J. Chem. Thermodynamics 1997, 29, 1047–1054

The coexistence curves of{xC6H5NO2 + (1− x)CH3(CH2)12CH3} in thecritical region

Xueqin An, Haihong Zhao, a Fuguo Jiang, Chunfeng Mao,and Weiguo Shen b

Department of Chemistry, Lanzhou University, Lanzhou, Gansu 730000,P.R. China

Coexistence curves for {xC6H5NO2 + (1− x)CH3(CH2)12CH3} have been determined bymeasurements of refractive index. The critical amplitude B and the critical exponent b havebeen deduced from (T, n), (T, x), and (T, f) curves, where n, x, and f are refractive index,mole fraction, and volume fraction, respectively. The values of b are consistent with thetheoretical ones. The experimental results have also been analysed to examine Wegnercorrection terms and the behaviour of the diameter of the coexistence curve in various unitsover a temperature range of 10 K from the critical temperature. The coexistence curves havebeen successfully described by a combination of the Wegner equation and the expression forthe diameter. The molar mass (M)-dependence of critical volume fraction fc may be expressedas (1−fc)/fc A M−r and −r is found to be 0.40. 7 1997 Academic Press Limited

KEYWORDS: coexistence curve; critical phenomena; refractive index; nitrobenzene;tetradecane

1. Introduction

In previous work, the coexistence curves were measured for five mixturesnear the critical points by measurements of refractive index using a techniqueof ‘‘minimum deviations’’.(1–4) In those mixtures, one of the components wasN,N-dimethylacetamide, and the other was an n-alkane with varying chainlength. The critical volume fraction of the polar solvent fc, and the difference ofvolume fractions with the Df between two coexisting phases at various temperaturesfor each mixture were fitted with the following equations, with b fixed at thetheoretical value 0.327:

=Df =/fc A M−btb, (1)

(1−fc)/fc A M−r, (2)

where t is the related temperature: t=(Tc −T )/Tc, Tc is the critical temperature,and M is the molar mass. The values of (0.412 0.01) and (0.062 0.04) for r andb were obtained for mixtures of N,N-dimethylacetamide in n-alkane. For polymer

a Present address: Chendu Organic Chemistry Institute, The Chinese Academy of Science.b To whom correspondence should be addressed.

0021–9614/97/091047+08 $25.00/0/ct970222 7 1997 Academic Press Limited

X. An et al.1048

solutions, assuming fc : 1, then equations (1) and (2) become:

=Df = A M−btb (3)

(1−fc) A M−r. (4)

The value of r is (0.382 0.01) from experimental studies of polymer solutions,which is close to the value of 0.41. However the value of 0.06 for b from our previousstudies is significantly smaller than 0.25 from the classical theory, and 0.28 from theexperimental value for polymer solutions.

As a continuing part of the study of the molar mass dependence of criticalbehaviour in binary mixtures of chain molecules, this paper reports the measurementsof coexistence curves for (nitrobenzene+ tetradecane) and the critical volumefractions of a series of mixtures of nitrobenzene in n-alkanes with different chainlengths.

2. Experimental

The nitrobenzene obtained from Beijing Chemical Factory (A.R. grade) was driedover calcium chloride for 5 days and distilled slowly under reduced pressure.The middle part of the distillate was collected and passed through a column ofdried chromatographic alumina. The purified nitrobenzene was stored in darknessover a 0.4 mm molecular sieve. The pentane, heptane, octane, tridecane, andhexadecane (mass fraction: 0.99) were supplied by the Aldrich Chemical Co.The hexane, undecane, dodecane, and tetradecane (mass fraction: 0.99) were suppliedby the Shanghai Chemical Reagent Factory. The n-alkanes were dried and storedover a 0.4 mm molecular sieve.

The critical compositions were determined by the technique of ‘‘equal volumesof two coexisting phases’’.(1–5) The apparatus and experimental procedure formeasurement of refractive index have been described previously.(1−4) Duringmeasurements, the temperature was constant to 20.002 K. The accuracy andthe precision in measurements of temperature were better than 20.01 K and20.001 K, respectively. The accuracy of measurement was 20.003 K for thetemperature difference (Tc −T), 20.0002 for the refractive index in each coexistingphase, and 20.001 for the critical mole fraction xc.

3. Results and discussion

The refractive indices n were measured for each coexisting phase of{xC6H5NO2 + (1− x)CH3(CH2)12CH3} at various temperatures. The results arelisted in columns 2 and 3 of table 1 and are also shown in figure 1a.

In previous papers,(1–4) we assumed that the refractive index n of a liquid ora mixture may be expressed as a linear function of temperature in a certaintemperature range:

n(T, x)= n(T°, x)+R(x)·(T−T°), (5)

R(x)= xRA + (1− x)RB, (6)

Coexistence curves of 4xC6H5NO2 + (1− x)CH3(CH2)12CH35 1049

TABLE 1. Coexistence curves of (T, n), (T, x), (T, f) for {xC6H5NO2 + (1− x)CH3(CH2)12CH3}.Refractive indices n were measured at wavelength l=632.8 nm, and Tc =304.940 K. Volume fraction is

denoted by f. Subscripts 1 and 2 relate to upper and lower phases, respectively

T/K n1 n2 x1 x2 f1 f2

304.939 1.4734 1.4771 0.662 0.689 0.436 0.468304.935 1.4730 1.4775 0.659 0.692 0.433 0.471304.929 1.4724 1.4784 0.654 0.698 0.428 0.478304.925 1.4716 1.4790 0.648 0.702 0.422 0.483304.919 1.4711 1.4795 0.644 0.706 0.417 0.488304.911 1.4707 1.4801 0.641 0.710 0.414 0.492304.902 1.4702 1.4804 0.637 0.712 0.409 0.495304.888 1.4697 1.4810 0.633 0.716 0.406 0.500304.872 1.4692 1.4815 0.629 0.719 0.402 0.503304.857 1.4687 1.4820 0.625 0.722 0.398 0.507304.840 1.4684 1.4826 0.622 0.726 0.394 0.512304.813 1.4678 1.4832 0.617 0.730 0.389 0.517304.776 1.4671 1.4838 0.611 0.734 0.384 0.522304.727 1.4664 1.4845 0.605 0.738 0.378 0.527304.669 1.4659 1.4853 0.600 0.743 0.373 0.533304.597 1.4651 1.4861 0.593 0.748 0.366 0.540304.487 1.4641 1.4872 0.584 0.754 0.358 0.548304.371 1.4633 1.4882 0.576 0.760 0.349 0.556304.252 1.4626 1.4891 0.569 0.765 0.343 0.563304.051 1.4616 1.4902 0.559 0.771 0.334 0.572304.017 1.4614 1.4906 0.557 0.773 0.332 0.574303.793 1.4602 1.4919 0.545 0.780 0.322 0.584303.354 1.4588 1.4941 0.529 0.791 0.308 0.599303.060 1.4579 1.4951 0.519 0.796 0.299 0.608302.654 1.4570 1.4968 0.508 0.804 0.290 0.619302.038 1.4556 1.4989 0.490 0.814 0.276 0.634301.277 1.4543 1.5010 0.472 0.823 0.262 0.648300.445 1.4532 1.5032 0.455 0.832 0.248 0.662299.464 1.4519 1.5055 0.436 0.841 0.234 0.677298.558 1.4513 1.5073 0.423 0.848 0.225 0.688297.723 1.4503 1.5090 0.407 0.854 0.214 0.698296.852 1.4497 1.5107 0.395 0.860 0.205 0.708296.084 1.4493 1.5121 0.385 0.865 0.199 0.717294.858 1.4486 1.5141 0.369 0.871 0.188 0.728

where R(x) is the derivative of n with respect to T for a particular composition x,RA and RB are the values of R(x) for x=1 and x=0, respectively. It allowedus to obtain n(T, x) as a function of T and x simply by measuring R(x) fortwo pure components and the refractive indices for mixtures with various knowncompositions at a fixed temperature. In this paper, as a further test, we measuredthe refractive indices of a series of mixtures of (nitrobenzene+ tetradecane)with known compositions at various temperatures. The results are listed in table 2.Fitting equation (5) with pure nitrobenzene and tetradecane gives RA =−5.13·10−4·K−1 and RB =−4.57·10−4·K−1. The results for n(T, x) at eachcomposition were fitted with equations (5) and (6) to obtain n(T°, x) with standarddeviation of 0.0002 in n. The value of T° was chosen as 305.677 K. The smallstandard deviation indicates that equations (5) and (6) are valid. A polynomial form

X. An et al.1050

FIGURE 1. Coexistence curves of: a, temperature against refractive index (T, n); b, temperatureagainst mole fraction (T, x); and c, temperature against volume fraction (T, f) for {xC6H5NO2 +(1− x(CH3(CH2)12CH3}. W, Experimental values of concentration variables (r) of the coexisting phases;R, Experimental values of diameter (rd) of the coexisting phases; - - - -, Concentration variables(rcal) of coexisting phases from calculation.

of n(T°, x) for various x at T° was obtained as following:

n(T°, x)=1.4221+0.0358·x+0.0795·x2 −0.0849·x3 +0.0889·x4, (7)

with a standard deviation of less than 0.0002. It was used to convert refractive indicesto mole fractions. The volume fraction f of C6H5NO2 was then calculated fromthe mole fraction by:

1/f=(1−K)+K/x, (8)

K= dAMB/dBMA, (9)

where d is mass density, M is the molar mass and subscript A and B relateto nitrobenzene and tetradecane, respectively. The values of dA and dB wereobtained from references 6 and 7. The values of (x, f) for coexisting phases atvarious temperature are listed in columns 4 to 7 of table 1, and are shown infigure 1b and c.

Coexistence curves of 4xC6H5NO2 + (1− x)CH3(CH2)12CH35 1051

TABLE 2. Refractive indices n at wavelength l=632.8 nm for {xC6H5NO2 + (1− x)CH3(CH2)12CH3)}at various temperatures

x T/K n T/K n T/K n T/K n

0.0000 295.010 1.4267 298.094 1.4253 301.619 1.4238 305.089 1.4222296.140 1.4262 299.164 1.4249 302.716 1.4233 306.344 1.4216297.056 1.4258 300.401 1.4244 303.884 1.4228 307.821 1.4211

0.0996 295.758 1.4312 300.092 1.4291 303.941 1.4274 305.285 1.4269297.691 1.4302 301.560 1.4285 304.577 1.4272 305.671 1.4267

0.2070 295.200 1.4371 301.359 1.4343 304.853 1.4328297.057 1.4362 303.744 1.4332 305.298 1.4326299.205 1.4353 303.251 1.4330 305.682 1.4324

0.2946 294.799 1.4428 298.796 1.4410 302.935 1.4392 305.237 1.4382296.691 1.4419 301.089 1.4400 304.679 1.4384 305.681 1.4380

0.3971 297.567 1.4496 301.030 1.4479 303.925 1.4465 305.081 1.4460299.343 1.4487 302.812 1.4471 304.548 1.4463 305.666 1.4458

0.4998 302.612 1.4563 304.515 1.4554 305.683 1.4548 308.028 1.4537304.008 1.4556 305.105 1.4551 306.731 1.4543 310.318 1.4527

0.6018 304.784 1.4661 305.662 1.4656 306.987 1.4649 308.963 1.4640305.354 1.4658 306.334 1.4653 308.014 1.4645 309.986 1.4635

0.7113 305.034 1.4804 306.371 1.4798 309.402 1.4784 312.057 1.4772305.687 1.4801 307.874 1.4791 311.104 1.4777

0.8033 303.112 1.4965 305.684 1.4951 309.266 1.7933 312.562 1.4917304.481 1.4958 307.692 1.4941 310.818 1.4925

0.8891 295.254 1.5172 300.384 1.5148 303.724 1.5132 305.682 1.5123297.705 1.5161 301.905 1.5142 304.934 1.5127 306.090 1.5121

1.0000 285.008 1.5515 293.312 1.5472 299.062 1.5449 305.096 1.5417287.515 1.5501 295.136 1.5468 301.799 1.5435 306.751 1.5409290.264 1.5487 297.105 1.5459 303.885 1.5423 307.669 1.5404

In the region sufficiently close to the critical temperature, the coexistence curve canbe represented by:(1–4)

r2 − r1 =Btb, (10)

where B is the critical amplitude, b is the critical exponent, r is the ‘‘density’’variable, r1 and r2 are the values of r in upper and lower coexistence phases.The differences (r2 − r1) obtained in this work were fitted with equation (10),with all points equally weighted, to obtain b and B. The results are listed intable 3. The values of b and B slightly depend on the cutoff values of tmax butfor tmax Q 0.003, the values of the exponent b approach the theoretical value of0.327.

With the critical exponents b and D to be fixed at the theoretical values (b=0.327,D=0.50),(8,9) a weighted non-linear least-squares programme was used to fit theWegner equation:(10)

r2 − r1 =Btb +B1tb+D +B2t

b+2D +· · ·, (11)

to obtain the parameters B and B1. The total uncertainties used in weighting includedthe errors in measurements of (r2 − r1), and standard deviations propagated frommeasurements of (Tc −T ). The results are summarized in table 4.

X. An et al.1052

TABLE 3. Values of critical amplitudes B and critical exponents b for coexistence curves of (T, n), (T, x),and (T, f) for {xC6H5NO2 + (1− x)CH3(CH2)12CH3} in equation (10); t=(Tc −T )/Tc

tQ 0.030 tQ 0.003

Order parameter B b B b

n 0.2002 0.004 0.3312 0.003 0.192 0.01 0.3252 0.005x 1.542 0.05 0.3362 0.003 1.402 0.06 0.3262 0.005f 1.652 0.03 0.3302 0.003 1.572 0.07 0.3252 0.005

TABLE 4. Coefficients of equation (11) and x2v for coexistence curves of (T, n), (T, x), and (T, f) for

{xC6H5NO2 + (1− x)CH3(CH2)12CH3}; x2v is a measure of accuracy of fit

Order parameter B B1 x2v

n 0.19792 0.0002 2.80.19322 0.0006 0.0382 0.004 0.4

x 1.4932 0.003 3.91.4082 0.008 0.702 0.06 0.1

f 1.6362 0.004 0.41.602 0.01 0.272 0.08 0.1

Diameters rd of coexistence curves for the three choices of the ‘‘density’’ variablewere fitted with the form:

rd = rc +Dtz, (12)

to test the presence of a 2b term and to examine the goodness of the selected orderparameter. The apparent exponent Z in equation (12) was fixed at the values(1− a)=0.89, and 2b=0.654,(8,9) in separate fitting procedures. The results arecompared in table 5. It is found that a fit with t(1− a) is better than that with t2b forthe order parameter f, but the best fit is that with t2b for the order parameter x.This is consistent with that indicated by the symmetry of the coexistence curves,because the (T, f) curve is more symmetrical than (T, x) curve for this binarymixture.

The experimental value of nc in table 5 was obtained by extrapolating refractiveindices against temperatures in the one-phase region to the critical temperature.The experimental values of xc and fc were determined by the ‘‘equalvolume’’ technique and calculated by equations (8) and (9). The uncertainties ofoptimal parameters reported in table 5 include no systematic uncertaintiescontributed by converting n to x, an x to f. Such uncertainties in x and f wereestimated to be about 20.002. Therefore, the values of xc and fc obtained fromextrapolation of equation (12) are consistent with those from observations. Thisis evidence that no significant critical anomaly is present in refractive indices andthat they were properly converted to mole fractions and volume fractions in ourtreatments.

Coexistence curves of 4xC6H5NO2 + (1− x)CH3(CH2)12CH35 1053

TABLE 5. Coefficients of equation (10) and standard deviations s in rd, for diameter of coexistencecurves of (T, n), (T, x), and (T, f), for {xC6H5NO2 + (1− x)CH3(CH2)12CH3}. (rc)exp is the critical

value of the order parameter determined by the techniques described in the text

(T, n) (T, x) (T, f)

(rc)exp 1.47522 0.0002 0.6762 0.001 0.452 2 0.001

Z=0.89

rc 1.47522 0.0001 0.6742 0.001 0.453 2 0.001D 0.1252 0.001 −1.192 0.02 0.107 2 0.004s 4.8·10−4 1.9·10−3 3.1·10−4

Z=0.654

rc 1.47502 0.0001 0.6762 0.001 0.452 2 0.001D 0.0542 0.001 −0.5222 0.002 0.046 2 0.002s 2.5·10−4 3.8·10−4 3.4·10−4

TABLE 6. Critical mole fractions xc, critical volume fractions fc, and critical temperatures Tc of{xC6H5NO2 + (1− x)CH3(CH2)mCH3} at various m. The number of (CH2) is denoted by m

m Tc/K xc fc m Tc/K xc fc

3 297.1 0.386 0.358 9 298.0 0.606 0.4274 231.1 0.427 0.366 10 300.4 0.630 0.4545 291.9 0.472 0.384 11 303.0 0.654 0.4416 293.1 0.505 0.392 12 304.9 0.676 0.4528 296.0 0.574 0.415 14 309.7 0.713 0.464

Combination of equations (11) and (12) yields:

r1 = rc +Dtz −(1/2)·Btb −(1/2)·B1tb+D, (13)

r2 = rc +Dtz +(1/2)·Btb +(1/2)·B1tb+D. (14)

When Z, b, D, and Tc were fixed at 0.654, 0.327, 0.5, and 304.940 K, and thevalues of D, rc, B, and B1 were taken from tables 4 and 5, the values of r1

and r2 were calculated from equations (13) and (14); the results are shown aslines in figure 1. The calculated values are in good agreement with experimentalresults.

The critical mole fractions, the critical volume fractions, and the criticaltemperatures of mixtures of nitrobenzene in n-alkanes with various chainlengths were determined, and are reported in table 6. A ln–ln plot of (1−fc)/fc

against molar mass M of n-alkane is shown in figure 2. The value of r wasobtained from the slope of the straight line and is (0.40 2 0.01), which is inexcellent agreement with the values of 0.41 for (N,N-dimethylacetamide +n-alkane). We also reanalysed the experimental data of coexistence curves for(methylcyclohexane +polystyrene) by using equations (1) and (2) instead ofequations (3) and (4), and found the value of r to be 0.40. This implies that equation(1) is probably universal for chain-molecule solutions of both small molecules andpolymers.

X. An et al.1054

FIGURE 2. An ln–ln plot of (1− fc)/fc against molar mass M of n-alkane, where fc = criticalvolume fraction.

This work was supported by the National Natural Science Foundation and theNatural Science Foundation of Gansu province, P.R. China.

REFERENCES

1. An, X.; Zhao, H.; Jiang, F.; Shen, W. J. Chem. Thermodynamics 1996, 28, 1221–1231.2. An, X.; Zhao, H.; Shen, W. J. Chem. Thermodynamics 1995, 27, 1241–1247.3. An, X.; Shen, W. J. Chem. Thermodynamics 1994, 26, 461–468.4. An, X.; Shen, W.; Wang, H.; Zheng, G. J. Chem. Thermodynamics 1993, 25, 1373–1381.5. Shen, W.; Smith, G. R.; Knobler, C. M.; Scott, R. L. J. Phys. Chem. 1990, 94, 7943–7949.6. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Solvents: 4th edition. Techniques of Chemistry,

Vol. II. Wiley-Interscience: New York. 1986.7. Thermodynamics Research Centre. API 44 Tables, Selected Values of Properties of Hydrocarbons

and Related Components, Vol.I. 1972.8. Alpert, D. Z. Phys. Rev. 1982, B25, 4810–4814.9. Le Guillou, J. C.; Zinn-Justin, J. Phys. Rev. 1980, B21, 3976–3998.

10. Wegner, F. J. Phys. Rev. B 1972, 5, 4529–4536.

(Received 3 January 1997; in final form 4 March 1997)

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