The liquid–liquid coexistence curves of {x dimethyl adipate + (1 − x) n-octane} and {x dimethyl adipate + (1 − x) n-nonane} in the critical region

J. Chem. Thermodynamics 51 (2012) 132–138

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The liquid–liquid coexistence curves of {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) n-nonane} in the critical region Zhiyun Chen a, Li Cai a, Meijun Huang a, Tianxiang Yin a, Xueqin An a, Weiguo Shen a,b,⇑ a b

School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China Department of Chemistry, Lanzhou University, Lanzhou 730000, China

a r t i c l e

i n f o

Article history: Received 20 January 2012 Received in revised form 18 February 2012 Accepted 29 February 2012 Available online 10 March 2012 Keywords: Critical behaviour Coexistence curve Dimethyl adipate N-alkane

a b s t r a c t The liquid–liquid coexistence curves of (dimethyl adipate + n-octane) and (dimethyl adipate + n-nonane) have been determined within about 10 K from the critical temperatures, from which the critical amplitudes and the critical exponents are deduced. The critical exponents corresponding to the coexistence curve b are consistent with the 3D-Ising values. The experimental results have been analyzed to determine Wegner-correction terms and to discuss the asymmetric behaviour of the diameters of the coexistence curves by the complete scaling theory. Molar mass-dependences of the critical amplitude and the critical volume fraction have been shown to be consistent with the theoretical prediction. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction

dence of amplitudes for chain molecule solutions was given by An et al. [13]:

Recently, Anisimov and co-workers [1–3] extended the complete scaling to binary mixtures for both incompressible and weakly compressible liquid mixtures. Although the tests of the complete scaling have been carried out by analyzing the limited experimental data of the coexistence curves for binary liquid mixtures [1–6], more accurate coexistence-curve data are required. It is commonly accepted that critical behaviour of fluids and fluid mixtures belong to the universality class of the three-dimensional Ising model [7]. In the region sufficiently close to the critical point, the differences of the general density variable of two coexisting phases Dq may be described by a universal power-law of the reduced temperature s near the critical point (s = |T  Tc|/Tc, Tc is the critical temperature):

ð1  /c Þ=/c / M r ;

ð2Þ

B/ /c1:865 / Mb ;

ð3Þ

Dq ¼ jq2  q1 j ¼ Bsb ;

ð1Þ

where b is the corresponding critical exponent; the subscripts 1 and 2 denote each of the two coexisting phases; B is the critical amplitude. The current theoretical value of b is 0.326 from the Renormalization Group calculations [8]. The dependences of the critical behaviour of binary solutions with one component being the chain-molecule on the number N of monomer units or the molar mass M of the chain molecule have been paid much attention [9–16]. A formulation of the M-depen⇑ Corresponding author at: School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China. Tel.: +86 931 8912541/+86 21 64250804; fax: +86 21 64250804. E-mail address: [email protected] (W. Shen). 0021-9614/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jct.2012.02.038

where /c is the critical volume fraction of the non-chain-molecule component; B/ is the critical amplitude relating to the volume fraction /; r and b are the universal exponents with r = 0.41 and b = 0.29, respectively for the chain-molecule solutions of both small molecules and polymers. As a part of our continuous studies on the critical behaviour of the binary mixtures of (dimethyl adipate + alkanes) [6], in this paper, we report the coexistence curves of {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) n-nontane} (with x being the mole fraction of dimethyl adipate) determined by measurements of the refractive index n. The experimental results are analyzed to determine the critical exponent b and the critical amplitude B, to examine the Wegner correction terms and the diameters of the coexistence curves, and furthermore to discuss the asymmetry of the coexistence curve through the complete scaling theory proposed by Anisimov and co-workers [1–3]. The dependences of the critical amplitudes on the molar mass of nalkane for a series of system {x dimethyl adipate + (1  x) nalkanes} are also discussed. 2. Experimental Table 1 lists the purities and suppliers of the chemicals used in this work.

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Z. Chen et al. / J. Chem. Thermodynamics 51 (2012) 132–138 TABLE 1 Purities and suppliers of chemicals. Chemical

Supplier

Purity, mass fraction

Dried and stored method

Dimethyl adipate n-Octane n-Nonane

Alfa Aesar Alfa Aesar Alfa Aesar

0.99 0.99 0.99

0.4 nm molecular sieves 0.4 nm molecular sieves 0.4 nm molecular sieves

TABLE 3 Coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1  x) noctane}. Refractive indexes were measured at wavelength k = 632.8 nm, pressure p = 0.1 MPa and Tc = 299.609 K. Mole fraction and volume fraction are denoted by x and /. Subscripts 1 and 2 relate to upper and lower phases, respectively.a

A series of binary solutions of {x dimethyl adipate + (1  x) noctane} and {x dimethyl adipate + (1  x) n-nonane} was prepared in glass tubes provided with Ace-thread connections, and the tubes were placed into the water bath. The critical mole fraction xc was determined with an accuracy of ±0.001 by the technique of ‘‘equal volume’’ [17]. A binary solution with the critical composition was prepared in a rectangular fluorimeter cell provided also with an Ace-thread connection and placed in the water bath for measurements of the refractive indexes of two coexisting phases. The phase separation temperature was carefully determined and taken as the critical temperature. The refractive indexes were measured according to the ‘‘minimum deviation angle’’ technique described previously [18]. During measurements, the temperature in water bath was constant within ±0.002 K. The accuracy of measurement was ±0.02 K for temperature T, ±0.003 K for the temperature difference (T  Tc), and ±0.0001 for the refractive-index in each coexisting phase.

3. Results and discussion The critical mole fractions and critical temperatures for {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) n-nonane} were determined and are shown in table 2, where x and xc are the mole fraction of dimethyl adipate and its critical value, respectively. The refractive indexes n were measured for each coexisting phase at various temperatures and the wavelength k = 632.8 nm. The results are listed in columns 2 and 3 of table 3 for {x dimethyl adipate + (1  x) n-octane} and table 4 for {x dimethyl adipate + (1  x) n-nonane}, respectively. They are also shown in figures 1a and 2a, as the plots of temperature against refractive index and denoted as the (T, n) coexistence curve. To convert the refractive index n to the mole fraction x to obtain the coexistence curves of temperature against mole fraction (T, x), the refractive indexes n of pure dimethyl adipate, n-octane and nnonane at various temperatures were measured and are listed in table 5. It has been shown that the refractive index of the binary mixtures of {x polar solvent + (1  x) n-alkane} may be expressed as a linear empirical function [19] of temperature in a certain temperature range by using the equations

RðxÞ ¼ x  R1 þ ð1  xÞ  R2 ;

ð4Þ

0

0

nðT; xÞ ¼ nðT ; xÞ þ RðxÞ  ðT  T Þ;

ð5Þ

T/K

n1

n2

x1

x2

/1

/2

299.597 299.593 299.590 299.585 299.578 299.569 299.558 299.537 299.518 299.499 299.481 299.461 299.431 299.405 299.364 299.323 299.270 299.173 299.018 298.843 298.697 298.187 297.778 297.166 296.547 295.759 294.686 293.697 292.707 291.459 290.325

1.4045 1.4044 1.4044 1.4043 1.4041 1.4040 1.4038 1.4037 1.4035 1.4035 1.4034 1.4032 1.4031 1.4030 1.4029 1.4028 1.4026 1.4024 1.4021 1.4019 1.4014 1.4014 1.4012 1.4010 1.4009 1.4008 1.4008 1.4009 1.4011 1.4013 1.4015

1.4066 1.4067 1.4068 1.4069 1.4070 1.4071 1.4073 1.4075 1.4077 1.4078 1.4079 1.4080 1.4082 1.4083 1.4085 1.4086 1.4089 1.4092 1.4096 1.4101 1.4107 1.4113 1.4119 1.4129 1.4135 1.4145 1.4155 1.4164 1.4172 1.4183 1.4191

0.393 0.390 0.390 0.387 0.380 0.377 0.371 0.367 0.361 0.361 0.357 0.351 0.347 0.344 0.340 0.336 0.329 0.322 0.310 0.301 0.280 0.275 0.262 0.247 0.234 0.218 0.201 0.189 0.180 0.166 0.155

0.457 0.460 0.463 0.466 0.469 0.472 0.478 0.484 0.490 0.492 0.495 0.498 0.504 0.506 0.512 0.514 0.523 0.530 0.540 0.553 0.567 0.581 0.593 0.616 0.626 0.646 0.662 0.676 0.688 0.705 0.715

0.395 0.392 0.392 0.389 0.383 0.379 0.373 0.370 0.363 0.363 0.359 0.353 0.349 0.346 0.342 0.338 0.331 0.323 0.312 0.303 0.281 0.277 0.264 0.248 0.235 0.220 0.203 0.190 0.181 0.168 0.156

0.460 0.463 0.466 0.469 0.471 0.474 0.480 0.486 0.492 0.495 0.497 0.500 0.506 0.509 0.514 0.516 0.525 0.533 0.543 0.555 0.569 0.583 0.596 0.618 0.628 0.648 0.664 0.678 0.690 0.707 0.717

a

Standard uncertainties u are u(p) = 10 kPa, u(T) = 0.02 K, u(T  Tc) = 0.002 K, u(n) = 0.0001, u(x) = 0.003, u(/) = 0.003.

where R(x) is the derivative of n with respect to T for a particular composition x; and R1 and R2 are the values of R(x) for x = 1 and x = 0, respectively. The values of n for binary mixtures {x dimethyl adipate + (1  x) n-octane} at T = 299.79 K and {x dimethyl adipate + (1  x) n-nonane} at T = 305.07 K with various compositions were measured and are listed in table 6. Fitting equation (4) to the values listed in table 5 gives R1 = 4.04  104 K1 for dimethyl adipate, R2 = 4.75  104 K1 for n-octane, and R2 = 4.83  104 K1 for n-nonane. Then R(x) and n(T0, x) for {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) n-nonane} at various compositions and at their middle temperatures T0 = 294.961 and 300.420 K of the coexistence curves were calculated from the data listed in table 6 by using equations (4) and (5); finally n(T0, x) were fitted into the polynomial form to obtain

nðT 0 ¼ 294:961 K;xÞ ¼ 1:3955 þ 0:0216x þ 0:0249x2  0:0244x3 þ 0:0092x4 ;

ð6Þ

for {x dimethyl adipate + (1  x) n-octane} and

nðT 0 ¼ 300:420 K;xÞ ¼ 1:4009 þ 0:0104x þ 0:0295x2  TABLE 2 Critical temperatures Tc, critical mole fractions xc, critical amplitudes B/ and critical volume fractions /c for {x dimethyl adipate + (1  x) n-alkanes}.

a

n-Alkane

Tc/K

xc

B/

/c

n-Hexanea n-Heptanea n-Octane n-Nonane

289.3 ± 0.1 294.2 ± 0.1 299.6 ± 0.1 305.0 ± 0.1

0.353 ± 0.002 0.393 ± 0.002 0.429 ± 0.001 0.466 ± 0.001

1.693 ± 0.003 1.704 ± 0.002 1.753 ± 0.003 1.779 ± 0.003

0.406 ± 0.002 0.420 ± 0.002 0.431 ± 0.001 0.444 ± 0.001

Reference [6].

0:0277x3 þ 0:0114x4

ð7Þ

for {x dimethyl adipate + (1  x) n-nonane} with a standard deviation of less than ±0.0001. The refractive indices measured for the coexisting phases at various temperatures were converted to the mole fractions by simultaneously solving equations (4)–(6) for {x dimethyl adipate + (1  x) n-octane} or equations (4), (5), and (7) for {x dimethyl adipate + (1  x) n-nonane} by the Newton iteration method.

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Z. Chen et al. / J. Chem. Thermodynamics 51 (2012) 132–138

TABLE 4 Coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1  x) nnonane}. Refractive indexes were measured at wavelength k = 632.8 nm, pressure p = 0.1 MPa and Tc = 305.029 K. Mole fraction and volume fraction are denoted by x and /. Subscripts 1 and 2 relate to upper and lower phases, respectively.a T/ K

n1

n2

x1

x2

/1

/2

305.026 305.024 305.021 305.018 305.014 305.009 305.007 304.996 304.985 304.968 304.949 304.926 304.905 304.888 304.858 304.825 304.785 304.726 304.662 304.581 304.442 304.275 304.114 303.906 303.569 303.176 302.567 302.04 301.227 300.196 299.256 297.986 296.791 295.815

1.4073 1.4071 1.4070 1.4069 1.4068 1.4067 1.4067 1.4066 1.4065 1.4063 1.4062 1.4061 1.4060 1.4060 1.4059 1.4058 1.4057 1.4055 1.4054 1.4053 1.4051 1.4050 1.4049 1.4048 1.4046 1.4045 1.4045 1.4045 1.4045 1.4047 1.4048 1.4050 1.4054 1.4057

1.4082 1.4083 1.4085 1.4086 1.4087 1.4087 1.4087 1.4089 1.4090 1.4092 1.4093 1.4094 1.4095 1.4096 1.4098 1.4099 1.4100 1.4102 1.4104 1.4106 1.4109 1.4112 1.4115 1.4118 1.4124 1.4129 1.4135 1.4141 1.4148 1.4158 1.4165 1.4175 1.4184 1.4191

0.445 0.437 0.433 0.428 0.424 0.420 0.420 0.416 0.411 0.403 0.398 0.394 0.389 0.389 0.384 0.379 0.374 0.364 0.359 0.353 0.341 0.334 0.326 0.317 0.302 0.289 0.276 0.264 0.246 0.233 0.216 0.195 0.187 0.178

0.481 0.485 0.493 0.497 0.501 0.501 0.501 0.509 0.512 0.520 0.524 0.527 0.531 0.534 0.542 0.545 0.548 0.555 0.562 0.568 0.577 0.586 0.595 0.603 0.621 0.633 0.646 0.660 0.673 0.694 0.706 0.723 0.738 0.748

0.424 0.416 0.412 0.408 0.404 0.400 0.400 0.395 0.391 0.382 0.378 0.373 0.369 0.369 0.364 0.359 0.354 0.345 0.339 0.334 0.323 0.315 0.308 0.299 0.284 0.272 0.259 0.248 0.231 0.218 0.202 0.182 0.174 0.166

0.460 0.464 0.472 0.476 0.480 0.480 0.480 0.487 0.491 0.499 0.502 0.506 0.510 0.513 0.521 0.524 0.527 0.534 0.541 0.547 0.557 0.565 0.574 0.583 0.600 0.613 0.626 0.641 0.654 0.676 0.688 0.706 0.721 0.732

1=/ ¼ ð1  KÞ þ K=x;

ð8Þ

K ¼ d1 M 2 =d2 M1 ;

ð9Þ

where d is mass density, M is the molar mass and subscripts 1 and 2 relate to dimethyl adipate and n-alkane, respectively. The values of d1 and d2 were obtained from Refs. [6] and [20], respectively. The values of x and / of coexisting phases at various temperatures are listed in columns 4–7 of tables 3 and 4, and are shown in figures 1b, c, 2b and c, respectively. In the region sufficiently close to the critical point, the coexistence curve can be represented by equation (1). The differences of general density variables, (q2  q1), for n, x, and / of coexisting phases obtained in this work were fitted to equation (1), with all points equally weighted to obtain b and B. The results are listed in table 7. The values of b are in very good agreement with that of other (polar liquid + n-alkane) systems we reported previously, and the theoretical prediction of 0.326. With the critical exponents b and D to be fixed at the theoretical values (b = 0.326, D = 0.5), a least-squares program was used to fit the Wegner equation to obtain the parameters B and B1 [21].

jq2  q1 j ¼ B  sb þ B1  sbþD þ   

a

Standard uncertainties u are u(p) = 10 kPa, u(T) = 0.02 K, u(T  Tc) = 0.002 K, u(n) = 0.0001, u(x) = 0.004, u(/) = 0.004.

ð10Þ

The results are summarized in table 8, from which it may be seen that the contributions of B1  sb+D term for the variables x and / are negligible. It was known that the diameter qd of a coexistence curve at least is the sum of the three terms proportional to s1a, s2b and s, where a characterizes the divergence of the heat capacity at constant volume for pure fluids as the critical point is approached. These terms were thought as a direct consequence of complete scaling [1], while the presence of the term s2b was attributed to a wrong choice for the order parameters in the past [22,23]. It is almost impossible to simultaneously obtain the coefficients of the above three terms by fitting the experimental data of the coexistence curves to a corresponding equation; however the diameter qd may be fitted to the form:

qd ¼ qc þ D  sz To convert the (T, x) coexistence curve to the coexistence curve of temperature against volume fraction (T, /), the volume fraction /0 of dimethyl adipate was calculated from the mole fraction by

with an apparent exponent Z being fixed at the values 1, 1  a = 0.89 or 2b = 0.652 in separate fitting procedures. The results are compared in table 9.

a

300

ð11Þ

c

b

298

T/K

296

294

292

290 1.40

1.41

n

1.42 0.2

0.3

0.4 0.5

x

0.6

0.7

0.2

0.3

0.4 0.5

0.6

0.7

φ

FIGURE 1. Coexistence curves of (a) temperature against refractive index (T, n); (b) temperature against mole fraction (T, x); (c) temperature against volume fraction (T, /) for{x dimethyl adipate + (1  x) n-octane}. N, experimental values of diameter (qd) of the coexisting curves; d, experimental values of density variables (q) of the coexisting phases; —, density variables q (calc.) and diameters qd (calc.) of the coexisting phases calculated from a combination of equations (12) and (13) with coefficients listed in tables 8 and 9.

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Z. Chen et al. / J. Chem. Thermodynamics 51 (2012) 132–138

306

a

c

b

304

T/K

302

300

298

296 1.41

1.42 0.2

0.3

n

0.4 0.5

0.6

0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

φ

x

FIGURE 2. Coexistence curves of (a) temperature against refractive index (T, n); (b) temperature against mole fraction (T, x); (c) temperature against volume fraction (T, /) for {x dimethyl adipate + (1  x) n-nonane}. N, experimental values of diameter (qd) of the coexisting curves; d, experimental values of density variables (q) of the coexisting phases; —, density variables q (calc.) and diameters qd (calc.) of the coexisting phases calculated from a combination of equations (12) and (13) with coefficients listed in tables 8 and 9.

TABLE 5 Refractive indexes n at wavelength k = 632.8 nm for pure dimethyl adipate, n-octane and n-nonane at pressure p = 0.1 MPa and various temperatures T.a T/K

a

n

T/K

n

T/K

n

adipate 298.35 299.40 302.58 303.19

1.4254 1.4250 1.4236 1.4234

T/K

n

303.88

1.4231

293.15 294.14 299.40 300.18

1.4275 1.4271 1.4249 1.4246

295.19 296.25 301.00 301.84

Dimethyl 1.4267 1.4263 1.4243 1.4239

293.23 294.23

1.3964 1.3959

295.29 296.27

n-Octane 1.3954 297.41 1.3949 298.48

1.3944 1.3939

299.50

1.3934

299.26 300.07

1.4015 1.4011

300.85 301.85

n-Nonane 1.4007 302.58 1.4003 303.30

1.3999 1.3995

303.83

1.3993

Standard uncertainties u are u(n) = 0.0001, u(T) = 0.02 K, and u(p) = 10 kPa.

TABLE 6 Refractive indexes n at wavelength k = 632.8 nm and pressure p = 0.1 MPa for {x dimethyl adipate + (1  x) n-octane} (at 299.79 K) and {x dimethyl adipate + (1  x) nnonane} (at 305.07 K).a x

n

0.099 0.200 0.302

x Dimethyl adipate + (1  x) n-octane 1.3955 0.400 1.4047 0.701 1.3985 0.500 1.408 0.800 1.4014 0.602 1.4113 0.899

x

n

x

n 1.4146 1.4180 1.4214

0.101 0.200 0.295

x Dimethyl adipate + (1  x) n-nonane 1.4000 0.390 1.4060 0.713 1.4017 0.492 1.4085 0.790 1.4040 0.594 1.4110 0.902

1.4143 1.4163 1.4197

a

Standard uncertainties u are u(n) = 0.0001, u(x) = 0.001, u(T) = 0.02 K and u(p) = 10 kPa.

In table 9, the experimental values of nc were obtained by extrapolating refractive indexes against temperatures in the onephase region to the critical temperature. The experimental values of xc and /c were determined by the ‘‘equal volume’’ technique and calculated by using equations (8) and (9), respectively. The uncertainties of optimal parameters reported in table 9 are the

standard errors of fits, and include no systematic uncertainties resulting from converting n to x, or x to /. Such uncertainties in x and / were estimated to be about ±0.003 and ±0.004 for (dimethyl adipate + n-octane) and (dimethyl adipate n-nonane), respectively. Thus, the values of xc and /c obtained from extrapolation of equation (11) are consistent with those from the observations. The goodness of fit to equation (11) may be indicated by the reduced chi-squared value v2/N (N is the number of degree of freedom of the fitting) [24], which are also listed in table 9. The smaller the value of v2/N, the more significant the contribution of the term is. It shows almost the same significance for x and / for (dimethyl adipate + n-octane), while the fit with Z = 1  a for / is better than that for x for (dimethyl adipate n-nonane). It implies that the term s1a is possibly dominant for variable / for the latter mixture. The ratio |D/B| is a measure of the symmetry of the coexistence curve; the smaller the value of the ratio, the better the symmetry is [25]. The values of these ratios calculated from the values of D and B listed in tables 9 and 7, respectively, show that both the (T, x) and (T, /) coexistence curves should be significantly more symmetric than that of (T, n), which is consistent with what is observed in figures 1 and 2. Combination of equations (10) and (11) yields:

q1 ¼ qc þ Dsz  ð1=2ÞBsb  ð1=2ÞB1 sbþD ;

ð12Þ

q2 ¼ qc þ Dsz þ ð1=2ÞBsb þ ð1=2ÞB1 sbþD :

ð13Þ

Fixing Z, b and D at 0.89, 0.326 and 0.5, respectively for two mixture systems; and taking the values of D, qc, B and B1 from tables 8 and 9, the qd, q1 and q2 were calculated from equations (12) and (13). The values are shown as lines in figures 1 and 2. The values from calculation are in good agreement with the experimental results. According to the theory proposed by Perez-Sanchez et al. [3], the width of the phase boundary Dx2,cxc and the deviation function of the diameter Dx2,d of a coexistence curve can be expressed by:

b x2 jsjb ð1 þ B b x2 jsjD þ B b x2 jsj2b Þ; Dx2;cxc  jxþ2  x2 j=2  B 0 1 2

ð14Þ

b x2 jsj2b þ Dx2;d  ðxþ2 þ x2 Þ=2  x2;c  D 2 b  jsj1a =ð1  aÞ þ B b cr jsjÞ; b x2 ð A D

ð15Þ

1

0

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Z. Chen et al. / J. Chem. Thermodynamics 51 (2012) 132–138

TABLE 7 Values of critical amplitude B and critical exponent b for coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) nnonane} in equation (1). Order parameter

Tc  T < 1 K

Tc  T < 10 K

B

b

n x /

0.058 ± 0.001 1.79 ± 0.04 1.79 ± 0.04

x Dimethyl adipate + (1  x) n-octane 0.327 ± 0.002 0.328 ± 0.003 0.328 ± 0.003

n x /

0.045 ± 0.002 1.83 ± 0.08 1.82 ± 0.08

x Dimethyl adipate + (1  x) n-nonane 0.326 ± 0.006 0.327 ± 0.006 0.327 ± 0.006

B

b

0.055 ± 0.001 1.76 ± 0.01 1.76 ± 0.01

0.321 ± 0.001 0.326 ± 0.001 0.326 ± 0.001

0.042 ± 0.001 1.79 ± 0.03 1.77 ± 0.03

0.319 ± 0.003 0.324 ± 0.003 0.324 ± 0.003

TABLE 8 Parameters B and B1 of equation (6) for coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) n-nonane}. Parameter

x Dimethyl adipate + (1  x) n-octane B

n x

B

B1

0.0561 ± 0.0002 0.0580 ± 0.0002

0.018 ± 0.002

0.0432 ± 0.0002 0.0452 ± 0.0002

0.018 ± 0.001

1.752 ± 0.003 1.765 ± 0.006

0.12 ± 0.05

1.794 ± 0.003 1.810 ± 0.007

0.15 ± 0.06

0.12 ± 0.05

1.779 ± 0.003 1.794 ± 0.007

0.14 ± 0.06

1.754 ± 0.003 1.767 ± 0.006

/

x Dimethyl adipate + (1  x) n-nonane B1

TABLE 9 Parameters of equation (11) and the reduced Chi-squared values v2/N in qd for diameters of coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1  x) n-octane} and {x dimethyl adipate + (1  x) n-nonane}. (T, n)

qc,expta

(T, x)

x Dimethyl adipate + (1  x) n-octane 1.4056 ± 0.0001 0.429 ± 0.001

(T, /) 0.431 ± 0.001

Z=1

qc

0.426 ± 0.001 0.36 ± 0.02 0.151

0.428 ± 0.001 0.34 ± 0.02 0.149

Z = 2b = 0.652 1.4053 ± 0.0001 0.425 ± 0.001 0.044 ± 0.001 0.10 ± 0.01 4.438 0.149

0.427 ± 0.001 0.10 ± 0.01 0.149

v /N

Z = 1  a = 0.89 1.4055 ± 0.0001 0.425 ± 0.001 0.104 ± 0.001 0.24 ± 0.01 0.284 0.142

0.428 ± 0.001 0.23 ± 0.01 0.142

qc,expta

x Dimethyl adipate + (1  x) n-nonane 1.4078 ± 0.0001 0.466 ± 0.001

0.444 ± 0.004

D

v2/N

qc D 2

v /N

qc D 2

1.4056 ± 0.0001 0.156 ± 0.001 0.191

Z=1

qc D

v2/N

qc D 2

v /N

qc D

v2/N

1.4077 ± 0.0001 0.155 ± 0.001 0.078

0.461 ± 0.001 0.01 ± 0.03 0.092

0.441 ± 0.001 0.25 ± 0.02 0.076

Z = 2b = 0.652 1.4075 ± 0.0001 0.461 ± 0.001 0.043 ± 0.001 0.00 ± 0.01 4.777 0.092

0.440 ± 0.001 0.07 ± 0.01 0.09

Z = 1  a = 0.89 1.4077 ± 0.0001 0.461 ± 0.001 0.10 ± 0.001 0.00 ± 0.02 0.343 0.092

0.441 ± 0.001 0.16 ± 0.02 0.078

a

qc,expt is the critical value of the order parameter determined by the techniques described the Section 3 of the text.

where x2 is the mole fraction of the component with larger molecular volume, i.e., the mole fraction of dimethyl adipate for (dimethyl adipate + n-octane) or mole fraction of the n-nonane for (dimethyl

adipate + n-nonane); the superscripts ‘‘+’’ and ‘‘’’ relate to upper b x2 , B b x2 , B b x2 , D b x2 and D b x2 are the system dependent and lower phases; B 0 1 2 2 1  b amplitudes; A 0 is a reduced variable of the critical amplitude correb cr ¼ Bcr V c =R is a sponding to the heat capacity in two-phase region; B reduced variable with Bcr being the critical fluctuation-induced contribution to the background of heat capacity and R being the gas constant. As discussed above the contribution of the |s|b term in equation (14) is dominant [4]. Pérez-Sánchez and co-workers [3] pointed that the net effect of the contributions proportional to |s|1a and |s| in equation (15), which have opposite sign, appears to be of minor b x2 in equation significance for Dx2,d. Thus we sat the parameter D 1 (15) to be zero and neglected the terms of |s|D and |s|2b in equation (14). Equations (14) and (15) may be expressed simply by

b x2 jsjb ; Dx2;cxc  jxþ2  x2 j=2  B 0

ð16Þ

b x2 jsj2b : x2;d  ðxþ2 þ x2 Þ=2  x2;c þ D 2

ð17Þ

The experimental data of (T, x) coexistence curves within Tc  T < 10 K were fitted by equations (16) and (17) with b being fixed at the theoretical value 0.326 to obtain the amplitudes x2,c, b x2 and D b x2 , which are listed in columns 2, 4 and 5 of table 10. figure B 0 2 3a and b show the plots of x2,d vs. |s|2b for (dimethyl adipate + n-octane) and (dimethyl adipate + n-nonane), respectively. The differences between the optimized values and the experimental ones for x2,c are as much as 0.004–0.005, which are within the estimated uncertainties including that from n converting to x, and x to /. The asymmetric coefficient a1 of the coexistence curve was related to b x2 and ð B b x2 Þ2 , and the complete scaling theory gives the ratio of D 2 0 [3]:

b x2 =ð B b x2 Þ2 ¼ a1 =ð1  a1 x2;c Þ: D 2 0

ð18Þ

The values of a1 were calculated by equation (18) and are listed in column 6 of table 10. In this column we also list the calculated results for {x2 dimethyl adipate + (1  x2) n-hexane} and {x2 dimethyl adipate + (1  x2) n-heptane} we reported previously [6]. With an assumption that the volume change upon mixing is neglected and the system is incompressible, it has been shown [3] that the asymmetric coefficient a1 of the coexistence curve is

137

Z. Chen et al. / J. Chem. Thermodynamics 51 (2012) 132–138

TABLE 10 b x2 and D b x2 , and the asymmetric coefficients a1 for binary solutions of Critical mole fractions x2,c, the molecular volume ratios V 02;c =V 01;c at critical temperature, parameters B 0 2 dimethyl adipate + n-alkanes.

a b c d e

n-Alkane

x2,c

V 02;c =V 02;c

b x2 b B 0

b x2 c D 2

a1d

a1e

n-Hexanea n-Heptanea n-Octane n-Nonane

0.353 0.393 0.425c 0.539c

1.258 1.120 1.008 1.091

0.805 ± 0.001 0.836 ± 0.001 0.877 ± 0.003 0.897 ± 0.002

0.209 ± 0.009 0.190 ± 0.005 0.104 ± 0.007 0.001 ± 0.007

0.36 ± 0.02 0.30 ± 0.01 0.14 ± 0.01 0.00 ± 0.01

0.26 0.12 0.01 0.09

Reference [6]. Calculated from equation (16). Calculated from equation (17). Calculated from equation (18). Calculated from equation (19).

to increase with decrease of the molecular volume ratio V 02;c =V 01;c and approaches to zero as the ratio is close to 1, which is consistent with the prediction of equation (19) and the perfect symmetries of the coexistence curves shown in figures 1b and 2b. However the differences between the values listed in columns 6 and 7 of table 10 are significant. One possible way to solve this deviation is to take b  jsj1a =ð1  aÞ þ B b x ðA b cr jsjÞ in equation the heat capacity term D 1 0 (15) into account [5], which requires the precise measurements of

0.45

a

x2,d

0.44

0.43

0.42

0.41

0.17

0.00

0.02

0.04

0.06

lτ l

0.08

0.10

0.12 0.16

2β

0.15

log {(1−φc)/φc}

0.555

b

0.550 0.545

x2,d

0.540

0.14 0.13 0.12

0.535

0.11

0.530

0.10

0.525

0.09 1.92

0.520

0.00

0.02

0.04

0.06

lτ l

0.08

0.10

1.94

1.96

1.98

2.00

2.02

2.04

2.06

2.08

2.10

2.12

-1

log {M/(g mol )}

0.12

2β

FIGURE 3. Plots of x2,d vs. |s|2b for (a) (dimethyl adipate + n-octane); (b) (dimethyl adipate + n-nonane).

FIGURE 4. A plot of logfð1  /c Þ=/c g vs. log{M/(g mol1)} for (dimethyl adipate + nalkane).

dependent on the molar volume ratio V 02;c =V 01;c of the solvent and the solute:

0.96

a1 ¼ 1  V 02;c =V 01;c ;

0.95

-1.865

log {(Bφ φc)

where V 01;c and V 02;c are the molar volumes of the components 1 and 2; the components 1 and 2 refer to that with the smaller molar volume and the larger molar volume at the critical temperature, respectively. The value of V 01;c and V 02;c were obtained by V 0i;c ¼ Mi =q0i;c , where Mi and q0i;c are the molar mass and the density of the pure component i (i = 1 or 2) at the critical temperature, respectively. The densities of dimethyl adipate were obtained from reference [6], and the densities n-octane and n-nonane were taken from the database of Cibulka [20]. The theoretical values of a1 were calculated by equation (19) are also listed in column 7 of table 10 for comparison. It can be seen from table 10 that the values of a1 calculated by b x2 and D b x2 obequation (18) with the values of the amplitudes B 0 2 tained from fitting equations (16) and (17) show a general tendency

}

ð19Þ

0.94 0.93 0.92 0.91 0.90 1.92

1.94

1.96

1.98

2.00

2.02

2.04

2.06

2.08

2.10

2.12

-1

log{M/(g mol )} FIGURE 5. A plot of logfðB/ /c Þ1:865 g vs. log{M/(g mol1)} for (dimethyl adipate + nalkane).

138

Z. Chen et al. / J. Chem. Thermodynamics 51 (2012) 132–138

the heat capacities of the corresponding binary solutions both in the critical and non-critical regions. Table 2 summarizes the experimental values of B/ and /c for a series of solutions (dimethyl adipate + n-alkanes) we determined. Log-log plots of equations (2) and (3) are shown in figures 4 and 5, respectively. Least-squares fits gave r = 0.39 ± 0.02 and b = 0.26 ± 0.04. They agree with the theoretical predictions and support the assumption that equations (2) and (3) are universal for chain molecule solutions of small molecules. Acknowledgements This work was supported by the National Natural Science Foundation of China (Projects 20973061, 21073063 and 21173080). References [1] C.A. Cerdeiriña, M.A. Anisimov, J.V. Sengers, Chem. Phys. Lett. 424 (2006) 414– 419. [2] J.T. Wang, C.A. Cerdeiriña, M.A. Anisimov, J.V. Sengers, Phys. Rev. E 77 (2008) 031127. [3] G. Pérez-Sánchez, P. Losada-Pérez, C.A. Cerdeiriña, J.V. Sengers, M.A. Anisimov, J. Chem. Phys. 132 (2010) 154502. [4] M.J. Huang, Z.Y. Chen, T.X. Yin, X.Q. An, W.G. Shen, J. Chem. Eng. Data 56 (2011) 2349–2355. [5] M.J. Huang, Y.T. Lei, T.X. Yin, Z.Y. Chen, X.Q. An, W.G. Shen, J. Phys. Chem. B 115 (2011) 13608–13616.

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JCT 12-20