J. Chem. Thermodynamics 35 (2003) 1751–1762

www.elsevier.com/locate/jct

The coexistence curves of{xN-methyl-2-pyrrolidone+ (1� x)cyclohexane}

and {xN-methyl-2-pyrrolidone +(1� x)cyclooctane} in the critical region

Xueqin An a,b, Chunfeng Mao a, Weijiang Ying a, Lixin Han a,Weiguo Shen a,*

a Department of Chemistry, Lanzhou University, Lanzhou, Gansu 730000, Chinab College of Chemistry and Environment Science, Nanjing Normal University, Nanjing, 210097, China

Received 9 June 2003; accepted 26 June 2003

Abstract

Coexistence curves of ðT ; nÞ, ðT ; xÞ and ðT ;/Þ, where n, x, and / are refractive index, mole

fraction and volume fraction, respectively, for the binary mixtures {xN-methyl-2-pyrrolidone

(NMP)+ (1� x)cyclohexane} and {xNMP+ (1� x)cyclooctane} have been determined in the

critical regions by measurements of n. The critical amplitude B and the critical exponent bhave been deduced and the values of b are consistent with the theoretical ones. The experimen-

tal results have been analysed to examine the Wegner correction terms and the behaviour of

the rectilinear diameter of the coexistence curves. The coexistence curves have been success-

fully described by a combination of the Wegner equation and the expression for the diameter.

The power law dependence of critical behaviour either on the molar volume of cycloalkane or

on the interaction parameter per unit volume has also been discussed.

� 2003 Elsevier Ltd. All rights reserved.

Keywords: Coexistence curve; Refractive index; Critical phenomena; N-methyl-2-pyrrolidone; Cyclohex-

ane; Cyclooctane

* Corresponding author. Fax: +86-931-862-5576.

E-mail address: shenwg@lzu.edu.cn (W. Shen).

0021-9614/$ - see front matter � 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0021-9614(03)00152-6

1752 X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762

1. Introduction

In previous work, we reported the critical temperatures and the critical composi-

tions for binary solutions of {N-methyl-2-pyrrolidone (NMP)+methylcyclohexane},

{NMP+cyclohexane}, {NMP+ cyclooctane}, and {NMP+propylcyclopentane}and the coexistence curves for {NMP+methylcyclohexane} near the critical point

by using a refractive-index technique. It was found that the critical volume fractions

of the above solutions might be well described by:

ð1� /cÞ=/c / V �rB ; ð1Þ

where r ¼ 0:41 is a constant exponent [1,2], /c is the critical volume fraction of

NMP, and VB is the molar volume of cycloalkane. As a continuing part of the studyof the critical phenomena in binary mixtures of cycloalkane in a polar solvent, we

report the measurements of coexistence curves for (NMP+ cyclohexane),

(NMP+ cyclooctane). The experimental results are analysed to determine the critical

exponents b and the critical amplitudes B, and to examine the behaviour of the

differences (q2 � q1) of general ‘‘densities’’ of lower and upper coexisting phases and

the diameters qd of the coexistence curves. The power law dependence of critical

behaviour either on the molar volume of cycloalkane or on the interaction parameter

per unit volume are then discussed.

2. Experimental

The NMP (0.99 mass fraction) purchased from Carl Roth Inc. was purified by

fractional distillation at decreased pressure. Cyclohexane (0.99 mass fraction) from

Beijing Chemical Factory was passed through a column of dried chromatographic

alumina and distilled slowly under reduced pressure. Cyclooctane (0.99 mass frac-tion) was supplied by Fluka Chemical Company Inc. All materials were dried and

stored over 4 � 10�10 m molecular sieves.

The coexistence curves were determined by measurements of the refractive indices.

The apparatus and experimental procedure for refractive index measurements, and

the techniques for determinations of the critical mole fractions xc, and the critical tem-

peratures Tc have been described in detail previously [3]. During measurements, the

temperature was constant to 0.002 K. The accuracy and the precision in measure-

ments of temperature were 0.01 K and 0.001 K, respectively. The accuracy of mea-surement was T ¼ 0:003 K for the temperature difference (T � Tc), 0.0001 for the

refractive index in each coexisting phase, and 0.001 for the critical mole fraction xc.

3. Results and discussion

The critical mole fractions and the critical temperatures were determined to be

xc ¼ ð0:384� 0:001Þ, and Tc ¼ ð289:2� 0:2Þ K for {xNMP+ (1� x)cyclohexane},

X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762 1753

and xc ¼ ð0:455� 0:001Þ, and Tc ¼ ð291:5� 0:2Þ K for {xNMP+ (1� x)cyclooc-tane}, which are in good agreement with the previous measurements [1].

The refractive indices n were measured for each coexisting phase at various tem-

peratures. The results are listed in columns 2 and 3 of table 1 for {xNMP+ (1� x)cy-clohexane}, and of table 2 for {xNMP+ (1� x)cyclooclane}. They are also shown infigure 1(a) and figure 2(a).

We assumed that the refractive index n of a pure liquid or a mixture can be ex-

pressed as a linear function of temperature in a certain temperature range:

TABLE 1

Coexistence curves of ðT ; nÞ, ðT ; xÞ and (T ;/) for {xNMP+ (1� x)cyclohexane}

(Tc � T )/K n1 n2 x1 x2 /1 /2

0.024 1.4397 1.4430 0.346 0.426 0.320 0.397

0.034 1.4395 1.4433 0.341 0.433 0.315 0.404

0.044 1.4394 1.4434 0.338 0.435 0.312 0.406

0.052 1.4393 1.4436 0.336 0.440 0.310 0.411

0.068 1.4391 1.4438 0.330 0.444 0.305 0.415

0.079 1.4391 1.4439 0.330 0.446 0.305 0.417

0.096 1.4389 1.4440 0.325 0.448 0.300 0.4200

0.114 1.4388 1.4442 0.322 0.453 0.297 0.424

0.133 1.4387 1.4444 0.320 0.457 0.294 0.428

0.172 1.4384 1.4447 0.312 0.464 0.287 0.435

0.221 1.4381 1.4450 0.303 0.470 0.279 0.441

0.284 1.4379 1.4453 0.297 0.477 0.273 0.447

0.347 1.4376 1.4456 0.289 0.483 0.265 0.454

0.443 1.4374 1.4460 0.283 0.491 0.259 0.462

0.538 1.4372 1.4464 0.276 0.499 0.253 0.470

0.661 1.4371 1.4467 0.272 0.505 0.249 0.475

0.785 1.4367 1.4471 0.260 0.512 0.238 0.483

0.907 1.4366 1.4474 0.256 0.518 0.234 0.488

1.026 1.4364 1.4477 0.249 0.524 0.228 0.494

1.148 1.4363 1.4479 0.245 0.527 0.224 0.497

1.424 1.4360 1.4486 0.233 0.540 0.213 0.510

1.776 1.4358 1.4492 0.223 0.549 0.203 0.520

2.346 1.4356 1.4503 0.210 0.568 0.191 0.539

2.902 1.4353 1.4511 0.194 0.580 0.176 0.551

3.444 1.4354 1.4520 0.190 0.594 0.172 0.566

3.977 1.4353 1.4526 0.179 0.602 0.162 0.573

4.497 1.4353 1.4533 0.172 0.612 0.156 0.584

5.004 1.4353 1.4539 0.165 0.620 0.149 0.592

5.502 1.4353 1.4545 0.158 0.628 0.142 0.600

5.988 1.4354 1.4551 0.153 0.636 0.138 0.609

6.465 1.4355 1.4556 0.149 0.643 0.135 0.615

7.160 1.4354 1.4563 0.136 0.651 0.123 0.624

8.051 1.4356 1.4572 0.128 0.662 0.116 0.635

8.907 1.4358 1.4580 0.121 0.670 0.109 0.644

9.957 1.4361 1.4590 0.114 0.682 0.103 0.656

11.128 1.4364 1.4600 0.105 0.692 0.094 0.666

Refractive indices were measured at wavelength k ¼ 632:8 nm. Subscripts 1 and 2 relate to upper and

lower phases, Tc ¼ 289:153 K.

TABLE 2

Coexistence curves of ðT ; nÞ, ðT ; xÞ and ðT ;/Þ for {xNMP+ (1� x)cyclooctane}

(Tc � T )/K n1 n2 x1 x2 /1 /2

0.017 1.4598 1.4606 0.422 0.501 0.344 0.419

0.060 1.4597 1.4609 0.408 0.526 0.332 0.444

0.084 1.4597 1.4610 0.407 0.534 0.331 0.452

0.108 1.4596 1.4610 0.395 0.533 0.319 0.451

0.156 1.4595 1.4611 0.380 0.540 0.306 0.457

0.257 1.4595 1.4612 0.375 0.544 0.301 0.462

0.347 1.4594 1.4614 0.357 0.558 0.285 0.475

0.478 1.4593 1.4616 0.336 0.569 0.267 0.487

0.715 1.4593 1.4619 0.321 0.584 0.254 0.503

1.031 1.4594 1.4622 0.315 0.597 0.249 0.515

1.444 1.4594 1.4626 0.287 0.613 0.224 0.532

2.014 1.4594 1.4631 0.244 0.631 0.189 0.552

2.809 1.4597 1.4637 0.232 0.649 0.178 0.570

3.837 1.4601 1.4644 0.217 0.666 0.166 0.589

5.154 1.4605 1.4652 0.173 0.681 0.131 0.605

Refractive indices were measured at wavelength k ¼ 632:8 nm. Subscripts 1 and 2 relate to upper and

lower phases, Tc ¼ 291:503 K.

FIGURE 1. Coexistence curves of: (a) (T ; n); (b) (T ; x); and (c) (T ;/) for {xNMP+ (1� x)cyclohexane};d, experimental values of concentration variables (q) of the coexisting phases; N, experimental values of

diameter (qd) of the coexisting phases; –––, values of (qcal) and (qdcal) calculated from a combination of

equations (10) to (12) with coefficients listed in tables 6 and 7.

1754 X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762

nðT ; xÞ ¼ nðT 0; xÞ þ RðxÞðT � T 0Þ; ð2Þ

RðxÞ ¼ xRA þ ð1� xÞRB; ð3Þ

where RðxÞ is the derivative of n with respect to T for a particular composition x, andRA and RB are the values of RðxÞ for x ¼ 1 and x ¼ 0, respectively. In previous papers

[1,3–5], we made a series of measurements of refractive indices of binary solutions

FIGURE 2. Coexistence curves of: (a) (T ; n); (b) (T ; x); and (c) (T ;/) for {xNMP+ (1� x)cyclooctane};d, experimental values of concentration variables (q) of the coexisting phases; N, experimental values

of diameter (qd) of the coexisting phases; –––, values of (qcal) and (qdcal) calculated from a combination

of equations (10) to (12) with coefficients listed in tables 6 and 7.

X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762 1755

with various known compositions at various temperatures, and found that equations

(2) and (3) fitted the experimental data very well with a standard deviation of less

than 0.0002 in refractive index. The validity of equations (2) and (3) allowed us to

obtain nðT ; xÞ as a function of T and x simply from measurements of RðxÞ for twopure components at various temperatures, and measurements of the refractive in-

dices for mixtures with various known compositions at a fixed temperature above the

critical point. The refractive indices n of pure NMP, cyclohexane and cyclooctane

at various temperatures were measured and are listed in table 3. Two batches of

NMP were used in studies, batch 1 for (NMP+ cyclohexane) and batch 2 for

(NMP+ cyclooctane). The difference in refractive index of two batches of NMP was

about 5 � 10�4; however it does not influence the final results because the same batch

of NMP was used for measurements of (n; x) standard curve and the coexistencecurve. Table 4 gives the measured refractive indices of a series of binary mixtures

with known values of mole fraction x in the one phase region at T ¼ 289:65 K for

{xNMP+ (1� x)cyclohexane}, and T ¼ 292:60 K for {xNMP+ (1� x)cyclooctane}.Fitting equation (2) to the results listed in table 3 gives RA ¼ �4:15 � 10�4 K�1 for

NMP, RB ¼ �5:34 � 10�4 K�1 for cyclohexane, and RB ¼ �4:81 � 10�4 K�1 for cyc-

looctane. The values of n at various compositions and given temperatures listed in

table 4 were fitted with a polynomial form. We obtained equation (4) for

{xNMP+ (1� x)cyclohexane} and equation (5) for {xNMP+ (1� x)cyclooctane},respectively:

nð289:65 K; xÞ ¼ 1:4268þ 0:0328xþ 0:0111x2; ð4Þ

nð292:60 K; xÞ ¼ 1:4569þ 0:0029xþ 0:0047x2 þ 0:0045x3; ð5Þ

TABLE 4

Refractive indices n at wavelength k ¼ 632:8 nm for {xNMP+ (1� x)cyclohexane} and {xNMP+

(1� x)cyclooctane} at T ¼ 289:65 K and 292.60 K

x n x n x n

x NMP+ (1� x)cyclohexane at T ¼ 289:65 K

0.000 1.4269 0.439 1.4434 0.812 1.4606

0.101 1.4300 0.517 1.4468 0.909 1.4655

0.200 1.4336 0.611 1.4509 1.000 1.4708

0.304 1.4378 0.682 1.4543

x NMP+ (1� x)cyclooctane at T ¼ 292:60 K

0.000 1.4569 0.350 1.4587 0.798 1.4645

0.100 1.4572 0.500 1.4601 0.898 1.4666

0.193 1.4577 0.601 1.4613 1.000 1.4690

0.300 1.4583 0.687 1.4626

TABLE 3

Refractive indices n at wavelength k ¼ 632:8 nm for pure NMP, cyclohexane and cyclooctane at various

temperatures

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

NMP (batch 1)

289.650 1.4708 287.150 1.4718 282.150 1.4739

289.150 1.4710 286.150 1.4723 280.150 1.4747

288.150 1.4714 284.150 1.4731

NMP (batch 2)

292.346 1.4691 290.160 1.4700 287.631 1.4710

291.344 1.4695 288.926 1.4705 286.466 1.4716

Cyclohexane

289.650 1.4269 287.150 1.4282 284.150 1.4298

289.150 1.4271 286.150 1.4287 282.150 1.4309

288.150 1.4276 285.150 1.4292 280.150 1.4319

Cyclooctane

292.348 1.4572 290.162 1.4582 287.630 1.4594

291.290 1.4576 288.924 1.4588 286.467 1.4600

1756 X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762

with a standard deviation of less than 0.0001. Equations (2) to (5) were used to

convert refractive indices of the coexistence curve to mole fractions. The volume

fraction / of NMP for both solutions was then calculated from the mole fraction by

1=/ ¼ ð1� KÞ þ K=x; ð6Þ

K ¼ VB=VA; ð7Þ

where VA and VB are molar volume of NMP and cycloalkane, respectively, which

were obtained from reference [6]. The values of (x;/) of coexisting phases at various

temperatures are listed in columns 4 to 7 of tables 1 and 2, and are shown in (b) and

(c) of figures 1 and 2.

X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762 1757

In the region sufficiently close to the critical temperature, the coexistence curve

can be represented by:

TABL

Values

equatio

Den

vari

nx/

nx/

q2 � q1 ¼ Bsb; ð8Þ

where s ¼ ðTc � T Þ=Tc; Tc is the critical temperature, B is the critical amplitude, b is

the critical exponent, q is the ‘‘density’’ variable, q1 and q2 are the values of q in

upper and lower coexistence phases. The differences (q2 � q1) obtained in this work

were fitted with equation (8) to obtain b and B. The results are listed in table 5.

Larger errors in exponents b and amplitudes B for (NMP+ cyclooctane) are ob-served due to the fact that the measured variable n is insensitive to variation of the

mole fractions x. The values of b and B depend on the cutoff values of (Tc � T ), butfor ðTc � T Þ < 1 K, the values of the exponent b are consistent with the theoretical

one of 0.3265 within the experimental uncertainties.

With the critical exponents b and D being fixed at the theoretical values

(b ¼ 0:3265;D ¼ 0:50) [7,8] a non-linear least-squares programme was used to fit

the Wegner equation [9],

q2 � q1 ¼ Bsb þ B1sbþD þ B2s

bþ2D þ � � � ; ð9Þ

to obtain the parameters B and B1. The results are summarized in table 6.

Diameter qd of coexistence curves for the three choices of the ‘‘density’’ variable

were fitted to the form:

qd ¼ ðq2 þ q1Þ=2 ¼ qc þ Dsz; ð10Þ

to test the presence of a 2b term and to examine the goodness of the selected orderparameters. The apparent exponent Z in equation (10) was fixed at the values

ð1� aÞ ¼ 0:89, and 2b ¼ 0:653 in least-separate fitting procedures, where a char-

acterizes the divergence, as the critical point is approached, of the heat capacity at

constant volume for a pure fluid. The results are compared in table 7. The quality of

the fit with equation (10) is indicated by the values of standard deviations S listed in

table 7. The difference DS of the standard deviations between the fits with Z ¼ 1� a

E 5

of critical amplitudes B and critical exponents b for coexistence curves of ðT ; nÞ, ðT ; xÞ, ðT ;/Þ inn (8) for {xNMP+ (1� x)cyclohexane} and {xNMP+ (1� x)cyclooctane}

sity

ables

ðT � TcÞ=K < 1 ðT � TcÞ=K < 11:2

B b B b

xNMP+ (1� x)cyclohexane0.071� 0.002 0.326� 0.003 0.069� 0.001 0.322� 0.001

1.72� 0.04 0.327� 0.003 1.71� 0.01 0.326� 0.001

1.67� 0.04 0.327� 0.003 1.67� 0.01 0.326� 0.001

xNMP+ (1� x)cyclooctane0.017� 0.002 0.320� 0.014 0.017� 0.001 0.316� 0.006

1.87� 0.19 0.331� 0.015 1.95� 0.07 0.337� 0.007

1.74� 0.18 0.328� 0.015 1.80� 0.06 0.333� 0.006

TABLE 6

Values of critical amplitudes B and B1 for coexistence curves of ðT ; nÞ, ðT ; xÞ, ðT ;/Þ in equation (9) for

{xNMP+ (1� x)cyclohexane} and {xNMP+ (1� x)cyclooctane}

Density variable B B1

xNMP+ (1� x)cyclohexanen 0.0699� 0.0002

0.0720� 0.0002 )0.016� 0.001

x 1.715� 0.002

1.731� 0.004 )0.13� 0.03

/ 1.668� 0.002

1.678� 0.004 )0.08� 0.03

xNMP+ (1� x)cyclooctanen 0.0178� 0.0001

0.0184� 0.0003 )0.007� 0.003

x 1.848� 0.015

1.797� 0.037 0.58� 0.38

/ 1.740� 0.013

1.712� 0.032 0.33� 0.34

1758 X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762

and with Z ¼ 2b was used to show the goodness of the order parameter [10]. The

smaller the value of DS, the better is the order parameter. For the systems of

(NMP+ cyclohexane) and (NMP+ cyclooctane), the values of DS are almost the

same for / and x, which indicate that parameters / and x are almost equally good.

This is consistent with what the symmetry of the coexistence curves shows: the

difference in the symmetry between the (T ; x) curve and (T ;/) curve is too small to

be observed in figures 1 and 2. The experimental values of nc listed in table 7 were

obtained by extrapolating refractive indices against temperatures in the one-phaseregion to the critical temperatures. The experimental values of xc and /c listed in

table 7 were determined by the technique of ‘‘equal volumes of two coexistence

phases’’ and calculated by equations (6) and (7). The uncertainties of optimal pa-

rameters reported in table 7 include no systematic uncertainties contributed by

converting n to x, and x to /. Such uncertainties in x and / were estimated to

be about �0.003 for {xNMP+ (1� x)cyclohexane} and �0.011 for {xNMP+

(1� x)cyclooctane}, respectively. Thus, the values of xc and /c obtained from

extrapolation of equation (10) are consistent with those from observations. It isevidence that no significant critical anomaly is present in refractive indices and

that they were properly converted to mole fractions and volume fractions in our

treatments.

Combination of equations (9) and (10) yields:

q1 ¼ qc þ Dsz � ð1=2ÞBsb � ð1=2ÞB1sbþD; ð11Þ

q2 ¼ qc þ Dsz þ ð1=2ÞBsb þ ð1=2ÞB1sbþD: ð12Þ

Fixing Z, b, D and Tc, at temperatures (0.89, 0.3265, 0.5, and 289.153) K for{xNMP+ (1� x)cyclohexane} or 291.503 K for {xNMP+ (1� x)cyclooctane}, re-

TABLE 7

Parameters of equation (10) and standard deviations S in qd for diameters of coexistence curves of ðT ; nÞ,ðT ; xÞ and (T ;/) for {xNMP+ (1� x)cyclohexane} and {xNMP+ (1� x)cyclooctane}

(T ; n) (T ; x) (T ;/)

xNMP+ (1� x)cyclohexaneqc;expt 1.4414� 0.0001 0.384� 0.001 0.357� 0.001

Z ¼ 0:89

qc 1.4413� 0.0001 0.386� 0.001 0.359� 0.001

D 0.123� 0.001 0.22� 0.01 0.40� 0.01

S 7.2� 10�5 1.2� 10�3 1.1� 10�3

Z ¼ 0:653

qc 1.4409� 0.0001 0.386� 0.001 0.358� 0.001

D 0.055� 0.001 0.10� 0.01 0.18� 0.01

S 3.0� 10�4 1.5� 10�3 1.4� 10�3

xNMP+ (1� x)cyclooctaneqc;expt 1.4601� 0.0001 0.455� 0.002 0.374� 0.002

Z ¼ 0:89

qc 1.4602� 0.0001 0.463� 0.002 0.384� 0.001

D 0.093� 0.001 )1.28� 0.15 )0.54� 0.12

S 4.8� 10�5 4.8� 10�3 4.0� 10�3

Z ¼ 0:653

qc 1.4600� 0.0001 0.466� 0.002 0.385� 0.001

D 0.036� 0.002 )0.51� 0.05 )0.22� 0.05

S 1.3� 10�4 4.2� 10�3 3.8� 10�3

X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762 1759

spectively; and taking the values of D, qc, B, and B1 from tables 6 and 7. qd, q1 and q2

were calculated from equations (10) to (12). The results are shown as lines in figures 1

and 2. The values from calculation are in good agreement with experimental results.

The critical parameters obtained from experiments including that reported in the

previous work [1] are summarized in the columns 2 to 5 of table 8.

According to Scatchard–Hildebrand theory, the free energy of mixing per unit

volume may be expressed as:

F ¼ RT fð/=VAÞ ln/þ ½ð1� /Þ=VB� lnð1� /Þ þ /ð1� /Þvg; ð13Þ

where v is an interaction parameter per unit volume. If equation (13) is applicable

and v is a constant to a series cycloalkanes, then the power-law dependence of critical

amplitude on molar volume for coexistence curve, or correlation length, or suscep-

tibility can be derived from a Landau–Ginsburg–Wilson type model for the binary

cycloalkane solutions [1,2]. Thus, the critical amplitude B/ relating to the coexistence

curve with / being the density variable may be expressed as:

B//�kc / V �b

B ; ð14Þ

where k and b are universal exponents. By using equation (1), the values of k and bwere calculated to be 1.865 and 0.29, respectively [2].

TABLE 8

Critical temperatures Tc, critical mole fractions xc, critical volume fractions /c, molar volume VB at Tc, in-teraction parameters per unit volume v, and amplitudes B/ for {xNMP+ (1� x)cycloalkanes}

Cycloalkanes Tc/K /c/K 103 � VB B/ v/dm�3

(dm3 �mol�1)

Methylcyclohexane 290.2 0.367 127.2 1.636 0.0246

Cyclohexane 289.2 0.357 107.6 1.668 0.0227

Cyclooctane 291.5 0.374 133.9 1.740 0.0206

1760 X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762

Our experimental results show that equation (1) is valid but equation (14) fails to

correlate the coexistence data for the cycloalkane solutions we studied. It could becaused by the fact that the interaction parameter per unit volume v varies for differ-

ent cycloalkanes, although we have selected a series of cycloalkanes with the values

of v as close as possible. Indeed, without the assumption of v being constant, equa-

tion (14) would not be obtained, but it does not affect the derivation of equation (1)

with exponent r being 0.5 from classical theory.

We estimated the values of v from solubility parameters of NMP and cycloalk-

anes by the relation proposed by Karger et al. [11]

TABL

Solubi

T ¼ 29

NM

Met

Cyc

Cyc

v ¼n�

ddA � ddB

�2

þ ðdoAÞ2 þ

�2dinAd

dA � 2dinAd

dB

�o.RT ; ð15Þ

where dd, do, and din are the solubility parameters for dispersion, orientation, and

induction, respectively. The subscripts A and B represent NMP and cycloalkane.Equation (15) assumes the polar contribution of cycloalkane molecules to v is

negligible. Table 9 lists the values of solubility parameters of NMP, methylcyclo-

hexane, cyclohexane, and cyclooctane. The solubility parameters of NMP listed in

table 9 were taken from reference [12] and those of cycloalkanes were calculated

from the heat of vaporization DHV by:

ddB ¼ fðDHV � RT Þ=VBg1=2; ð16Þ

where the values of DHV and VB were taken from references [13] and [6], respectively.

The values of v for three solutions of (NMP+methylcyclohexane), (NMP+ cyclo-

hexane), and (NMP+ cyclooctane) were then calculated by using equation (15) and

are listed in column 6 of table 8.

E 9

lity parameters dd, do, and din for NMP, methylcyclohexane, cyclohexane and cyclooctane at

8:15 K

dd do din

(J1=2 � cm�3=2) (J1=2 � cm�3=2) (J1=2 � cm�3=2)

P 17.43 7.28 2.13

hylcyclohexane 16.01

lohexane 16.77

looctane 17.93

FIGURE 3. A plot of ð/2 � /1Þ/�1:865c v0:32V 0:29

B against s0:3265 for (NMP+cycloalkane); d, methylcyclo-

hexane; �, cyclohexane; N, cyclooctane.

X. An et al. / J. Chem. Thermodynamics 35 (2003) 1751–1762 1761

By careful analysis of the experimental datas we found that the amplitude B/ is

proportional to v�0:32. Thus, the volume-fraction differences of two coexisting phases

should be representable by a scaling form:

/2 � /1 / /1:865c v�0:32V �0:29

B s0:3265; ð17Þ

and a plot of ð/2 � /1Þð/�1:865c v0:32V 0:29

B Þ against s0:3265 should yield a straight line

passing through the origin. Figure 3 shows such a plot and indeed all the experi-

mental points for three systems are on the universal line. It would be very interesting

to explore the scaling relation of v by further experimental and theoretical studies.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(Project 29873020, 20173024, 20273032), by the foundation for University Key

Teachers of China and by the Natural Science Foundation of Gansu Province.

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JCT 03-073