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The coexistence curves of {xN-methyl-2-pyrrolidone + (1−x)cyclohexane} and...
Transcript of The coexistence curves of {xN-methyl-2-pyrrolidone + (1−x)cyclohexane} and...
J. Chem. Thermodynamics 35 (2003) 1751–1762
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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: [email protected] (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, whichwere 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 valuesð1� 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