0.375 0.627 0.332 0.5811.053 1.4076 1.4162 0.351 0.652 0.309 0.6081.426 1.4073 1.4169 0.331 0.666 0.290 0.6231.646 1.4072 1.4173 0.323 0.675 0.283 0.6322.182 1.4070 1.4181 0.306 0.693 0.268 0.6512.962 1.4069 1.4191 0.286 0.711 0.249 0.6703.826 1.4070 1.4201 0.271 0.728 0.235 0.6894.674 1.4071 1.4209 0.256 0.741 0.222 0.7036.492 1.4073 1.4225 0.231 0.764 0.199 0.7288.665 1.4078 1.4243 0.204 0.787 0.175 0.754

11.212 1.4084 1.4263 0.181 0.811 0.154 0.780

Ž .The coexistence curves of diethyl maleate q decane 1201

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. y7 Ž .TABLE 2. Refractive indices n at wavelength l s 6.328 10 m for CHCOOC H2 5 2Ž .and CH CH CH at various temperatures T3 2 8 3

Ž . Ž .CHCOOC H CH CH CH2 5 2 3 2 8 3TrK n TrK n

307.749 1.4329 308.286 1.4043310.372 1.4317 310.297 1.4035312.339 1.4308 311.273 1.4030314.388 1.4300 312.309 1.4025316.328 1.4292 313.386 1.4019318.357 1.4283 314.288 1.4015320.358 1.4274 315.291 1.4011322.226 1.4266 317.298 1.4002324.021 1.4258 319.200 1.3993326.213 1.4249

Ž .In order to obtain the T , x coexistence curve, the refractive indices of purediethyl maleate and n-decane at various temperatures, and the refractive indices ofa series of binary mixtures with known values of the mole fractions in a singlephase region were also measured at T s 318.81 K . The results are listed in tables2 and 3. The refractive index may be expressed as a linear function of temperaturein a certain temperature range by:

. .n T , x s n T8, x q ­ nr­T T y T8 , 1aŽ . Ž . Ž . Ž . Ž .x

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

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 .x s 1 and x s 0, respectively. The validity of equation 1 has been tested inŽ1,4. Ž .previous work. The results listed in table 2 were fitted to equation 1a to obtain

. y4 . y4Ž . Ž .­ n r­ T s y4.31 10 and ­ n r­ T s y4.60 10 with a standard deviationA B. y4 Ž . Ž .of 1 10 in n. The values of n T8, x were calculated from equation 1 with the

Ž . Ž .values of n T , x listed in table 3, and a polynomial form of n T8, x for various xat T8 was obtained:

. . 2 . 3 . 4n T8, x s 1.4019 q 0.0282 x y 0.0400 x q 0.0807 x y 0.0403 x , 2Ž . Ž .. y4with a standard deviation of 1 10 . The value of T8 was chosen as T s 313.15 K,

corresponding to about the middle temperature for the coexistence curve

. y7 � Ž .TABLE 3. Refractive indices n at wavelength l s 6.328 10 m for x CHCOOC H q2 5 2Ž . Ž . 41 y x CH CH CH at the temperature T s 318.81 K3 2 8 3

x n x n x n

0.0000 1.3995 0.4002 1.4083 0.7985 1.41070.1000 1.4017 0.5050 1.4113 0.9004 1.42470.1988 1.4035 0.6013 1.4140 1.0000 1.42820.3003 1.4057 0.7006 1.4172

Ž .The coexistence curves of diethyl maleate q decane 1203

determined in this work. The values of the refractive indices for the coexistingŽ .phases were then converted to mole fractions by calculating n T8, x through

Ž . Ž .equation 1 and iteratively solving equation 2 . The results are listed in columns 4Ž .and 5 of table 1 and shown in figure 1 b .

The mole fraction was used to calculate the volume fraction using,

1rfs 1 y K q Krx , 3aŽ . Ž .Ks d M r d M , 3bŽ . Ž . Ž .A B B A

where M is the molar mass, d is the mass density, and subscripts A and B refer todiethyl maleate and n-decane, respectively. The values of d and d wereA Bobtained from references 5 and 6. 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:Ž7.

r y r s Bt b q B t bqD q ??? , 4Ž .2 1 1

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

Ž .Ž8,9.and 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 the1order parameter. In a region sufficiently close to the critical temperature, equationŽ .4 becomes:

r y r s Bt b. 5Ž .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 5 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 slightly on the cutoff values of T y T ,c

but the values of b are in good agreement with the theoretical prediction ofŽ . Ž8.0.3265 " 0.001 .

TABLE 4. Values of the critical amplitudes B and critical exponents b for the coexistence curves ofŽ . Ž . Ž . Ž . � Ž . Ž . Ž . 4T , n , T , x , and T , f in equation 5 for x CHCOOC H q 1 y x CH CH CH2 5 2 3 2 8 3

OrderŽ . Ž .parameter T y T - 1.06 K T y T - 11.3 Kc c

B b B b

n 0.055 " 0.004 0.328 " 0.002 0.055 " 0.003 0.328 " 0.002x 1.93 " 0.17 0.329 " 0.002 1.95 " 0.06 0.329 " 0.004f 1.92 " 0.18 0.329 " 0.002 1.93 " 0.06 0.329 " 0.004

X. An et al.1204

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

r s r q r r2 s r q Dt Z q ??? , 6Ž . 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 examined by testing the symmetry of the coexistence curve and by

Ž . Ž .comparing the goodness of the fitting equation 6 with Z s 1, 1 y a , and2b ,Ž10,11. where a s 0.11Ž9. characterizes the divergence, as the critical point isapproached, of the heat capacity at constant volume for the pure fluid. The resultsfor the system we studied are compared in table 5, where the experimental value ofn was obtained by extrapolating the refractive indices against temperatures inc, exptthe single-phase region up to the critical temperature, and the value of x wasc, exptdetermined by the technique of ‘‘equal volume’’. The value of f listed in tablec, expt

Ž .5 was then calculated from x through equation 3 . The uncertainties of rc, expt c, exptlisted in table 5 include no systematic errors arising from converting n to x, and x

. y3to f. Such errors in x and f were estimated to be "5 10 . The values of n , x ,c cŽ .and f obtained from the extrapolation of equation 6 are consistent with thosec

observed, as shown in table 5. 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 6 may be indicated by the values of the standarddeviation s listed in table 5. We have found that the fit with t 1ya is slightly betterthan the fit with t 2 b for f, but the best fit for the order parameter x is obtainedwith t 2 b. This indicates that f is a slightly better order parameter than x.

Ž .TABLE 5. Coefficients of equation 6 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 8 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 1.4113 " 0.0002 0.505 " 0.001 0.458 " 0.001c, expt

Z s 1r 1.4113 " 0.0001 0.505 " 0.001 0.459 " 0.001cD 0.176 " 0.001 y0.35 " 0.07 0.21 " 0.05

y4 y3 y3. . .s 1.1 10 3.0 10 2.1 10

Z s 0.89r 1.4112 " 0.0001 0.505 "0.001 0.459 " 0.001cD 0.1193 " 0.0007 y0.24 " 0.04 0.14 " 0.03

y5 y3 y3. . .s 9.2 10 2.8 10 2.1 10

Z s 0.654r 1.4110 " 0.0001 0.506 " 0.001 0.459 " 0.001cD 0.051 " 0.001 y0.11 " 0.02 0.06 " 0.02

y4 y3 y3. . .s 4.2 10 2.5 10 2.3 10

Ž .The coexistence curves of diethyl maleate q decane 1205

Ž . Ž .TABLE 6. Critical amplitudes B and B for the coexistence curves of T , n , T , x , and1Ž . Ž . � Ž . Ž . Ž . 4T , f from equation 4 for x CHCOOC H q 1 y x CH CH CH2 5 2 3 2 8 3

Order parameter B B1

n 0.0545 " 0.00040.0559 " 0.001 y0.011 " 0.008

x 1.919 " 0.0131.977 " 0.033 y0.47 "0.2

f 1.904 " 0.0131.959 " 0.032 y0.45 "0.2

When the critical exponents b and D are fixed at the theoretical valuesŽ . Ž7,8. Ž .b s 0.3265, D s 0.50 , equation 4 was used to fit the phase-separation datato obtain the parameters B and B . The results are listed in table 6. Combination1

Ž . Ž .of equations 4 and 6 yields:

r s r q Dt Z y 1r2 Bt b y 1r2 B t bqD , 7Ž . Ž . Ž .1 c 1

r s r q Dt Z q 1r2 Bt b q 1r2 B t bqD . 8Ž . Ž . Ž .2 c 1

When Z, b , D, and T are fixed at 0.89, 0.3265, 0.5, and 318.802 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

Ž . Ž . Ž .r , and r can be calculated from equations 7 , 8 , and 6 . The results are shown2 das lines in figure 1. The calculated values are in good agreement with theexperimental results.

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.

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Related Components, Vol. I. 1972.7. Wegner, F. J. Phys. Re¨ . 1972, B5, 4529]4536.8. Alpert, D. Z. Phys Re¨ . 1982, B25, 4810]4814.9. Le Guillou, J. C.; Zinn-Justin, J. Phys. Re¨ . 1980, B21, 3976]3998.

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( )Recei ed 3 December 1997; in final form 19 May 1998

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