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The measurement of coexistence curves for {x(CHCOOC2H5)2+(1−x) CH3(CH2)7CH3} in the critical region
J. Chem. Thermodynamics 1998, 30, 1253]1261 Article No. ct980394
The measurement of coexistence curves for { x( CHCOOC 2 H 5 ) 2 H ( 1 I x ) CH 3( CH 2 ) 7 CH 3 } in the critical region Xueqin An, Jiong Yang, and Weigou Shen a Department of Chemistry, Lanzhou Uni¨ ersity, Lanzhou, Gansu 730000, P. R. China
Coexistence curves for binary liquid mixtures of x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 have been determined by measurement of the refractive index. The experimental results have been used in the determination of the critical exponent b and the critical amplitude B, and the study of the diameters of the coexistence curves. The values of the critical exponent b are found to be consistent with theoretical predictions. The coexistence curves have been successfully described by a combination of the Wegner equation and the expression for the diameter. The exponents r and b in the relations Ž1 y fc .rfc A Myr and .865 Bf fy1 A Myb , where fc is the critical volume fraction of ŽCHCOOC 2 H 5 . 2 , have been c found to be Ž0.42 " 0.01. and Ž0.31 " 0.04., respectively, consistent with observations from experimental studies for other chain-molecule solution systems, and support the Landau]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 the molar mass dependence of the critical amplitudes for chain-molecule solutions of both small molecules and polymers.Ž1. From this model we derived a general form for the difference of the volume fractions of two coexisting phases as: .865 A Myb , Bf fy1 c
Ž 1.
where b s 0.29 and fc is the critical volume fraction of the non-chain-molecule component Ž 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, and Bf is the amplitude associated with the coexistence curve of temperature against volume fraction. In addition, fc was experimentally found to be dependent on the molar mass as follows:
Ž 1 y fc . rfc A Myr ,
Ž 2.
with the universal exponent r s 0.41Ž2. for the chain-molecule solutions for both small 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
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X. An, J. Yang, and W. Shen
Ž1. and Ž2., we have measured the coexistence curves near the critical points for Ždiethylmaleate q n-alkane. with alkane carbon numbers 6, 7, 8, and 10.Ž3 ] 6. In this paper we report the measurements of the coexistence curves for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 . The critical amplitude B and the critical exponent b have been deduced from ŽT, n., ŽT, x ., and ŽT, f . curves Žwhere n, x, and f are the refractive index, the mole fraction and the volume fraction, respectively., and the behaviour of the diameters r d of the coexistence curves have been examined. The coexistence curves have been successfully described by the combination of the Wegner equation and the expression for the diameter. The values the exponents b and r in equations Ž1. and Ž2. have been determined and compared with the values predicted by theory, or found in other experimental studies.
2. Experimental Diethyl maleate ŽCHCOOC 2 H 5 . 2 obtained from Beijing Chemicals Factory was purified by fractional distillation under vacuum. The n-nonane Žmass fraction s 0.99. was supplied by Aldrich Chemical Company Inc. and was stored over 4 . 10y1 0 m molecular sieves. The coexistence curves were determined by measurement of the refractive indices using the method of minimum deviation. The apparatus and the experimental procedure for the measurement of the refractive index, and the techniques for the determination of the critical temperature and the critical concentration have been described previously.Ž7. During the measurements, the temperature was constant to within "2 . 10y3 K. The accuracy and precision in the measurement of temperature were better than "1 . 10y2 K and "1 . 10y3 K, respectively. The accuracy of measurement was "3 . 10y3 K for the temperature difference ŽT y Tc ., "2 . 10y4 for the refractive index in each coexisting phase, and "1 . 10y3 for the critical mole fraction x c .
3. Results and discussion The critical mole fractions and the critical temperature of x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 34 were determined to be Ž0.473 " 0.001. and Ž310.1 " 0.1. K, respectively. 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 1Ža.. In order to obtain the ŽT, x . coexistence curve, the refractive indices of pure diethyl maleate and nonane at various temperatures, and the refractive indices of a series of binary mixtures with known mole fractions in the single phase region at T s 310.29 K were also measured. The results are listed in tables 2 and 3. The refractive index may be expressed as a linear function of temperature in a certain temperature range by: n Ž T , x . s n . Ž T8, x . q Ž nr T . x . Ž T y T8 . , Ž 3.
Ž nr T . x s x . Ž n Ar T . q Ž 1 y x . . Ž n Br T . ,
Ž 4.
The coexistence curves of Ždiethyl maleate q nonane.
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TABLE 1. Coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4. Refractive indices n were measured at wavelength l s 6.328 . 10y7 m, and Tc s 310.064 K. Volume fraction is denoted by f 0 . Subscripts 1 and 2 relate to upper and lower phases, respectively ŽTc y T .rK
n1
n2
x1
x2
f1
f2
0.002 0.004 0.008 0.013 0.020 0.032 0.041 0.051 0.064 0.081 0.104 0.124 0.151 0.187 0.230 0.277 0.320 0.368 0.421 0.481 0.552 0.631 0.720 0.810 0.926 1.061 1.213 1.370 1.550 1.749 1.982 2.356 2.742 3.263 4.064 4.902 6.006 7.189 8.605 10.235
1.4093 1.4091 1.4089 1.4087 1.4084 1.4082 1.4081 1.4080 1.4079 1.4077 1.4075 1.4074 1.4073 1.4071 1.4070 1.4068 1.4067 1.4066 1.4065 1.4064 1.4062 1.4060 1.4058 1.4057 1.4055 1.4054 1.4053 1.4052 1.4051 1.4050 1.4049 1.4047 1.4045 1.4044 1.4044 1.4042 1.4044 1.4046 1.4047 1.4051
1.4106 1.4108 1.4110 1.4112 1.4115 1.4117 1.4119 1.4119 1.4121 1.4123 1.4125 1.4127 1.4128 1.4131 1.4134 1.4136 1.4137 1.4139 1.4141 1.4143 1.4146 1.4148 1.4150 1.4152 1.4155 1.4157 1.4161 1.4164 1.4168 1.4171 1.4175 1.4180 1.4185 1.4192 1.4202 1.4212 1.4222 1.4234 1.4246 1.4259
0.457 0.451 0.445 0.439 0.431 0.425 0.422 0.419 0.416 0.410 0.403 0.400 0.397 0.391 0.387 0.381 0.377 0.373 0.370 0.366 0.359 0.352 0.344 0.340 0.332 0.327 0.322 0.317 0.311 0.305 0.298 0.287 0.274 0.263 0.251 0.231 0.221 0.208 0.189 0.175
0.493 0.499 0.504 0.510 0.518 0.523 0.529 0.528 0.534 0.539 0.544 0.549 0.552 0.559 0.567 0.572 0.574 0.579 0.583 0.588 0.595 0.599 0.604 0.608 0.614 0.618 0.627 0.633 0.641 0.646 0.654 0.663 0.671 0.683 0.699 0.714 0.727 0.743 0.757 0.771
0.431 0.426 0.420 0.414 0.406 0.400 0.397 0.394 0.391 0.385 0.379 0.376 0.373 0.366 0.363 0.357 0.353 0.350 0.346 0.342 0.336 0.329 0.322 0.317 0.310 0.305 0.300 0.295 0.289 0.284 0.277 0.266 0.254 0.244 0.231 0.213 0.203 0.192 0.173 0.161
0.467 0.473 0.478 0.484 0.492 0.497 0.503 0.503 0.508 0.513 0.519 0.524 0.526 0.534 0.542 0.546 0.549 0.553 0.558 0.563 0.570 0.575 0.579 0.583 0.590 0.593 0.602 0.608 0.617 0.623 0.630 0.639 0.648 0.660 0.677 0.693 0.706 0.723 0.737 0.752
where Ž nr T . x is the derivative of n with respect to T for a particular composition x, and Ž n Ar T . and Ž n Br T . are the values of Ž nr T . x for x s 1 and x s 0, respectively. The validity of equations Ž3. and Ž4. have been confirmed in previous work.Ž3,7. The results listed in table 2 were fitted to equation Ž3. to obtain Ž n Ar T . s y4.56 . 10y4 and Ž n Br T . s y4.70 . 10y4 with a standard deviation of 1 . 10y4 in n. With a chosen value of T8 s 304.95 K, corresponding to about the middle temperature for the coexistence curve determined in this work, the values of nŽT8, x . were calculated from equations Ž3.
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X. An, J. Yang, and W. Shen
FIGURE 1. Coexistence curves of: a, ŽT, n.; b, ŽT, x .; and c, ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 where T s temperature, n s refractive index, x s mole fraction, and f s volume fraction. v, experimental values of the concentration variables r of the coexisting phases; ', experimental values of the diameter rd of the coexisting phases; }}}, calculated concentration variables and diameter of coexisting phases.
and Ž4. with the nŽT, x . data listed in table 3. A polynomial form of nŽT8, x . as a function of x at T8 was obtained: n Ž T8, x . s 1.3987 q 0.0177 . x q 0.0353 . x 2 y 0.0324 . x 3 q 0.0144 . x 4 ,
Ž 5.
with a standard deviation of 1 . 10y4 . The values of the refractive indices for the coexisting phases were then converted to mole fractions by calculating nŽT8 , x .
The coexistence curves of Ždiethyl maleate q nonane.
1257
TABLE 2. Refractive index n at wavelength l s 6.328 . 10y7 m for ŽCHCOOC 2 H 5 . 2 and CH 3 ŽCH 2 . 7 CH 3 at various temperature T ŽCHCOOC 2 H 5 . 2
CH 3 ŽCH 2 . 7 CH 3
TrK
n
TrK
n
299.059 300.807 302.239 303.149 304.157 306.079 307.770 309.287 310.457
1.4365 1.4356 1.4350 1.4345 1.4340 1.4331 1.4324 1.4318 1.4313
310.255 309.591 308.793 307.696 306.741 305.606 304.383 300.943
1.3962 1.3965 1.3969 1.3974 1.3978 1.3984 1.3989 1.4006
through equations Ž3. and Ž4., and iteratively solving equation Ž5.. The results are listed in columns 4 and 5 of table 1 and shown in figure 1Žb.. The mole fraction was used to calculate the volume fraction through: 1rf s Ž 1 y K . q Krx,
Ž 6.
K s Ž dA M B . r Ž d B MA . ,
Ž 7.
where M is the molar mass, d is the mass density, and subscript A and B refer to diethyl maleate and n-nonane, respectively. The values of dA and d B were obtained from references 8 and 9. The values of f of the coexisting phases for this system at various temperatures are listed in columns 6 and 7 of table 1 and shown in figure 1Žc.. The differences Ž r 2 y r 1 . of the ‘‘density’’ variables of the coexisting phases may be expressed by the Wegner expression: Ž10.
r 2 y r 1 s Bt b q B1t bqD q . . . ,
Ž 8.
where r 1 and r 2 are the ‘‘density’’ variables in the upper and lower coexisting phases; t s ŽTc y T .rTc , Tc is the critical temperature; and b Ž0.3265. and DŽ0.50.Ž11,12. are universal critical exponents which describe the shape of the coexistence curve. Values of B and B1 depend on the system and the choice of the TABLE 3. Refractive indices n at wavelength l s 6.328 . 10y7 m for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 at T s 310.29 K x
n
x
n
x
n
0.000 0.100 0.200 0.300
1.3962 1.3982 1.4009 1.4040
0.399 0.500 0.599 0.701
1.4071 1.4108 1.4144 1.4180
0.800 0.900 1.000
1.4226 1.4266 1.4313
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TABLE 4. Values of the critical amplitudes B and critical exponents b for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . in equation Ž9. Order parameter
n x f
ŽTc y T . - 1 K
ŽTc y T . - 10.3 K
B
b
B
b
0.066 " 0.002 1.86 " 0.05 1.85 " 0.04
0.324 " 0.003 0.324 " 0.004 0.325 " 0.004
0.064 " 0.001 1.81 " 0.02 1.81 " 0.02
0.320 " 0.002 0.322 " 0.002 0.322 " 0.002
order parameter. In a region sufficiently close to the critical temperature, equation Ž8. becomes:
r 2 y r 1 s Bt b .
Ž 9.
The differences Ž r 2 y r 1 . of the ‘‘density’’ variables of the coexisting phases were calculated from the results listed in table 1, and were fitted to equation Ž9. to obtain the critical exponent b and the critical amplitude B. The results are shown in table 4. The values of b and B depend on the cutoff values of ŽTc y T ., but for ŽTc y T . - 1 K, the values of b for three choices of variables are all in good agreement with the theoretical prediction of Ž0.3265 " 0.001.Ž11. within the experimental uncertainties. The diameter r d of the coexistence curve may be expressed by:
r d s Ž r 2 q r 1 . r2 s rc q Dt Z q . . . ,
Ž 10 .
where rc is the value of r at the critical point, D is the system-dependent parameter, and Z is the apparent exponent. For a binary mixture there is no a priori reason for choosing one ‘‘density’’ variable rather than another, but the variables may be tested by examining the symmetry of the coexistence curve and by comparing the goodness of fits of equation Ž10. with Ž1 y a . and 2 b ,Ž13,14. where a s 0.11Ž12. characterizes the divergence, as the critical point is approached, of the heat capacity at constant volume for the pure fluid. The experimental diameter data were fitted to equation Ž10.. The results of the fits are compared in table 5, where the experimental value of n c, expt was obtained by extrapolating the refractive indices against temperatures in the single phase region up to the critical temperature; the value of x c, expt was determined by the technique of ‘‘equal volume’’, and the value of fc, expt was then calculated from x c, expt through equations Ž6. and Ž7.. The uncertainties of the optimal parameters reported in table 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 values of n c , x c , and fc obtained from the extrapolation of equation Ž10. are consistent with those observed. This constitutes evidence that no significant critical anomaly was present in the refractive indices, and that the refractive indices were properly converted to mole fractions and volume fractions in our treatment. The quality of the fit of equation Ž10. may be indicated by the values of the standard
The coexistence curves of Ždiethyl maleate q nonane.
1259
TABLE 5. Parameters of equation Ž10. and standard deviation s in r d for the diameters of the coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4. rc, expt is the critical value of the order parameter determined by the techniques described in the text ŽT, n.
rc, expt
1.4099 " 0.0002
rc D s
1.4099 " 0.0001 0.1141 " 0.0008 5.2 . 10y5
ŽT, x .
ŽT, f .
0.473 " 0.005
0.447 " 0.005
0.475 " 0.001 y0.04 " 0.02 1.2 . 10y3
0.450 " 0.001 0.15 " 0.02 1.3 . 10y3
Z s 0.89
Z s 0.653
rc D s
1.4097 " 0.0001 0.048 " 0.002 2.3 . 10y4
0.475 " 0.001 y0.016 " 0.007 1.2 . 10y3
0.450 " 0.001 0.068 " 0.007 1.2 . 10y3
deviation s listed in table 5. We have found no significant difference between fits with t 1y a and with t 2 b for both x and f . Therefore we conclude that f and x are almost equally good variables for the construction of order parameters for the system under study. This is consistent with what the symmetry indicates: no significant symmetry difference between ŽT, x . and ŽT, f . is observed in figure 1.
v,
FIGURE 2. A ln]ln plot of Ž1 y fc .rfc against molar mass M for Ždiethyl maleate q n-alkane.: experimental values; }}}, calculated values from equation Ž2. with r s 0.42.
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X. An, J. Yang, and W. Shen
TABLE 6. Parameters of equation Ž8. for the coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 Order parameter
B
B1
n
0.0653 " 0.0002 0.0675 " 0.0002
y0.021 " 0.002
x
1.850 " 0.004 1.890 " 0.006
y0.38 " 0.05
f
1.836 " 0.004 1.875 " 0.006
y0.37 " 0.05
When the critical exponents b and D are fixed at the theoretical values Ž b s 0.3265, D s 0.50.,Ž11,12. and equation Ž8. is used to fit the phase separation data, the parameters B and B1 can be obtained. The results are listed in table 6. Combination of equations Ž8. and Ž10. yields:
r 1 s rc q Dt Z y Ž 1r2 . Bt b y Ž 1r2 . B1t bqD ,
Ž 11 .
r 2 s rc q Dt Z q Ž 1r2 . Bt b q Ž 1r2 . B1t bqD .
Ž 12 .
When Z, b , D, and Tc are fixed at 0.89, 0.3265, 0.5, and 310.064 K, respectively, and the values of D, rc , B, and B1 are taken from tables 5 and 6, the values of r 1 ,
v,
.865 FIGURE 3. A ln]ln plot of Bf fy1 against molar mass M for Ždiethyl maleate q n-alkane.: c experimental values; }}}, calculated values from equation Ž1. with b s 0.31.
The coexistence curves of Ždiethyl maleate q nonane.
1261
TABLE 7. Critical amplitudes Bf and critical volume fractions fc for ŽCHCOOC 2 H 5 . 2 q n-alkane4. M is the molar mass of n-alkane n-alkane Hexane Heptane Octane Nonane Decane
MrŽg . moly1 . 86.178 100.206 114.233 128.260 142.287
fc
Bf
0.406 0.421 0.436 0.447 0.458
1.749 1.824 1.816 1.836 1.904
r 2 , and r d can be calculated from equations Ž11., Ž12., and Ž10.. The results are shown as lines in figure 1. The calculated values are in good agreement with the experimental results. Table 7 summarizes the values of Bf and fc for five experimentally investigated systems of Ždiethyl maleate q n-alkane.. According to equations Ž1. and Ž2., ln]ln .865 plots of Ž1 y fc .rfc and Bf fy1 against the molar mass of n-alkane M will c yield two straight lines. These two plots are shown in figures 2 and 3. A least-squares fit results in the values of Ž0.42 " 0.01. and Ž0.31 " 0.04. for the exponents r and b, which are in excellent agreement with calculated values Ž b s 0.29.,Ž1. and observed values from experimental studies on chain-molecule solutions of both small molecules Ž2,15. and polymers.Ž2. This supports the universality of the exponents of M and the Landau]Ginsburg]Wilson type model we proposed recently for chain-molecule solutions. This work was supported by the National Natural Science Foundation ŽProject 29673019., the State Education Committee, and the Natural Science Foundation of Gansu Province ŽNo: ZR-96-009., P. R. China. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
An, X.; Jiang, F.; Chen, C; Shen, W. Chem. Phys. Letters 1998, 282, 403]408. An, X., Jiang, F.; Chen, C.; Shen, W. Pure and Applied Chem. 1998, 70, 609]614. An, X.; Lui, X.; Shen, W. J. Chem. Thermodynamics 1997, 29, 669]675. An, X.; Yang, J.; Shen, W. J. Chem. Thermodynamics 1998, 30, 13]19. An, X.; Mao, C.; Sun, G.; Shen, W. J. Chem. Thermodynamics 1998, 30, 689]695. An, X.; Chi, X.; Wang, T.; Shen, W. J. Chem. Thermodynamics 1998, 30, 1199]1205. An, X.; Shen, W.; Wang, H.; Zheng, G. J. Chem. Thermodynamics 1993, 25, 1373]1383. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Sol¨ ents: 4th edition. Techniques of Chemistry, Vol. II. Wiley-Interscience: New York. 1986. Thermodynamics Research Centre, API 44 Tables, Selected Values of Properties of Hydrocarbons and Related Components, Vol. I. 1972. Wegner, F. J. Phys. Re¨ . 1972, B5, 4529]4536. Alpert, D. Z. Phys. Re¨ . 1982, B25, 4810]4814. Le Guillou, J. C.; Zinn-Justin, J. Phys. Re¨ . 1980, B21, 3976]3998. Ewing, M. B.; Johnson, K. A.; McGlashan, M. L. J. Chem. Thermodynamics, 1988, 20, 49]62. Greer, S. C.; Das, B. K.; Kumar, A.; Gopal, E. S. R. J. Chem. Phys. 1983, 79, 4545]4552. 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)
WA98rO12
The measurement of coexistence curves for { x( CHCOOC 2 H 5 ) 2 H ( 1 I x ) CH 3( CH 2 ) 7 CH 3 } in the critical region Xueqin An, Jiong Yang, and Weigou Shen a Department of Chemistry, Lanzhou Uni¨ ersity, Lanzhou, Gansu 730000, P. R. China
Coexistence curves for binary liquid mixtures of x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 have been determined by measurement of the refractive index. The experimental results have been used in the determination of the critical exponent b and the critical amplitude B, and the study of the diameters of the coexistence curves. The values of the critical exponent b are found to be consistent with theoretical predictions. The coexistence curves have been successfully described by a combination of the Wegner equation and the expression for the diameter. The exponents r and b in the relations Ž1 y fc .rfc A Myr and .865 Bf fy1 A Myb , where fc is the critical volume fraction of ŽCHCOOC 2 H 5 . 2 , have been c found to be Ž0.42 " 0.01. and Ž0.31 " 0.04., respectively, consistent with observations from experimental studies for other chain-molecule solution systems, and support the Landau]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 the molar mass dependence of the critical amplitudes for chain-molecule solutions of both small molecules and polymers.Ž1. From this model we derived a general form for the difference of the volume fractions of two coexisting phases as: .865 A Myb , Bf fy1 c
Ž 1.
where b s 0.29 and fc is the critical volume fraction of the non-chain-molecule component Ž 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, and Bf is the amplitude associated with the coexistence curve of temperature against volume fraction. In addition, fc was experimentally found to be dependent on the molar mass as follows:
Ž 1 y fc . rfc A Myr ,
Ž 2.
with the universal exponent r s 0.41Ž2. for the chain-molecule solutions for both small 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
1254
X. An, J. Yang, and W. Shen
Ž1. and Ž2., we have measured the coexistence curves near the critical points for Ždiethylmaleate q n-alkane. with alkane carbon numbers 6, 7, 8, and 10.Ž3 ] 6. In this paper we report the measurements of the coexistence curves for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 . The critical amplitude B and the critical exponent b have been deduced from ŽT, n., ŽT, x ., and ŽT, f . curves Žwhere n, x, and f are the refractive index, the mole fraction and the volume fraction, respectively., and the behaviour of the diameters r d of the coexistence curves have been examined. The coexistence curves have been successfully described by the combination of the Wegner equation and the expression for the diameter. The values the exponents b and r in equations Ž1. and Ž2. have been determined and compared with the values predicted by theory, or found in other experimental studies.
2. Experimental Diethyl maleate ŽCHCOOC 2 H 5 . 2 obtained from Beijing Chemicals Factory was purified by fractional distillation under vacuum. The n-nonane Žmass fraction s 0.99. was supplied by Aldrich Chemical Company Inc. and was stored over 4 . 10y1 0 m molecular sieves. The coexistence curves were determined by measurement of the refractive indices using the method of minimum deviation. The apparatus and the experimental procedure for the measurement of the refractive index, and the techniques for the determination of the critical temperature and the critical concentration have been described previously.Ž7. During the measurements, the temperature was constant to within "2 . 10y3 K. The accuracy and precision in the measurement of temperature were better than "1 . 10y2 K and "1 . 10y3 K, respectively. The accuracy of measurement was "3 . 10y3 K for the temperature difference ŽT y Tc ., "2 . 10y4 for the refractive index in each coexisting phase, and "1 . 10y3 for the critical mole fraction x c .
3. Results and discussion The critical mole fractions and the critical temperature of x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 34 were determined to be Ž0.473 " 0.001. and Ž310.1 " 0.1. K, respectively. 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 1Ža.. In order to obtain the ŽT, x . coexistence curve, the refractive indices of pure diethyl maleate and nonane at various temperatures, and the refractive indices of a series of binary mixtures with known mole fractions in the single phase region at T s 310.29 K were also measured. The results are listed in tables 2 and 3. The refractive index may be expressed as a linear function of temperature in a certain temperature range by: n Ž T , x . s n . Ž T8, x . q Ž nr T . x . Ž T y T8 . , Ž 3.
Ž nr T . x s x . Ž n Ar T . q Ž 1 y x . . Ž n Br T . ,
Ž 4.
The coexistence curves of Ždiethyl maleate q nonane.
1255
TABLE 1. Coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4. Refractive indices n were measured at wavelength l s 6.328 . 10y7 m, and Tc s 310.064 K. Volume fraction is denoted by f 0 . Subscripts 1 and 2 relate to upper and lower phases, respectively ŽTc y T .rK
n1
n2
x1
x2
f1
f2
0.002 0.004 0.008 0.013 0.020 0.032 0.041 0.051 0.064 0.081 0.104 0.124 0.151 0.187 0.230 0.277 0.320 0.368 0.421 0.481 0.552 0.631 0.720 0.810 0.926 1.061 1.213 1.370 1.550 1.749 1.982 2.356 2.742 3.263 4.064 4.902 6.006 7.189 8.605 10.235
1.4093 1.4091 1.4089 1.4087 1.4084 1.4082 1.4081 1.4080 1.4079 1.4077 1.4075 1.4074 1.4073 1.4071 1.4070 1.4068 1.4067 1.4066 1.4065 1.4064 1.4062 1.4060 1.4058 1.4057 1.4055 1.4054 1.4053 1.4052 1.4051 1.4050 1.4049 1.4047 1.4045 1.4044 1.4044 1.4042 1.4044 1.4046 1.4047 1.4051
1.4106 1.4108 1.4110 1.4112 1.4115 1.4117 1.4119 1.4119 1.4121 1.4123 1.4125 1.4127 1.4128 1.4131 1.4134 1.4136 1.4137 1.4139 1.4141 1.4143 1.4146 1.4148 1.4150 1.4152 1.4155 1.4157 1.4161 1.4164 1.4168 1.4171 1.4175 1.4180 1.4185 1.4192 1.4202 1.4212 1.4222 1.4234 1.4246 1.4259
0.457 0.451 0.445 0.439 0.431 0.425 0.422 0.419 0.416 0.410 0.403 0.400 0.397 0.391 0.387 0.381 0.377 0.373 0.370 0.366 0.359 0.352 0.344 0.340 0.332 0.327 0.322 0.317 0.311 0.305 0.298 0.287 0.274 0.263 0.251 0.231 0.221 0.208 0.189 0.175
0.493 0.499 0.504 0.510 0.518 0.523 0.529 0.528 0.534 0.539 0.544 0.549 0.552 0.559 0.567 0.572 0.574 0.579 0.583 0.588 0.595 0.599 0.604 0.608 0.614 0.618 0.627 0.633 0.641 0.646 0.654 0.663 0.671 0.683 0.699 0.714 0.727 0.743 0.757 0.771
0.431 0.426 0.420 0.414 0.406 0.400 0.397 0.394 0.391 0.385 0.379 0.376 0.373 0.366 0.363 0.357 0.353 0.350 0.346 0.342 0.336 0.329 0.322 0.317 0.310 0.305 0.300 0.295 0.289 0.284 0.277 0.266 0.254 0.244 0.231 0.213 0.203 0.192 0.173 0.161
0.467 0.473 0.478 0.484 0.492 0.497 0.503 0.503 0.508 0.513 0.519 0.524 0.526 0.534 0.542 0.546 0.549 0.553 0.558 0.563 0.570 0.575 0.579 0.583 0.590 0.593 0.602 0.608 0.617 0.623 0.630 0.639 0.648 0.660 0.677 0.693 0.706 0.723 0.737 0.752
where Ž nr T . x is the derivative of n with respect to T for a particular composition x, and Ž n Ar T . and Ž n Br T . are the values of Ž nr T . x for x s 1 and x s 0, respectively. The validity of equations Ž3. and Ž4. have been confirmed in previous work.Ž3,7. The results listed in table 2 were fitted to equation Ž3. to obtain Ž n Ar T . s y4.56 . 10y4 and Ž n Br T . s y4.70 . 10y4 with a standard deviation of 1 . 10y4 in n. With a chosen value of T8 s 304.95 K, corresponding to about the middle temperature for the coexistence curve determined in this work, the values of nŽT8, x . were calculated from equations Ž3.
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X. An, J. Yang, and W. Shen
FIGURE 1. Coexistence curves of: a, ŽT, n.; b, ŽT, x .; and c, ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 where T s temperature, n s refractive index, x s mole fraction, and f s volume fraction. v, experimental values of the concentration variables r of the coexisting phases; ', experimental values of the diameter rd of the coexisting phases; }}}, calculated concentration variables and diameter of coexisting phases.
and Ž4. with the nŽT, x . data listed in table 3. A polynomial form of nŽT8, x . as a function of x at T8 was obtained: n Ž T8, x . s 1.3987 q 0.0177 . x q 0.0353 . x 2 y 0.0324 . x 3 q 0.0144 . x 4 ,
Ž 5.
with a standard deviation of 1 . 10y4 . The values of the refractive indices for the coexisting phases were then converted to mole fractions by calculating nŽT8 , x .
The coexistence curves of Ždiethyl maleate q nonane.
1257
TABLE 2. Refractive index n at wavelength l s 6.328 . 10y7 m for ŽCHCOOC 2 H 5 . 2 and CH 3 ŽCH 2 . 7 CH 3 at various temperature T ŽCHCOOC 2 H 5 . 2
CH 3 ŽCH 2 . 7 CH 3
TrK
n
TrK
n
299.059 300.807 302.239 303.149 304.157 306.079 307.770 309.287 310.457
1.4365 1.4356 1.4350 1.4345 1.4340 1.4331 1.4324 1.4318 1.4313
310.255 309.591 308.793 307.696 306.741 305.606 304.383 300.943
1.3962 1.3965 1.3969 1.3974 1.3978 1.3984 1.3989 1.4006
through equations Ž3. and Ž4., and iteratively solving equation Ž5.. The results are listed in columns 4 and 5 of table 1 and shown in figure 1Žb.. The mole fraction was used to calculate the volume fraction through: 1rf s Ž 1 y K . q Krx,
Ž 6.
K s Ž dA M B . r Ž d B MA . ,
Ž 7.
where M is the molar mass, d is the mass density, and subscript A and B refer to diethyl maleate and n-nonane, respectively. The values of dA and d B were obtained from references 8 and 9. The values of f of the coexisting phases for this system at various temperatures are listed in columns 6 and 7 of table 1 and shown in figure 1Žc.. The differences Ž r 2 y r 1 . of the ‘‘density’’ variables of the coexisting phases may be expressed by the Wegner expression: Ž10.
r 2 y r 1 s Bt b q B1t bqD q . . . ,
Ž 8.
where r 1 and r 2 are the ‘‘density’’ variables in the upper and lower coexisting phases; t s ŽTc y T .rTc , Tc is the critical temperature; and b Ž0.3265. and DŽ0.50.Ž11,12. are universal critical exponents which describe the shape of the coexistence curve. Values of B and B1 depend on the system and the choice of the TABLE 3. Refractive indices n at wavelength l s 6.328 . 10y7 m for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 at T s 310.29 K x
n
x
n
x
n
0.000 0.100 0.200 0.300
1.3962 1.3982 1.4009 1.4040
0.399 0.500 0.599 0.701
1.4071 1.4108 1.4144 1.4180
0.800 0.900 1.000
1.4226 1.4266 1.4313
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TABLE 4. Values of the critical amplitudes B and critical exponents b for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . in equation Ž9. Order parameter
n x f
ŽTc y T . - 1 K
ŽTc y T . - 10.3 K
B
b
B
b
0.066 " 0.002 1.86 " 0.05 1.85 " 0.04
0.324 " 0.003 0.324 " 0.004 0.325 " 0.004
0.064 " 0.001 1.81 " 0.02 1.81 " 0.02
0.320 " 0.002 0.322 " 0.002 0.322 " 0.002
order parameter. In a region sufficiently close to the critical temperature, equation Ž8. becomes:
r 2 y r 1 s Bt b .
Ž 9.
The differences Ž r 2 y r 1 . of the ‘‘density’’ variables of the coexisting phases were calculated from the results listed in table 1, and were fitted to equation Ž9. to obtain the critical exponent b and the critical amplitude B. The results are shown in table 4. The values of b and B depend on the cutoff values of ŽTc y T ., but for ŽTc y T . - 1 K, the values of b for three choices of variables are all in good agreement with the theoretical prediction of Ž0.3265 " 0.001.Ž11. within the experimental uncertainties. The diameter r d of the coexistence curve may be expressed by:
r d s Ž r 2 q r 1 . r2 s rc q Dt Z q . . . ,
Ž 10 .
where rc is the value of r at the critical point, D is the system-dependent parameter, and Z is the apparent exponent. For a binary mixture there is no a priori reason for choosing one ‘‘density’’ variable rather than another, but the variables may be tested by examining the symmetry of the coexistence curve and by comparing the goodness of fits of equation Ž10. with Ž1 y a . and 2 b ,Ž13,14. where a s 0.11Ž12. characterizes the divergence, as the critical point is approached, of the heat capacity at constant volume for the pure fluid. The experimental diameter data were fitted to equation Ž10.. The results of the fits are compared in table 5, where the experimental value of n c, expt was obtained by extrapolating the refractive indices against temperatures in the single phase region up to the critical temperature; the value of x c, expt was determined by the technique of ‘‘equal volume’’, and the value of fc, expt was then calculated from x c, expt through equations Ž6. and Ž7.. The uncertainties of the optimal parameters reported in table 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 values of n c , x c , and fc obtained from the extrapolation of equation Ž10. are consistent with those observed. This constitutes evidence that no significant critical anomaly was present in the refractive indices, and that the refractive indices were properly converted to mole fractions and volume fractions in our treatment. The quality of the fit of equation Ž10. may be indicated by the values of the standard
The coexistence curves of Ždiethyl maleate q nonane.
1259
TABLE 5. Parameters of equation Ž10. and standard deviation s in r d for the diameters of the coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4. rc, expt is the critical value of the order parameter determined by the techniques described in the text ŽT, n.
rc, expt
1.4099 " 0.0002
rc D s
1.4099 " 0.0001 0.1141 " 0.0008 5.2 . 10y5
ŽT, x .
ŽT, f .
0.473 " 0.005
0.447 " 0.005
0.475 " 0.001 y0.04 " 0.02 1.2 . 10y3
0.450 " 0.001 0.15 " 0.02 1.3 . 10y3
Z s 0.89
Z s 0.653
rc D s
1.4097 " 0.0001 0.048 " 0.002 2.3 . 10y4
0.475 " 0.001 y0.016 " 0.007 1.2 . 10y3
0.450 " 0.001 0.068 " 0.007 1.2 . 10y3
deviation s listed in table 5. We have found no significant difference between fits with t 1y a and with t 2 b for both x and f . Therefore we conclude that f and x are almost equally good variables for the construction of order parameters for the system under study. This is consistent with what the symmetry indicates: no significant symmetry difference between ŽT, x . and ŽT, f . is observed in figure 1.
v,
FIGURE 2. A ln]ln plot of Ž1 y fc .rfc against molar mass M for Ždiethyl maleate q n-alkane.: experimental values; }}}, calculated values from equation Ž2. with r s 0.42.
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TABLE 6. Parameters of equation Ž8. for the coexistence curves of ŽT, n., ŽT, x ., and ŽT, f . for x ŽCHCOOC 2 H 5 . 2 q Ž1 y x .CH 3 ŽCH 2 . 7 CH 3 4 Order parameter
B
B1
n
0.0653 " 0.0002 0.0675 " 0.0002
y0.021 " 0.002
x
1.850 " 0.004 1.890 " 0.006
y0.38 " 0.05
f
1.836 " 0.004 1.875 " 0.006
y0.37 " 0.05
When the critical exponents b and D are fixed at the theoretical values Ž b s 0.3265, D s 0.50.,Ž11,12. and equation Ž8. is used to fit the phase separation data, the parameters B and B1 can be obtained. The results are listed in table 6. Combination of equations Ž8. and Ž10. yields:
r 1 s rc q Dt Z y Ž 1r2 . Bt b y Ž 1r2 . B1t bqD ,
Ž 11 .
r 2 s rc q Dt Z q Ž 1r2 . Bt b q Ž 1r2 . B1t bqD .
Ž 12 .
When Z, b , D, and Tc are fixed at 0.89, 0.3265, 0.5, and 310.064 K, respectively, and the values of D, rc , B, and B1 are taken from tables 5 and 6, the values of r 1 ,
v,
.865 FIGURE 3. A ln]ln plot of Bf fy1 against molar mass M for Ždiethyl maleate q n-alkane.: c experimental values; }}}, calculated values from equation Ž1. with b s 0.31.
The coexistence curves of Ždiethyl maleate q nonane.
1261
TABLE 7. Critical amplitudes Bf and critical volume fractions fc for ŽCHCOOC 2 H 5 . 2 q n-alkane4. M is the molar mass of n-alkane n-alkane Hexane Heptane Octane Nonane Decane
MrŽg . moly1 . 86.178 100.206 114.233 128.260 142.287
fc
Bf
0.406 0.421 0.436 0.447 0.458
1.749 1.824 1.816 1.836 1.904
r 2 , and r d can be calculated from equations Ž11., Ž12., and Ž10.. The results are shown as lines in figure 1. The calculated values are in good agreement with the experimental results. Table 7 summarizes the values of Bf and fc for five experimentally investigated systems of Ždiethyl maleate q n-alkane.. According to equations Ž1. and Ž2., ln]ln .865 plots of Ž1 y fc .rfc and Bf fy1 against the molar mass of n-alkane M will c yield two straight lines. These two plots are shown in figures 2 and 3. A least-squares fit results in the values of Ž0.42 " 0.01. and Ž0.31 " 0.04. for the exponents r and b, which are in excellent agreement with calculated values Ž b s 0.29.,Ž1. and observed values from experimental studies on chain-molecule solutions of both small molecules Ž2,15. and polymers.Ž2. This supports the universality of the exponents of M and the Landau]Ginsburg]Wilson type model we proposed recently for chain-molecule solutions. This work was supported by the National Natural Science Foundation ŽProject 29673019., the State Education Committee, and the Natural Science Foundation of Gansu Province ŽNo: ZR-96-009., P. R. China. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
An, X.; Jiang, F.; Chen, C; Shen, W. Chem. Phys. Letters 1998, 282, 403]408. An, X., Jiang, F.; Chen, C.; Shen, W. Pure and Applied Chem. 1998, 70, 609]614. An, X.; Lui, X.; Shen, W. J. Chem. Thermodynamics 1997, 29, 669]675. An, X.; Yang, J.; Shen, W. J. Chem. Thermodynamics 1998, 30, 13]19. An, X.; Mao, C.; Sun, G.; Shen, W. J. Chem. Thermodynamics 1998, 30, 689]695. An, X.; Chi, X.; Wang, T.; Shen, W. J. Chem. Thermodynamics 1998, 30, 1199]1205. An, X.; Shen, W.; Wang, H.; Zheng, G. J. Chem. Thermodynamics 1993, 25, 1373]1383. Riddick, J. A.; Bunger, W. B.; Sakano, T. K. Organic Sol¨ ents: 4th edition. Techniques of Chemistry, Vol. II. Wiley-Interscience: New York. 1986. Thermodynamics Research Centre, API 44 Tables, Selected Values of Properties of Hydrocarbons and Related Components, Vol. I. 1972. Wegner, F. J. Phys. Re¨ . 1972, B5, 4529]4536. Alpert, D. Z. Phys. Re¨ . 1982, B25, 4810]4814. Le Guillou, J. C.; Zinn-Justin, J. Phys. Re¨ . 1980, B21, 3976]3998. Ewing, M. B.; Johnson, K. A.; McGlashan, M. L. J. Chem. Thermodynamics, 1988, 20, 49]62. Greer, S. C.; Das, B. K.; Kumar, A.; Gopal, E. S. R. J. Chem. Phys. 1983, 79, 4545]4552. 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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