J. Chem. Thermodynamics 1998, 30, 751]760Article No. ct970341

Measurements of coexistence curves and{turbidity for xC H NO H6 5 2

( ) ( ) }1 I x CH CH CH3 2 6 3in the critical region

Xueqin An, Fuguo Jiang, Haihong Zhao,a Chuanyin Chen,and Weiguo Shenb

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

Ž .The coexistence curves and the turbidity of nitrobenzene q octane have been measured andthe critical amplitudes and exponents have been deduced. The values of exponents areconsistent with theoretical predictions. The experimental results of coexistence curves havealso been analysed to examine Wegner correction terms and the behaviour of the diameter ofthe coexistence curves in various units. A combination of the Wegner equation and theexpression for the diameter has been used to describe successfully the coexistence curves.Q 1998 Academic Press

KEYWORDS: critical phenomena; coexistence curve; turbidity; nitrobenzene; octane

1. Introduction

Polymer solutions near their critical points are known to exhibit power-law� Ž .dependence on the reduced temperature t t s T y T rT , where T is the criticalc c c

4temperature and chain length N, or molar mass M of a polymer molecule. Thus,apart from a system-dependent parameter, two scaling exponents may fullycharacterize any thermodynamic property of a polymer solution near its criticalpoint. This may be described as the general forms for difference of volumefractions of two coexisting phases Df, correlation length j , and osmoticcompressibility of susceptibility x :

< < b yb < < bDf s B t A M t , 1Ž .

< <yn f < <ynj s j t A M t , 2Ž .0

< <yg g < <ygx s x t A M t , 3Ž .0

where, b , n , and g are t exponents, and well established as universal indices by

a Present address: Chendu Organic Chemistry Institute, The Chinese Academy of Science.b Ž .To whom correspondence should be addressed E-mail: Shenwg@Lzu.edu.cn .

0021]9614r98r060751 q 10 $30.00r0 Q 1998 Academic Press

X. An et al.752

experiments and theories; and B, j , and x are the amplitudes that are dependent0 0on M. This dependence is characterized by M exponents b, f , and g, and thevalues of M exponents are generally believed to be universal.

The value of B may be deduced by determination of coexistence curves of aŽ1. Ž . Ž .binary solution. The amplitudes j and x in equations 2 and 3 may be0 0

extracted from turbidity measurements.Ž2,3. The turbidity t , the correlation length,and the osmotic compressibility are related by an integrated form of the Ornstein-Zernike equation,Ž2 ] 4.

23 4 2 xt s p rl ­ n r­f k T f a , 4Ž . Ž .Ž .Ž .0 b

where l is the wavelength of light in a vacuum, k is Boltzmann’s constant, and n0 band f are, respectively, the refractive index and the volume fraction of the

Ž .solution. The correlation length enters through the function f a :

f a s 2a 2 q 2a q 1 ln 1 q 2a ra 3 y 2 1 q a ra 2 , 5� 4Ž . Ž . Ž . Ž . Ž .

Ž .2with a s 2 2npjrl .0From the Flory-Huggins theory and the symmetric Landau-type expansion of

free energy within the limit of f < 1, the values of b s 0.25, f s 0.25, and g s 0cwere obtained.Ž5. The Flory-Huggins theory has been widely used in predicting theproperties of small-chain molecule solutions. Thus, a similar dependence of thethermodynamic properties on the molar mass of n-alkane near the critical solutionpoints was expected.Ž6. In the previous work, the coexistence curves ofŽ . Ž . Ž7,8.nitrobenzene q tetradecane and nitrobenzene q dodecane were measured.As a continuing part of the study of the universal molar mass dependence ofcritical behaviours in binary liquid mixtures of both small and polymer molecules,in this paper we present measurements of coexistence curves and the turbidity forŽ .nitrobenzene q octane . The experimental results of coexistence curves areanalysed to determine the critical exponent b and the critical amplitudes B, and to

Ž .examine the behaviour of the difference r y r of general ‘‘density’’ of two2 1coexisting phases and the diameters r of the coexistence curves over a temperaturedrange of about 10 K from the critical temperature. The values of exponents g andn , and the amplitudes j and x are extracted from results of turbidity0 0measurements in one phase region.

2. Experimental

Ž .The nitrobenzene obtained from Beijing Chemical Factory A.R. grade was driedover calcium chloride for 5 d and distilled slowly under reduced pressure. Themiddle part of the distillate was collected and passed through a column of driedchromatographic alumina. The purified nitrobenzene was stored in the dark over a

Ž .0.4 nm molecular sieve. The octane mass fraction 0.99 supplied by Fluka Co. wasdried and stored over a 0.4 nm molecular sieve.

The apparatus and experimental procedure for measurements of the coexistence

� Ž . Ž . 4Coexistence curves and turbidity of xC H NO q 1 y x CH CH CH6 5 2 3 2 6 3 753

curves have been described previously.Ž1. During measurements, the temperaturewas constant to "0.002 K. The accuracy and the precision in measurements oftemperature were better than "0.01 K and "0.001 K, respectively. The accuracy of

Ž .measurement was "0.003 K for the temperature difference T y T , "0.0002 forcthe refractive index in each coexisting phase, and "0.001 for the critical molefraction x .c

The experimental details for measurements of turbidity have been reportedrecently.Ž9. A sample with critical composition was prepared in a 10 nm path-lengthrectangular cell provided with an Ace-thread connection, which allowed it to besealed with a Teflon cap. The sample cell was placed in a metal block, where thetemperature was constant to within "0.003 K. The block was sat in a spectrometerŽ .Beckmann DU-7 , by which the turbidity was measured with an accuracy of0.005 cmy1. The critical temperature was determined by observation of the phaseseparation temperature through the light-beam-path hole. The accuracy in

Ž .measurement of temperature difference T y T was about "0.005 K.c

3. Results and discussion

COEXISTENCE CURVES

�The critical mole fraction and the critical temperature of xC H NO q6 5 2Ž . Ž . 4 Ž .1 y x CH CH CH were determined to be x s 0.505 " 0.001 and T s3 2 6 3 c cŽ .293.1 " 0.2 K, respectively.

The refractive indices n were measured for each coexisting phase at varioustemperatures. The results are listed in columns 2 and 3 of table 1 and are shown in

Ž .figure 1 a .The refractive index n of a pure liquid or a mixture may be expressed as a linear

function of temperature in a certain temperature range:.n T , x s n T8, x q R x T y T8 , 6Ž . Ž . Ž . Ž . Ž .

R x s xR q 1 y x R , 7Ž . Ž . Ž .A B

Ž .where R x is the derivative of n with respect to T for a particular composition x,Ž .and R and R are the values of R x for x s 1 and x s 0, respectively.A B

Ž . Ž . Ž1,7,10,11.Equations 6 and 7 were tested in previous papers. It allowed us to obtainŽ . Ž .n T , x as a function of T and x simply by measurements of R x for two pure

components at various temperatures, and of the refractive indices for mixtures withvarious known compositions at a fixed temperature. The refractive indices n ofpure octane at various temperatures and the refractive indices of a series of binarymixtures with known values of mole fraction x in the one phase region at

Ž .T s 293.30 K were measured and are listed in tables 2 and 3. Fitting equation 6. y4 y1to the results listed in table 2 gives R s y4.78 10 K . The value R sB A

. y4 y1 Ž8. Ž .y5.13 10 K was reported previously. The results for n T8, x for variousŽ .compositions at T8 s 293.30 K see table 3 were fitted with a polynomial form to

obtain:. . 2 . 3 . 2n T8, x s 1.3957 q 0.0876 x q 0.0788 x y 0.0464 x q 0.0316 x , 8Ž . Ž .

X. An et al.754

Ž . Ž . Ž . � Ž . Ž . 4TABLE 1. Coexistence curves of T , n , T , x , and T , f for xC H NO q 1 y x CH CH CH .6 5 2 3 2 6 3Refractive indices n were measured at wavelength l s 632.8 nm, and T s 293.052 K. Volume fractionc

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

Ž .T y T rK n n x x f fc 1 2 1 2 1 2

0.003 1.4541 1.4585 0.490 0.520 0.377 0.4060.006 1.4531 1.4594 0.484 0.526 0.371 0.4110.009 1.4524 1.4602 0.479 0.532 0.366 0.4170.013 1.4519 1.4607 0.475 0.535 0.363 0.4200.018 1.4513 1.4613 0.471 0.539 0.359 0.4240.027 1.4506 1.4620 0.466 0.543 0.355 0.4280.035 1.4502 1.4624 0.463 0.546 0.352 0.4310.048 1.4496 1.4631 0.459 0.550 0.348 0.4350.065 1.4490 1.4637 0.455 0.554 0.344 0.4390.082 1.4483 1.4644 0.450 0.559 0.340 0.4440.113 1.4475 1.4653 0.444 0.565 0.355 0.4490.160 1.4465 1.4664 0.437 0.571 0.328 0.4560.219 1.4455 1.4675 0.430 0.578 0.322 0.4630.291 1.4444 1.4687 0.421 0.586 0.314 0.4710.389 1.4433 1.4700 0.413 0.594 0.307 0.4790.501 1.4423 1.4712 0.405 0.601 0.300 0.4860.681 1.4409 1.4729 0.395 0.611 0.291 0.4971.011 1.4390 1.4754 0.379 0.625 0.278 0.5121.362 1.4373 1.4776 0.365 0.638 0.266 0.5261.836 1.4355 1.4801 0.350 0.651 0.253 0.5412.342 1.4341 1.4824 0.337 0.664 0.243 0.5542.907 1.4327 1.4847 0.324 0.676 0.232 0.5673.536 1.4313 1.4869 0.311 0.687 0.221 0.5804.251 1.4301 1.4892 0.299 0.698 0.212 0.5935.129 1.4287 1.4917 0.284 0.710 0.200 0.6075.954 1.4278 1.4939 0.273 0.720 0.192 0.6186.939 1.4267 1.4964 0.261 0.731 0.182 0.6327.958 1.4258 1.4986 0.249 0.741 0.173 0.6438.963 1.4251 0.5008 0.239 0.750 0.166 0.6549.983 1.4243 1.5028 0.228 0.758 0.157 0.664

Ž . Ž .with a standard deviation of less than 0.0002. Equations 6 and 8 were used toconvert refractive indices of the coexisting curve to mole fractions. The volumefraction f of C H NO was then calculated from the mole fraction by:6 5 2

1rf s 1 y K q Krx , 9Ž . Ž .K s d M rd M , 10Ž .A B B A

where d is mass density, M is the molar mass, and subscripts A and B relate tonitrobenzene and octane, respectively. The values of d and d were obtainedB A

Ž . Ž . Ž .from references 12 and 13 . The values of x, f of coexisting phases at variousŽ .temperatures are listed in columns 4 to 7 of table 1, and are shown in figures 1 b

Ž .and 1 c .In the region sufficiently close to the critical temperature, the coexistence curve

can be represented by:

r y r s Bt b , 11Ž .2 1

� Ž . Ž . 4Coexistence curves and turbidity of xC H NO q 1 y x CH CH CH6 5 2 3 2 6 3 755

Ž .FIGURE 1. Coexistence curves of: a, temperature against refractive index T , n ; b, temperatureŽ . Ž . � Žagainst mole fraction T , x ; and c, temperature against volume fraction T , f for xC H NO q 1 y6 5 2

. Ž . 4 Ž .x CH CH CH . v, experimental values of concentration variables r of the coexisting phases; ',3 2 6 3Ž .experimental values of diameter r of the coexisting phases; , concentration variablesd

Ž .r of coexisting phases from calculation.cal

where r is the ‘‘density’’ variable, and r and r are the values of r in upper and1 2Ž .lower coexistence phases. The experimental values of the differences r y r2 1

Ž .were fitted with equation 11 , with all points equally weighted, to obtain b and B.The results are listed in table 4. The values of b and B slightly depend on the

Žcutoff values of t , but are in good agreement with that of nitrobenzene qmax. Ž . Ž7,8.dodecane and nitrobenzene q tetradecane reported previously, and the

theoretical prediction of 0.327.Ž14.

Ž .TABLE 2. Refractive indices n at wavelength l s 632.8 nm for CH CH CH3 2 6 3at various temperatures

TrK n TrK n TrK n TrK n

282.773 1.4008 285.575 1.3995 287.520 1.3986 289.802 1.3975290.980 1.3969 292.105 1.3964 292.524 1.3961 293.300 1.3958

X. An et al.756

TABLE 3. Refractive indices n at wavelength l s 632.8 nm for� Ž . Ž . 4xC H NO q 1 y x CH CH CH at the temperature 293.30 K6 5 2 3 2 6 3

x n x n x n x n

0.0000 1.3958 0.1004 1.4048 0.2002 1.4163 0.2944 1.42740.3892 1.4398 0.4995 1.4550 0.6067 1.4717 0.6962 1.48670.7924 1.5039 0.8969 1.5246 1.0000 1.5472

Ž .TABLE 4. Values of critical amplitudes B and critical exponents b for coexistence curves of T , n ,Ž . Ž . � Ž . Ž . 4 Ž . Ž .T , x , and T , f for xC H NO q 1 y x CH CH CH in equation 11 ; t s T y T rT6 5 2 3 2 6 3 c c

t - 0.033 t - 0.003Orderparameter B b B b

n 0.237 " 0.001 0.328 " 0.001 0.228 " 0.002 0.324 " 0.002x 1.598 " 0.008 0.328 " 0.001 1.54 " 0.02 0.324 " 0.003f 1.530 " 0.008 0.329 " 0.001 1.46 " 0.01 0.324 " 0.003

ŽWith the critical exponents b and D fixed at the theoretical values b s 0.327,. Ž14,15.D s 0.50 , a weighted non-linear least-squares program was used to fit the

Wegner equation,Ž16.

r y r s Bt bB t bqD q ??? , 12Ž .2 1 1

to obtain the parameters B and B . The total uncertainties used in weighting1Ž .included the errors in measurements of r y r and standard deviations2 1

Ž .propagated from measurements of T y T . The results are summarized in table 5.cDiameters r of coexistence curves for the two choices of the ‘‘density’’ variabled

were fitted with the form:

r s r q Dt z , 13Ž .d c

to test the presence of a 2b term and to examine the goodness of the selectedŽ .order parameter. The apparent exponent z in equation 13 was fixed at the values

Ž .1 y a s 0.89, and 2b s 0.654, in separate fitting procedures. The results areŽ .compared in table 6. The quality of the fit with equation 13 is indicated by the

values of standard deviation s listed in table 6. The quantity of D s s s y s ,1ya 2 b

Ž . Ž .TABLE 5. Coefficients of equation 12 for coexistence curves of T , n ,Ž . Ž . � Ž . Ž . 4T , x , and T , f for xC H NO q 1 y x CH CH CH6 5 2 3 2 6 3

Order parameter B B1

n 0.2358 " 0.00020.2320 " 0.0006 0.029 " 0.005

x 1.593 " 0.0031.570 " 0.009 0.18 " 0.07

f 1.521 " 0.0041.49 " 0.01 0.20 " 0.09

� Ž . Ž . 4Coexistence curves and turbidity of xC H NO q 1 y x CH CH CH6 5 2 3 2 6 3 757

Ž .TABLE 6. Coefficients of equation 13 and standard deviations s in r , for diameter of coexistencedŽ . Ž . Ž . � Ž . Ž . 4 Ž .curves of T , n , T , x , and T , f for xC H NO q 1 y x CH CH CH . r is the critical6 5 2 3 2 6 3 c exp

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

Ž . Ž . Ž .T , n T , x T , f

Ž .r 1.4653 " 0.0002 0.505 " 0.001 0.392 " 0.001c exp

z s 0.89r 1.4563 " 0.0001 0.504 " 0.001 0.393 " 0.001cD 0.1475 " 0.0004 y0.239 " 0.007 0.385 " 0.007

y5 y4 y4. . .s 3.2 10 6.1 10 5.5 10

z s 0.654r 1.4559 " 0.0001 0.505 " 0.001 0.392 " 0.001cD 0.064 " 0.002 y0.106 " 0.001 0.170 " 0.001

y4 y4 y4. . .s 2.9 10 2.3 10 2.6 10

where s and s are standard deviations in fitting experimental values of r to1ya 2 b dŽ .equation 13 with z s 1 y a and z s 2b , respectively, may be used to show the

goodness of order parameters x and f.Ž6. It is found that the value of D s for x isslightly larger than that for f, which indicates that f is a better order parameterthan x. This is consistent with what the symmetry of the coexistence curves

Ž . Ž .showed, because the T , f curve is slightly more symmetric than the T , x curvefor this binary mixture. The experimental value of n listed in table 6 was obtainedcby extrapolating refractive indices against temperatures in the one-phase region tothe critical temperature. The experimental values of x and f listed in table 6c cwere determined by the technique of ‘‘equal volumes of two coexistence phases’’

Ž . Ž .and calculated by equations 9 and 10 . The uncertainties of optimal parametersreported in table 6 included no systematic uncertainties contributed by convertingn to x, and x to f. Such uncertainties in x and f were estimated to be about

Ž ."0.002. The values of x and f obtained from extrapolation of equation 12 arec cconsistent with those from observations. This is evidence that no significant criticalanomaly is present in refractive indices and that they were properly converted tomole fractions and volume fractions in our treatments.

Ž . Ž .Combination of equations 12 and 13 yields:

r s r q Dt z y 1r2 Bt b y 1r2 B t bqD , 14Ž . Ž . Ž .1 c 1

r s r q Dt z q 1r2 Bt b q 1r2 B t bqD . 15Ž . Ž . Ž .2 c 1

When z, b , D, and T were fixed at 0.654, 0.327, 0.5, and 293.052 K, respectively,cand the values of D, r , B, and B were taken from tables 5 and 6, the values ofc 1

Ž . Ž .r and r were calculated from equations 14 and 15 . The results are shown as1 2lines in figure 1. The values from calculation are in good agreement withexperimental results.

TURBIDITY MEASUREMENTS

Besides the contribution to the turbidity from the critical concentration fluctuation

X. An et al.758

t , the total turbidity t experimentally determined for various wavelengths in thisTwork would include the background contributions t , such as absorption of absolution and the cell windows, the effects of the reflections from the cell windowsand the scattering from the dust in the sample, and the contribution of densityfluctuation of the fluids. The contribution from the critical concentration fluctuationto turbidity may be expressed by:Ž3,17.

t s t y t . 16Ž .T b

When the temperature is far from the critical solution point, the contribution fromconcentration fluctuation to the total turbidity may be negligible. We made a seriesof turbidity measurements at various temperatures and found that the totalturbidity decreased as the temperature increased and almost remained unchangedwhen the temperature was 10 K above the critical solution point. The unchanged

Ž .value of turbidity then was taken as t , and used to calculate t by equation 16 forbvarious wavelengths. The results are listed in table 7.

� Ž . Ž . 4TABLE 7. Turbidity t for the critical solution xC H NO q 1 y x CH CH CH with the critical6 5 2 3 2 6 3concentration x s 0.505 at various temperatures and various wavelengths

Ž .T y T rK 440rnm 450rnm 480rnm 510rnm 540rnm 570rnm 600rnm 633rnmc

1.614 0.035 0.036 0.026 0.019 0.014 0.011 0.009 0.0071.364 0.045 0.045 0.033 0.024 0.018 0.014 0.011 0.0091.165 0.056 0.055 0.040 0.029 0.022 0.017 0.013 0.0100.993 0.068 0.068 0.049 0.035 0.026 0.021 0.016 0.0130.835 0.091 0.085 0.062 0.043 0.033 0.027 0.021 0.0170.696 0.115 0.106 0.077 0.055 0.043 0.034 0.027 0.0210.585 0.141 0.131 0.094 0.069 0.052 0.042 0.033 0.0260.494 0.173 0.161 0.115 0.085 0.065 0.052 0.041 0.0330.420 0.209 0.192 0.139 0.104 0.080 0.064 0.051 0.0410.359 0.247 0.227 0.166 0.123 0.095 0.076 0.061 0.0480.307 0.291 0.266 0.195 0.146 0.112 0.089 0.072 0.0570.262 0.341 0.311 0.228 0.172 0.133 0.106 0.085 0.0680.230 0.381 0.347 0.256 0.195 0.151 0.119 0.096 0.0760.230 0.384 0.349 0.257 0.195 0.151 0.120 0.097 0.0770.214 0.414 0.377 0.278 0.212 0.165 0.131 0.107 0.0860.198 0.441 0.399 0.295 0.226 0.175 0.139 0.112 0.0910.172 0.499 0.452 0.335 0.256 0.199 0.159 0.129 0.1030.147 0.571 0.518 0.384 0.294 0.230 0.185 0.151 0.1210.126 0.650 0.591 0.441 0.338 0.265 0.213 0.173 0.1390.105 } 0.674 0.504 0.389 0.306 0.246 0.201 0.1630.093 } } 0.559 0.433 0.341 0.276 0.226 0.1840.083 } } } 0.473 0.375 0.304 0.252 0.2030.072 } } } 0.526 0.417 0.339 0.279 0.2280.061 } } } 0.597 0.475 0.388 0.319 0.2620.052 } } } } 0.538 0.438 0.362 0.2990.043 } } } } 0.607 0.497 0.411 0.3400.038 } } } } 0.658 0.541 0.448 0.3710.032 } } } 0.913 0.737 0.608 0.506 0.4200.027 } } 1.289 1.020 0.826 0.683 0.571 0.4750.024 } 1.810 1.398 1.109 0.901 0.746 0.626 0.5230.021 2.129 1.947 1.509 1.199 0.976 0.809 0.680 0.5700.019 2.259 2.067 1.612 1.278 1.044 0.869 0.731 0.613

� Ž . Ž . 4Coexistence curves and turbidity of xC H NO q 1 y x CH CH CH6 5 2 3 2 6 3 759

Ž 2 . Ž .The ­ n r­f and n in equation 4 are dependent on temperature, volumeŽ 2 .fraction, and wavelength. The values of ­ n r­f and n for given temperature and

Ž . Ž .volume fraction may be calculated by using equations 6 and 10 . The dependenceof refractive index on wavelength may be expressed as:Ž18.

n s a q a rl2 q ??? . 17Ž .0 1

Ž .We assumed that equation 17 is valid to solutions with the following relation:

. .a s a x q a 1 y x , 18Ž . Ž .1s 1A 1B

Ž .where a , a , and a are the values of a in equation 17 for a solution at given1s 1A 1B 1x, for x s 1, and x s 0, respectively. The refractive indices of nitrobenzene andoctane for different wavelengths at given temperatures measured by us and

Ž13,19. Ž .reported by references were used to estimate a through equation 18 . The1sŽ . Ž .value of n for a solution then was calculated by equations 17 and 18 . Finally,

Ž 2 .the dependence of ­ n r­f on wavelength may be estimated by using equationsŽ . Ž . Ž . Ž .17 , 18 , and 6 through 10 .

Ž . Ž .In principle, the exponents of f , g, g , and n in equations 2 and 3 may beobtained by fitting the experimental values of turbidity at various temperatures and

Ž . Ž .wavelengths with equations 4 and 5 . However, in practice, simultaneousdetermination of these four exponents are difficult because of the strongcorrelations between them.Ž3,17,20.

Ž . Ž .It has been shown that the amplitudes in equations 1 through 3 were relatedby:

1r32R s j Br2 r k T x , 19Ž . Ž . Ž .� 40 B c 0

with a predicted valueŽ21,22. of 0.65 from series expansions, or 0.67 from e expansionsŽ .in a renormalization group setting. With the help of equation 19 , j may be0

Ž .expressed as a function of x and B. The value of B for nitrobenzene q octane0was taken as 1.521 listed in table 5, the number of optimal parameters then wasreduced to 3. A weighted non-linear least-squares program was used to fit the

Ž . Ž .turbidity data with equations 4 and 5 to obtain n , g , and j . The value of x0 0Ž .was then calculated by equation 19 . The total uncertainties of t used in weighting

include the uncertainties in measurements of t and t , the standard deviationsT bŽ .propagated from measurements of T y T , and the errors in the estimation of thec

Ž 2 .dependence of n and ­ n r­f on wavelength. The values of j , x , n , and g are0 0listed in lines 1 and 2 of table 8 for R s 0.65 and R s 0.67, respectively. The valueof x 2 in table 8 is a measure of accuracy of the fit. The optimal values ofn

exponents n and g are in good agreements with the theoretical predictions andalmost no differences for R s 0.65 and R s 0.67. With the critical exponents n

Ž . Ž15.and g to be fixed at the theoretical values n s 0.63, g s 1.241 , the turbidityŽ . Ž .data were fitted with equations 4 and 5 to determine j and further to calculate0

Ž .x by using equation 19 . The values of j and x are listed in lines 3 and 4 of0 0 0table 8 for R s 0.65 and R s 0.67, respectively. The numbers in the brackets intable 8 refer to the theoretical predictions. The values of x 2 indicate that the fitn

with R s 0.67 is slightly better than the fit with R s 0.65.

X. An et al.760

Ž . Ž . � Ž . Ž . 4TABLE 8. Critical parameters in equations 2 and 3 for xC H NO q 1 y x CH CH CH6 5 2 3 2 6 3

y9 3 y1 2. .Ž .R j rnm x r 1 10 m J n g x0 0 n

0.65 0.162 " 0.003 2.22 " 0.12 0.68 " 0.01 1.27 " 0.01 0.450.67 0.166 " 0.003 2.18 " 0.12 0.68 " 0.01 1.28 " 0.01 0.44

Ž . Ž .0.65 0.1681 " 0.0002 2.469 " 0.008 0.63 1.241 1.15Ž . Ž .0.67 0.1740 " 0.0002 2.500 " 0.008 0.63 1.241 1.05

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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( )Recei ed 29 August 1997; in final form 22 December 1997

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