doi: 10.1006/jcht.1999.0583Available online at http://www.idealibrary.com on

J. Chem. Thermodynamics 2000, 32, 187195

The coexistence curves for (diethyl maleate +undecane) in the critical region

Xueqin An, Xianlin Cui, and Weigou ShenaDepartment of Chemistry, Lanzhou University, Lanzhou, Gansu 730000 ,P.R.C.

Coexistence curves for {x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3} have been deter-mined by measurement of the refractive index. The experimental results have been used inthe determination of the critical exponent , the critical amplitude B, and the study of thediameters of the coexistence curves. The values of the critical exponent are found to beconsistent with theoretical predictions. The coexistence curves have been successfully de-scribed by a combination of the Wegner equation and the expression for the diameter. Theexponents r and b in the relations (1 c)/c Mr and B1.365c Mb, where cis the critical volume fraction of (CHCOOC2H5)2, have been found to be 0.42 0.01 and0.28 0.04, respectively, hence consistent with observations from experimental studieson other chain-molecule solution systems. It supports the LandauGinsburgWilson typemodel we proposed recently. c 2000 Academic Press

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

1. IntroductionIn the region sufficiently close to the critical solution temperature, the coexistence curve ofa binary solution may be represented by:

2 1 = B , (1)where = (Tc T )/Tc, Tc is the critical temperature, is the density variable orthe order parameter, 1 and 2 are the values of in the upper and lower coexistencephases, is a universal critical exponent with a theoretical value of 0.3265(1) whichdescribes the shape of the coexistence curve, and B is a critical amplitude which dependson the particular mixture. Recently, we proposed a LandauGinsburgWilson type modelto describe the dependence of the critical amplitudes on the molar mass of the chain-

aTo whom correspondence should be addressed.

00219614/00/020187 + 09 $35.00/0 c 2000 Academic Press

188 X. An, X. Cui, and W. Shen

molecule component, while the other component is fixed, for a series of binary chain-molecule solutions.(2) From this model we derived a general form:

B1.865c Mb, (2)where B is the amplitude in equation (1) when the volume fraction of the non-chain-molecule component is chosen as the order parameter, c is the critical value of , M isthe molar mass of the chain molecule, and b is the M exponent with a theoretical valueof 0.29. In addition, c was experimentally found to be dependent on the molar mass asfollows:

(1 c)/c Mr , (3)with the universal exponent r = 0.41(3) for the chain-molecule solutions of both smallmolecules and polymers.

In previous work,(4) we reported the coexistence curves near the critical pointsfor (diethyl maleate + n-alkane) with the alkane carbon numbers 6, 7, 8, 9,and 10. The coexistence curve measurements supported equations (2) and (3) andyielded r = 0.42 and b = 0.31. To further test the validity of the universalityof equations (2) and (3), we report measurements of the coexistence curves of{x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3}. The critical amplitude B and thecritical exponent are deduced from (T, n), (T, x), and (T, ) curves, where n and x arethe refractive index and the mole fraction, respectively, and the behaviour of the diametersd of the coexistence curves are examined. By adding the new experimental results to the(diethyl maleate + n-alkane) series, the values of the exponents b and r in equations (2)and (3) are determined and compared with the predicted ones by the theory, and the onesfound in other experimental studies.

2. ExperimentalDiethyl maleate (CHCOOC2H5)2 obtained from Beijing Chemicals Factory was purifiedby fractional distillation under vacuum. The undecane (mass fraction= 0.99) was suppliedby Aldrich Chemical Company Inc. and was stored over 4 1010 m molecular sieves.

The coexistence curves were determined by measurement of the refractive indices usingthe method of minimum deviation. The apparatus and the experimental procedure for themeasurement of the refractive index, and the techniques for the determination of the criticaltemperature and the critical composition have been described previously.(5) During themeasurements, the temperature was constant to within 2 103 K. The accuracy andprecision in the measurement of temperature were better than 102 K and 103 K,respectively. The accuracy of measurement was3 103 K for the temperature difference(T Tc),1 104 for the refractive index in each coexisting phase, and1 103 for thecritical mole fraction xc.

3. Results and discussionThe critical mole fractions and the critical temperatures of {x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3}were determined to be (0.5360.001) and (324.30.1)K, respectively.

The coexistence curves of (diethyl maleate + undecane) 189TABLE 1. Coexistence curves of (T, n), (T, x), and (T, ) for{x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3} where n is the re-fractive index, xi is the mole fraction, i is the volume fractionof substance i . Refractive indices were measured at wavelength = 6.328 107 m. Subscripts 1 and 2 relate to upper and lower

phases

(Tc T )/K n1 n2 x1 x2 1 20.002 1.4095 1.4103 0.517 0.555 0.450 0.4880.005 1.4093 1.4104 0.508 0.561 0.441 0.4950.010 1.4092 1.4107 0.501 0.571 0.434 0.5040.016 1.4091 1.4108 0.497 0.576 0.430 0.5100.028 1.4089 1.4110 0.488 0.584 0.422 0.5180.033 1.4090 1.4112 0.494 0.595 0.428 0.5290.043 1.4089 1.4112 0.486 0.593 0.419 0.5270.048 1.4090 1.4115 0.490 0.605 0.424 0.5380.062 1.4087 1.4113 0.476 0.597 0.410 0.5310.089 1.4086 1.4115 0.469 0.605 0.403 0.5390.156 1.4083 1.4119 0.454 0.620 0.389 0.5550.208 1.4082 1.4121 0.448 0.629 0.383 0.5650.270 1.4081 1.4124 0.439 0.636 0.375 0.5720.354 1.4079 1.4126 0.427 0.646 0.363 0.5820.435 1.4078 1.4127 0.420 0.648 0.357 0.5850.566 1.4077 1.4131 0.410 0.661 0.347 0.5980.773 1.4076 1.4137 0.398 0.679 0.336 0.6181.179 1.4074 1.4142 0.374 0.690 0.313 0.6291.645 1.4074 1.4149 0.361 0.709 0.301 0.6512.249 1.4073 1.4157 0.340 0.725 0.282 0.6692.900 1.4072 1.4163 0.316 0.737 0.261 0.6823.599 1.4073 1.4171 0.303 0.751 0.250 0.6984.690 1.4075 1.4180 0.280 0.765 0.230 0.7136.742 1.4080 1.4197 0.251 0.790 0.204 0.742

10.190 1.4091 1.4221 0.215 0.816 0.173 0.772

The refractive indices for each coexisting phase were measured at various temperatures.The results are listed in columns 2 and 3 of table 1, and shown in figure 1(a).

In order to obtain the (T, x) coexistence curve, the refractive indices of pure diethylmaleate and undecane at various temperatures and the refractive indices of a series ofbinary mixtures with known mole fractions in the single phase region were also measuredat T 0 = 324.4 K. The results are listed in tables 2 and 3. A polynomial form of n(T 0, x)

190 X. An, X. Cui, and W. Shen

1.405 1.410

n x

1.415 1.420 0.212

10

8

6

(Tc

T )/

K 4

2

0 a b c

0.4 0.6 0.8 0.2 0.4 0.6 0.8

FIGURE 1. Coexistence curves of a, (T, n); b, (T, x); and c, (T, ) for{x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3}, where T is the temperature, n is the refrac-tive index, x is the mole fraction, and is the volume fraction. , Experimental values of theconcentration variables of the coexisting phases; N, experimental values of the diameter d of thecoexisting phases; , calculated concentration variables and diameter of coexisting phases.

as a function of x was used to represent the data:

n(T 0, x) = 1.40228+ 0.00765 x + 0.01457 x2 0.01083 x3 + 0.01210 x4, (4)with a standard deviation of 2 104. The refractive index may be expressed as a linearfunction of temperature in a certain temperature range by:

n(T, x) = n (T 0, x)+ (n/T )x (T T 0), (5)(n/T )x = x (nA/T )+ (1 x) (nB/T ), (6)

where (n/T )x is the derivative of n with respect to T for a particular composition x ,(nA/T ) and (nB/T ) are the values of (n/T )x for x = 1 and x = 0, respectively.The validity of equations (4) and (5) has been confirmed in previous work.(6) The valuesof the refractive indices for the coexisting phases were then converted to mole fractionsby calculating n(T 0, x) through equations (5) and (6), and iteratively solving equation (4).The results are 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:

1/ = (1 K )+ K/x, (7)K = (dA MB/(dB MA), (8)

where M is the molar mass, d is the mass density, and the subscripts A and B refer todiethyl maleate and undecane, respectively. The values of dA and dB were obtained fromreferences 7 and 8. The values of of the coexisting phases for this system at varioustemperatures are listed in columns 6 and 7 of table 1 and shown in figure 1(c).

The differences (2 1) of the density variables of the coexisting phases werecalculated from the results listed in table 1 and were fitted to equation (1) to obtain thecritical exponent and the critical amplitude B. The results are shown in table 4. The

The coexistence curves of (diethyl maleate + undecane) 191TABLE 2. Refractive index n at wave-length = 6.328 107 m for diethylmaleate and undecane at various temper-

atures T

Diethyl maleate Undecane

T/K n T/K n

307.749 1.4329 314.283 1.4067310.372 1.4317 316.260 1.4057312.339 1.4308 318.268 1.4049314.388 1.4300 320.254 1.4037316.328 1.4292 322.305 1.4030318.357 1.4274 324.095 1.4023322.226 1.4266324.021 1.4258326.213 1.4249

TABLE 3. Refractive index n at wavelength = 6.328 107 m for {x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3} at

T = 324.4 K

x n x n x n

0.0000 1.4022 0.4004 1.4074 0.7991 1.41690.1000 1.4034 0.4987 1.4090 0.8997 1.42100.1996 1.4042 0.6008 1.4114 1.0000 1.42580.2998 1.4056 0.7001 1.4142

values of and B depend on the cut-off values of (Tc T ), but for (Tc T ) < 1 K, thevalues of for three choices of variables are all in good agreement with the theoreticalprediction of (0.3265 0.001) within the experimental uncertainties.

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

d = (2 + 1)/2 = c + D Z + , (9)where c is the value of at the critical point, D is a system-dependent parameter,and Z is the apparent exponent. For a binary mixture there is no a priori reason forchoosing one density variable rather than another, but the variables may be tested byexamining the symmetry of the coexistence curve and by comparing the goodness of fitsof equation (9) with (1) and 2,(9, 10) where = 0.11(11) characterizes the divergence,as the critical point is approached, of the heat capacity at constant volume for the purefluid. The experimental diameter data were fitted to equation (9). The results of the fits are

192 X. An, X. Cui, and W. Shen

TABLE 4. Values of the critical amplitudes B and critical expo-nents for the coexistence curves of (T, n), (T, x), and (T, ) for

{x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3} in equation (1)

Order (Tc T ) < 1 K (Tc T ) < 10 Kparameter B B

n 0.044 0.001 0.330 0.003 0.042 0.001 0.323 0.002x 2.01 0.03 0.327 0.003 1.90 0.03 0.321 0.001 2.03 0.02 0.328 0.001 1.89 0.04 0.321 0.001

compared in table 5, where the experimental value of nc,expt was obtained by extrapolatingthe refractive indices against temperature in the single phase region up to the criticaltemperature; the value of xc,expt was determined by the technique of equal volumes;and the value of c,expt was then calculated from xc,expt through equations (7) and (8).The uncertainties of the optimal parameters repeated in table 5 include no systematicerrors arising from converting n to x , and x to . Such uncertainties in x and wereestimated to be about 8 103. Therefore the values of nc, xc, and c obtained from theextrapolation of equation (9) are consistent with those observed. This constitutes evidencethat no significant critical anomaly was present in the refractive indices and that therefractive indices were properly converted to mole fractions and volume fractions in ourtreatment. The quality of the fit of equation (9) may be indicated by the values of thestandard deviation s listed in table 5. We have found no significant difference between fitswith 1 and with 2 for both x and . Therefore we conclude that and x are almostequally good variables for the construction of order parameters for the system under study.This is consistent with what the symmetry indicates: no significant symmetry differencebetween (T, x) and (T, ) is observed in figure 1.

It is necessary to add Wegner-correction terms(12) to equation (1) when they are appliedover a wider range of temperatures. Thus equation (1) becomes:

2 1 = B + B1+1 + . . . , (10)where 1 = 0.50(11) is the first Wegner exponent, and B1 depends on the system andthe choice of the order parameter. When the critical exponents and 1 are fixed at thetheoretical values ( = 0.3265,1 = 0.50) and equation (10) is used to fit the phase-separation data, the parameters B and B1 are obtained. The results are listed in table 6.

Combination of equations (9) and (10) yields:1 = c + D Z (1/2)B (1/2)B1+1, (11)2 = c + D Z + (1/2)B + (1/2)B1+1. (12)

When Z , , 1, and Tc are fixed at 0.89, 0.3265, 0.5, and 324.3 K, respectively, the valuesof D, c, B, and B1 are taken from tables 5 and 6, and the values of 1, 2, and d arecalculated from equations (11), (12), and (9). The results are shown as lines in figure 1.The calculated values are in good agreement with the experimental results.

The coexistence curves of (diethyl maleate + undecane) 193

4.40.10

0.15

0.20

0.25

ln{1

c)/

c}

0.30

0.35

0.40

4.5 4.6 4.7

ln (M /g mol1)4.8 4.9 5.0 5.1

FIGURE 2. A lnln plot of (1 c)/c against the molar mass M for (diethyl maleate + n-alkane);, experimental values; , calculated values from equation (3) with r = 0.422.

4.41.90

1.95

2.00

2.05

2.10

2.15

ln(B

c1.

865 ) 2.20

2.25

2.30

2.35

2.40

4.5 4.6 4.7ln (M /g mol1)

4.8 4.9 5.0 5.1

FIGURE 3. A lnln plot of B1.865c against the molar mass M for (diethyl maleate + n-alkane);, experimental values; , calculated values from equation (2) with b = 0.278.

194 X. An, X. Cui, and W. Shen

TABLE 5. Parameters of equation (9) and standard deviation sfor the diameters of the coexistence curves d of (T, n), (T, x),and (T, ) for {x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3}.c,expt is the critical value of the order parameter determined by

the techniques described in the text

(T, n) (T, x) (T, )c,expt 1.4099 0.536 0.467

Z = 0.89c 1.4100 0.0001 0.538 0.001 0.473 0.001D 0.117 0.001 0.55 0.05 0.017 0.095S 8.0 105 3.1 103 3.8 103

Z = 0.653c 1.4098 0.0001 0.539 0.001 0.473 0.001D 0.050 0.002 0.23 0.02 0.005 0.036S 2.0 104 3.1 103 3.7 103

TABLE 6. Parameters B and B1 of equation (10) forthe coexistence curves of (T, n), (T, x), and (T, )for {x(CHCOOC2H5)2 + (1 x)CH3(CH2)9CH3}where n is the refractive index, x is the mole fraction,

and is the volume fraction

Orderparameter B B1

n 0.0420 0.00020.0440 0.0002 0.018 0.002

x 1.937 0.0102.034 0.007 0.92 0.06

1.955 0.0112.019 0.023 0.73 0.24

According to equations (2) and (3), lnln plots of (1 c)/c and B1.865cagainst the molar mass of n-alkane M will yield two straight lines. These two plotsare shown in figures 2 and 3. A least-squares fit results in the values of 0.42 0.01and 0.28 0.04 for the exponents r and b. The lines in figures 2 and 3 represent:(1 c)/c = 9.56(M/g mol1)0.422 and B1.865c = 32.4(M/g mol1)0.278 withstandard deviations of about 1 103 and 3 103 in c and B, respectively. The valuesof r and b extracted from experimental studies of binary solutions of (diethyl maleate

The coexistence curves of (diethyl maleate + undecane) 195

+ n-alkanes) are in excellent agreement with the theoretical value (b = 0.29)(2) andthe observed values from experimental studies on chain-molecule solutions of both smallmolecules(2, 3, 13) and polymers.(2) This supports the universality of the exponents of Mand the LandauGinsburgWilson type model we proposed recently for chain-moleculesolutions.(2)

This work was supported by the National Natural Science Foundation (Project 29673019),the State Education Committee, and the Natural Science Foundation of Gansu Province,P.R.C.

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(Received 5 March 1999; in final form 22 July 1999)WA98/016

IntroductionExperimentalResults and discussionTable 1Fig. 1Table 2Table 3Table 4Fig. 2Fig. 3Table 5Table 6

References