The liquid–liquid coexistence curves of {x 1-octyl-3-methylimidazolium hexafluorophosphate + (1 − x) 1-butanol} and {x 1-octyl-3-methylimidazolium hexafluorophosphate + (1 − x) 2-butanol} in the critical region

Fluid Phase Equilibria 348 (2013) 52–59

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The liquid–liquid coexistence curves of {x 1-octyl-3-methylimidazolium hexafluorophosphate + (1 − x) 1-butanol} and {x 1-octyl-3-methylimidazolium hexafluorophosphate + (1 − x) 2-butanol} in the critical region Zhencun Cui a , Dongxing Cai a , Dashuang Fan a , Zhiyun Chen b , Weiguo Shen a,b,∗ a b

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

a r t i c l e

i n f o

Article history: Received 1 December 2012 Received in revised form 19 March 2013 Accepted 20 March 2013 Available online 1 April 2013 Keywords: Ionic solution Critical behavior Coexistence curve Refractive index Complete scaling theory

a b s t r a c t Coexistences curves were reported for the liquid–liquid phase transition of binary solutions of the room temperature ionic liquid (RTIL) 1-octyl-3-methylimidazolium hexafluorophosphate ([C8 mim][PF6 ]) in 1-butanol and 2-butanol. The critical amplitude B and the critical exponent ˇ corresponding to the coexistence curve were deduced and the values of ˇ were found to be consistent with the 3D-Ising value. The experimental results are also analyzed to examine the asymmetry of the diameters for the coexistence curves. These asymmetries were discussed by the complete scaling theory. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The room temperature ionic liquids (RTIL) have excellent properties, such as high stability, low volatility, high conductivity, large heat capacity, noninflammability, and good dissolving capacity, etc., which could be applied to many fields, among which the applications in chemical engineering as reaction media and in separation processes are more attractive [1,2]. Reactions have been proposed using the mixtures of RTIL/organic solvents as the media and taking advantage of liquid–liquid phase transitions, which enables elegant separation of products, catalyst and solvent by small changes of temperature or composition [3]. These applications require the knowledge of the liquid–liquid phase equilibriums of the RTILs solutions [4]. Liquid–liquid phase transitions in RTILs solutions have been known for quite a long time. Since Pitzer discovered the liquid–liquid phase transition of an ionic liquid in an organic solvent, the field became a prominent subject in statistical mechanics and related experiments. Due to the fundamental aspect of the Coulomb interactions in RTILs solutions, their critical behaviors

∗ Corresponding author at: Department of Chemistry, Lanzhou University, Lanzhou, Gansu 730000, China. Tel.: +86 21 64250804; fax: +86 21 64250804. E-mail addresses: [email protected], [email protected] (W. Shen). 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.03.020

of the liquid–liquid equilibriums near the critical points were expected to challenge the universality hypothesis, which postulates that the liquid–liquid phase transitions all belong to the Ising universality class independent of the molecular details of the system [5]. However computer simulations [6–8] and experimental studies [9–12] on phase separation near the critical point driven by the Coulombic force evidenced that it belongs to the Ising universality class. It was attributed to the shielding effect due to Debye-Hückel charge order in the ionic system, which makes longrange Coulomb interactions become effectively short-range. For fluid mixtures which belong to the Ising universality class, the differences of the values of the general density variable between the two coexisting phases  in the critical region may be expressed by





 = 2 − 1  = B ˇ

(1)

  where  is the reduced temperature ( = T − Tc  /Tc , Tc is the critical temperature);  is the general density variable and the subscript 1 or 2 indicates each of the two coexisting phases; ˇ is the critical exponent and B is the critical amplitude corresponding to the coexisting phases. The current theoretical value of ˇ is 0.327 based on the Renormalization Group calculations [13]. The asymmetric criticality of the liquid–liquid coexistence curve has been paid much attention recently. Fisher and co-workers

Z. Cui et al. / Fluid Phase Equilibria 348 (2013) 52–59

53

Table 1 Purities, suppliers and water contents of chemicals. Chemical name

Supplier

Purity, mass fraction

Dried and stored method

Water content

[C8 mim][PF6 ]

Chengjie Chemical

0.99

164 ppm

1-Butanol 2-Butanol

Alfa Aesar Alfa Aesar

0.999 0.999

Dried under an oil-pump vacuum at 330 K for 2 weeks and then stored in a desiccator over P2 O5 Stored in a desiccators over P2 O5 Stored in a desiccator over P2 O5

[14,15] proposed a general formulation of the complete scaling for one-component fluids, showing that the three scaling fields should be the linear mixture of all physical fields: the chemical potential u, the temperature T and the pressure P. Recently, Anisimov and co-workers [16–18] have extended the complete scaling to binary mixtures for incompressible or weakly compressible liquid mixtures. Although the tests of the complete scaling have been carried by analyzing the limited experimental data of the coexistence curves for binary liquid mixtures, more accurate coexistence-curve data for more complete fluids including RTILs solutions are highly required. In this paper, we report the liquid–liquid coexistence curves of (x 1-octyl-3-methylimidazolium hexafluorophosphate ([C8 mim][PF6 ]) + (1 − x) 1-butanol) and (x [C8 mim][PF6 ] + (1 − x) 2butanol). The experimental results are analyzed to determine the critical exponent ˇ and the critical amplitude B and to examine the asymmetric behavior of the diameters of the coexistence curves. These asymmetries are discussed according to the complete scaling theory [16–18]. 2. Experimental 2.1. Chemicals [C8 mim][PF6 ], 1-butanol and 2-butanol used in the experiments were purchased and purified. Table 1 lists the purities, suppliers and water contents of the chemicals used in this work. The mass fractions of water remaining in the dried samples were analyzed by the coulometric Karl Fischer titration. 2.2. Apparatus and procedure The critical composition was determined by adjusting the proportion of the two components to achieve “equal volume” of the two phases at the phase-separation point [19]. The samples with the critical concentrations were then prepared in the fluorescence cuvettes and placed into a water bath. The refractive indexes in the two coexisting phases of the samples were measured using “minimum deviation angle” technique which has been described previously [20]. During the measurements of the refractive indexes, the temperature was constant within ±0.002 K. The accuracy of the measurement was ±0.003 K for the temperature difference (T − Tc ), and ±0.0001 for the refractive index n at  = 632.8 nm in each coexisting phase. 3. Results and discussion The critical mole fractions of [C8 mim][PF6 ] and the critical temperatures were determined to be xc = (0.098 ± 0.001), Tc = (326.5 ± 0.1) K for (x [C8 mim][PF6 ] + (1 − x) 1-butanol) and xc = (0.084 ± 0.001), Tc = (324.5 ± 0.1) K for (x [C8 mim][PF6 ] + (1 − x) 2-butanol), respectively. It was observed that the critical temperature was influenced by the impurities introduced in the preparation of samples, nominally of the same composition, had different values of the critical temperatures, differing by as much as 0.2 K. However, the final results are not affected because only one sample was used

108 ppm 112 ppm

in the measurement of the whole coexistence curve and only the temperature difference (T − Tc ) was important in the data analysis to obtain the critical parameters. The refractive indexes n were measured for each coexisting phase at various temperatures. The results are listed in columns 2 and 3 for (x [C8 mim][PF6 ] + (1 − x) 1-butanol) in Table 2 and for (x [C8 mim][PF6 ] + (1 − x) 2-butanol) in Table 3. They are also plotted in Figs. 1a and 2a, as the plots of the temperature against the refractive index, denoted as the (T, n) coexistence curve. In order to convert the refractive index n to the mole fraction x to obtain the coexistence curves of the plot of temperature against mole fraction (T, x), a series of binary mixtures of (x [C8 mim][PF6 ] + (1 − x) 1-butanol) and (x [C8 mim][PF6 ] + (1 − x) 1-butanol) with known mole fractions were prepared, their refractive indexes were

Table 2 Coexistence curves of (T, n), (T, x), (T, ) for {x [C8 mim][PF6 ] + (1 − x) 1-butanol}. Refractive indexes were measured at wavelength  = 632.8 nm, pressure p = 85 kPa and Tc = 326.536 K. Mole fraction and volume fraction are denoted by x and . Subscripts 1 and 2 relate to upper and lower phases, respectivelya . T

n1

n2

x1

x2

1

2

326.526 326.517 326.511 326.495 326.480 326.463 326.441 326.427 326.413 326.385 326.352 326.326 326.276 326.249 326.221 326.166 326.135 326.063 325.987 325.946 325.824 325.749 325.658 325.635 325.406 325.252 325.048 324.798 324.489 324.112 323.677 323.156 322.540 321.830 321.018 320.113 319.111 318.006 316.811 315.525

1.3907 1.3906 1.3905 1.3903 1.3901 1.3900 1.3898 1.3897 1.3896 1.3895 1.3894 1.3893 1.3892 1.3891 1.3890 1.3889 1.3888 1.3887 1.3886 1.3886 1.3885 1.3884 1.3883 1.3883 1.3882 1.3881 1.3880 1.3879 1.3878 1.3878 1.3878 1.3878 1.3879 1.3880 1.3881 1.3883 1.3886 1.3889 1.3893 1.3897

1.3927 1.3930 1.3931 1.3934 1.3935 1.3936 1.3937 1.3939 1.3940 1.3943 1.3944 1.3945 1.3948 1.3949 1.3950 1.3951 1.3953 1.3955 1.3957 1.3958 1.3961 1.3963 1.3965 1.3965 1.3970 1.3972 1.3975 1.3978 1.3983 1.3988 1.3993 1.3998 1.4005 1.4012 1.4019 1.4025 1.4034 1.4042 1.4050 1.4059

0.082 0.081 0.079 0.076 0.073 0.072 0.069 0.067 0.066 0.064 0.063 0.061 0.060 0.058 0.057 0.055 0.054 0.052 0.050 0.050 0.048 0.047 0.045 0.045 0.042 0.040 0.038 0.035 0.033 0.031 0.029 0.026 0.024 0.022 0.019 0.017 0.016 0.014 0.013 0.011

0.113 0.117 0.119 0.124 0.126 0.127 0.129 0.132 0.134 0.139 0.140 0.142 0.147 0.149 0.150 0.152 0.155 0.158 0.162 0.163 0.168 0.172 0.175 0.175 0.183 0.186 0.191 0.195 0.204 0.212 0.219 0.227 0.238 0.249 0.260 0.267 0.282 0.294 0.305 0.319

0.209 0.207 0.203 0.196 0.189 0.187 0.180 0.175 0.173 0.168 0.166 0.161 0.159 0.154 0.152 0.147 0.145 0.140 0.135 0.135 0.130 0.127 0.122 0.123 0.115 0.110 0.105 0.097 0.092 0.087 0.081 0.073 0.068 0.063 0.054 0.049 0.046 0.040 0.038 0.032

0.274 0.282 0.286 0.295 0.299 0.301 0.305 0.311 0.314 0.323 0.325 0.329 0.338 0.342 0.343 0.347 0.352 0.357 0.364 0.366 0.374 0.381 0.386 0.386 0.399 0.404 0.412 0.418 0.432 0.444 0.454 0.465 0.481 0.496 0.510 0.520 0.538 0.553 0.566 0.582

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) = 0.002, u() = 0.002.

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Z. Cui et al. / Fluid Phase Equilibria 348 (2013) 52–59

Fig. 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 [C8 mim][PF6 ] + (1 − x) 1-butanol}., Experimental values of the diameter d of the coexisting phases; 䊉 experimental values of the density variables  of the coexisting phase; the lines represent the values of the density variables calc and the diameters d,calc of the coexisting phases calculated from Eqs. (16) and (17) with coefficients listed in Tables 7 and 8.

measured in one phase region at various temperatures, which are listed in Tables 4 and 5. With the assumption that no significant critical anomaly is present in the refractive index [20], the refractive index n of a mixture may be expressed as a linear function of temperature in a certain temperature range [21]: n(T, x) = n(T 0 , x) + R(x)(T − T 0 )

(2)

R(x) = (1 − x)R1 + xR2

(3)

where R(x) is the derivative of n with respect to T for a solution with particular composition x; R1 and R2 are the values of R(x) for x = 0 and x = 1, respectively. The refractive indexes of pure [C8 mim][PF6 ], 1-butanol and 2-butanol were measured and listed in Tables 4 and 5. Fitting Eq. (3) to the data for pure components gives R1 = −4.18 × 10−4 K−1 for 1-butanol, R2 = −2.62 × 10−4 K−1 for [C8 mim][PF6 ] of (x [C8 mim][PF6 ] + (1 − x) 1-butanol); and gives R1 = −4.85×10−4 K−1 for 2-butanol, R2 = −2.67 × 10−4 K−1 for [C8 mim][PF6 ] of (x [C8 mim][PF6 ] + (1 − x) 2-butanol). A series of binary mixtures with known mole fractions x were prepared, and

their refractive indexes in the single phase region at various temperatures were measured to find the dependence of n on x and T. The results are also listed in Tables 4 and 5, which were fitted by Eqs. (2) and (3) to obtain the values of n (T0 , x) with a standard deviation of 0.0001, where T0 is the middle temperature of the coexistence curves, T0 = 321.025 K for (x [C8 mim][PF6 ] + (1 − x) 1-butanol) and T0 = 319.571 K for (x [C8 mim][PF6 ] + (1 − x) 2-butanol), respectively. The value of n (T0 , x) were fitted to the polynomial forms to obtain n(T 0 = 321.025 K, x) = 1.3864 + 0.0922x − 0.1686x2 + 0.1967x3 − 0.1225x4 + 0.0309x5

(4)

n(T 0 = 319.571 K, x) = 1.3842 + 0.1071x − 0.0218x2 + 0.3027x3 − 0.2332x4 + 0.0736x5

(5)

with the standard deviations less than 0.0001. The coexistence curves (T, n) were then converted to mole fractions (T, x) by simultaneously solving Eqs. (2)–(5) using the Newton iteration method.

Fig. 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[C8 mim][PF6 ] + (1 − x) 2-butanol}., Experimental values of the diameter d of the coexisting phases;䊉 experimental values of the density variables  of the coexisting phase; the lines represent the values of the density variables calc and the diameters d,calc of the coexisting phases calculated from Eqs. (16) and (17) with coefficients listed in Tables 7 and 8.

Z. Cui et al. / Fluid Phase Equilibria 348 (2013) 52–59 Table 3 Coexistence curves of (T, n), (T, x), (T, ) for {x [C8 mim][PF6 ] + (1 − x) 2-butanol}. Refractive indexes were measured at wavelength  = 632.8 nm, pressure p = 85 kPa and Tc = 324.534 K. Mole fraction and volume fraction are denoted by x and . Subscripts 1 and 2 relate to upper and lower phases, respectivelya . T

n1

n2

x1

x2

1

2

324.530 324.528 324.524 324.518 324.511 324.502 324.492 324.484 324.473 324.464 324.455 324.447 324.433 324.419 324.391 324.377 324.353 324.325 324.294 324.258 324.215 324.165 324.105 324.032 323.943 323.844 323.719 323.579 323.394 323.177 322.917 322.608 322.244 321.793 321.223 320.524 319.670 318.643 317.430 316.006 314.613

1.3887 1.3886 1.3885 1.3884 1.3883 1.3882 1.3881 1.3880 1.3879 1.3878 1.3877 1.3876 1.3875 1.3875 1.3874 1.3873 1.3872 1.3871 1.3870 1.3869 1.3868 1.3867 1.3866 1.3865 1.3863 1.3862 1.3861 1.3860 1.3859 1.3858 1.3857 1.3856 1.3856 1.3856 1.3856 1.3858 1.3860 1.3862 1.3866 1.3870 1.3875

1.3902 1.3903 1.3905 1.3906 1.3908 1.3910 1.3912 1.3913 1.3914 1.3915 1.3916 1.3917 1.3918 1.3919 1.3920 1.3921 1.3922 1.3924 1.3925 1.3926 1.3928 1.3930 1.3932 1.3935 1.3937 1.3940 1.3942 1.3946 1.3949 1.3953 1.3957 1.3962 1.3968 1.3974 1.3979 1.3988 1.3996 1.4006 1.4017 1.4028 1.4039

0.074 0.073 0.071 0.070 0.069 0.067 0.066 0.065 0.064 0.062 0.061 0.060 0.059 0.059 0.057 0.056 0.055 0.054 0.052 0.051 0.050 0.048 0.047 0.045 0.043 0.041 0.039 0.037 0.035 0.033 0.031 0.028 0.026 0.024 0.021 0.020 0.018 0.015 0.013 0.010 0.009

0.093 0.094 0.097 0.098 0.101 0.104 0.107 0.108 0.109 0.111 0.112 0.113 0.115 0.116 0.117 0.119 0.120 0.123 0.124 0.125 0.128 0.131 0.134 0.138 0.140 0.145 0.147 0.152 0.156 0.161 0.166 0.173 0.181 0.188 0.193 0.205 0.214 0.226 0.239 0.251 0.263

0.190 0.188 0.185 0.182 0.179 0.176 0.173 0.170 0.167 0.164 0.161 0.159 0.156 0.155 0.152 0.149 0.146 0.143 0.140 0.137 0.133 0.130 0.126 0.123 0.116 0.112 0.107 0.103 0.098 0.092 0.086 0.079 0.074 0.068 0.061 0.057 0.051 0.043 0.038 0.030 0.025

0.233 0.235 0.241 0.244 0.249 0.255 0.260 0.263 0.266 0.269 0.271 0.274 0.277 0.280 0.282 0.285 0.287 0.293 0.295 0.298 0.303 0.308 0.313 0.321 0.325 0.333 0.337 0.347 0.353 0.362 0.371 0.382 0.395 0.407 0.415 0.433 0.446 0.464 0.482 0.498 0.514

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) = 0.002, u() = 0.002.

To convert the (T, x) coexistence curve to the curve of temperature against volume fraction (T, ), the values of the volume fraction  of two coexisting phases at various temperatures for the two corresponding systems were calculated from the mole fractions x through 1 K = (1 − K) + x 

K=

d2 M1 d1 M2

(6)

(7)

where d is mass density, M is the molar mass and subscripts 1 and 2 relate to butanol and [C8 mim][PF6 ], respectively. The densities of d1 and d2 were taken from references [22,23]. The values of x and  of the coexisting phases at various temperatures are listed in columns 4–7 in Tables 2 and 3, and shown in Figs. 1b, 1c, 2b and 2c, respectively. In the region sufficiently close to the critical point, the coexistence curve can be described by Eq. (1). The differences of the general density variables  for n, x, and  of the coexisting phases obtained in this work were fitted to Eq. (1), with all points equally weighted to obtain ˇ and B. The results are listed in Table 6. The

55

Table 4 Refractive indexes n at wavelength  = 632.8 nm for {x [C8 mim][PF6 ] + (1 − x) 1butanol} at pressure p = 85 kPa and at various compositions and temperatures Ta . T/K

n

T/K

n

T/K

n

x=0 316.977 319.448

1.3880 1.3871

321.915 324.339

1.3861 1.3850

326.405 328.396

1.3841 1.3833

x = 0.0999 328.871 328.489

1.3910 1.3912

327.850 327.313

1.3914 1.3916

327.099 326.766

1.3917 1.3918

x = 0.2002 324.873 325.795

1.3979 1.3977

326.956 328.187

1.3972 1.3968

329.453 330.773

1.3963 1.3959

x = 0.2999 320.038 321.548

1.4035 1.4030

322.909 324.417

1.4025 1.4021

326.004 327.636

1.4016 1.4010

x = 0.3999 324.796 325.963 327.162

1.4046 1.4043 1.4039

315.439 318.473

1.4078 1.4067

320.643 322.770

1.4061 1.4055

x = 0.4993 319.577 320.718

1.4086 1.4082

321.848 323.013

1.4079 1.4076

324.429 326.000

1.4072 1.4067

x = 0.5626 316.664 318.677

1.4107 1.4102

320.603 322.609

1.4096 1.4090

324.647 326.635

1.4084 1.4078

x = 0.6804 319.128 320.633

1.4118 1.4114

322.130 323.690

1.4109 1.4103

325.057 326.628

1.4099 1.4094

x = 0.7907 326.383 324.945

1.4113 1.4117

323.681 322.733

1.4120 1.4123

321.513 320.337

1.4126 1.4130

x = 0.8837 320.336 321.477

1.4140 1.4137

322.641 323.885

1.4134 1.4131

325.180 326.553

1.4127 1.4124

x=1 326.550 327.978 329.499

1.4137 1.4133 1.4128

325.208 323.913 322.667

1.4140 1.4143 1.4146

321.409 320.222

1.4150 1.4153

a

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

values of ˇ are in very good agreement with the theoretical prediction of 0.327. According to the formulism of the complete theory proposed by Perez-Sanchez and Coworkers [18], the width of the phase boundary Zcxc and the diameter Zd of a coexistence curve can be expressed by: Zcxc ≡

Zd ≡

  Z+ − Z− ≈ ±Bˆ 0Z ||ˇ 1 + Bˆ 1Z ||1 + Bˆ 2Z ||2ˇ 2

Z+ + Z− ˆ Z ||2ˇ + D ˆZ − Zc ≈ D 2 1 2



ˆ− A 0 1−˛

(8)

 ||1−˛ − Bˆ cr ||

(9)

where Z is any physical density, such as the mole fraction x, or the ˆ m ( ˆ m = m /mc = Vc /V , where m dimensionless molar density  is the molar density, i.e. the amount of the solute in a unit volume of a solution, mc is the critical value of m ), or the dimensionless partial molar density  ˆ m x (the product of  ˆ m and mole fraction x); the superscrips “+” and “−” relate to upper and lower phases; ˆ - is a ˆ Z and D ˆ Z are the system dependent amplitudes; A Bˆ 0Z , Bˆ 1Z , Bˆ 2Z , D 2 1 0 reduced variable of the critical amplitude corresponding to the heat capacity in the two-phase region, ˛ is the critical exponent characterizing the divergence of the heat capacity at constant volume for pure fluids as the critical point is approached; Bˆ cr = Bcr Vc /R is a reduced variable, and Vc , Bcr and R are the critical molar volume of the solution, the critical fluctuation-induced contribution to the background of the heat capacity and the gas constant, respectively.

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Z. Cui et al. / Fluid Phase Equilibria 348 (2013) 52–59

Table 5 Refractive indexes n at wavelength  = 632.8 nm for {x [C8 mim][PF6 ] + (1 − x) 2butanol} at pressure p = 85 kPa and at various compositions and temperatures Ta . T/K

n

T/K

n

T/K

n

x=0 314.990 317.549

1.3864 1.3851

320.094 322.670

1.3839 1.3827

325.207 327.735

1.3814 1.3802

x = 0.1001 325.926 324.429

1.3900 1.3906

325.097 326.857

1.3903 1.3896

328.587 329.973

1.3890 1.3884

x = 0.2003 322.103 324.069

1.3976 1.3969

325.148 326.189

1.3965 1.3961

327.417 328.518

1.3956 1.3952

1.4030 1.4024

322.562 324.315

1.4018 1.4012

326.109 328.130

1.4006 1.3999

x = 0.3667 319.043 320.989

1.4052 1.4046

322.598 324.242

1.4041 1.4036

326.150 328.143

1.4029 1.4021

x = 0.4898 318.449 320.126

1.4086 1.4081

321.839 323.983

1.4076 1.4070

325.531 327.714

1.4065 1.4058

x = 0.6004 319.353 321.091

1.4106 1.4101

322.811 324.208

1.4097 1.4093

326.011 327.728

1.4087 1.4081

x = 0.6987 317.628 319.326

1.4128 1.4123

320.930 323.141

1.4119 1.4113

324.689 326.702

1.4108 1.4103

x = 0.8011 318.109 319.684

1.4142 1.4138

321.533 323.153

1.4132 1.4127

324.817 326.694

1.4123 1.4118

x = 0.8993 318.106 319.389

1.4153 1.4149

321.250 322.883

1.4145 1.4140

324.352 326.190

1.4136 1.4131

x = 1.0000 315.850 317.547 319.265

1.4172 1.4167 1.4162

320.848 322.339 323.996

1.4158 1.4154 1.4150

326.079 328.136

1.4144 1.4139

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

ˆ Z in Eqs. (8) and (9) The expressions for the coefficients Bˆ 0Z , Bˆ 2Z and D 2 are given by Bˆ 0x = (1 − a1 xc ) B0 , Bˆ 0ˆ m = (a1 + a3 ) B0 , Bˆ 0ˆ m x = (1 + a3 xc ) B0



 2

Bˆ 2x =

a21 Bˆ 0x

(1 − a1 xc )2

, Bˆ 2ˆ m =

a1 Bˆ 0x

1 − a1 xc

 ˆ a21 Bˆ 0 m

ˆ ˆ m = ,D 2

2

(a1 + a3 )2



 2

ˆx = − D 2

V + = (1 − x+ )

M1 M2 + x+ d1 d2

(13)

V − = (1 − x− )

M1 M2 + x− d1 d2

(14)

The molar volume Vc at the critical point can be calculated by Vc = (1 − xc )

x = 0.2999 319.147 320.887

a

where a1 and a3 are the mixing coefficients in the complete scaling theory, B0 is a constant and the subscript “c” denotes the value at the critical point.With the assumption of ideal mixing, the molar volumes V of the solution in each of the coexistence phases can be expressed by

 ˆ a3 Bˆ 0 m



, Bˆ 2ˆ m x =

2

2

(1 + a3 xc )2



ˆ ˆ m x = ,D 2

a1 + a3

 ˆ x a23 Bˆ 0 m

 ˆ x a3 Bˆ 0 m

(10)

(11)

2

1 + a3 xc

(12)

M1 M2 + xc d1,c d2,c

(15)

The values of di under each required phase separation temperature and di,c at the critical point were calculated by the linear temperature dependence of the density reported in the references + − + − + [22,23]. Thus with  ˆm = Vc /V and  ˆm = Vc /V , the values of  ˆm , − + −  ˆm ,  ˆm x and  ˆm x were calculated for each temperature, which ˆ m ) and (T, allowed us to convert the (T, x) coexistence curves to (T,   ˆ m x) curves. These curves are shown as the plots in Figs. 3 and 4 for (x [C8 mim][PF6 ] + (1 − x) 1-butanol) and (x [C8 mim][PF6 ] + (1 − x) 2-butanol), respectively. With the critical exponents ˇ and 1 being fixed at the theoretical values (ˇ = 0.327, 1 ≈ 0.5), a least-squares fitting program was used to fit Eq. (8) with the first term, or the first two terms, or all the three terms separately for x,  ˆ m and  ˆ m x to obtain the parameters Bˆ 0Z , Bˆ 1Z and Bˆ 2Z . The results are summarized in Table 7, where the standard deviation s of the fit is the judgment of the goodness of the fit. It may be seen that the fits were almost not improved after the ||3ˇ term was added, indicating the insignificance of the 3ˇ term. Therefore first two terms in Eq. (8) are sufficient to fit the experimental data of the width of the coexistence curves. As shown in Eq. (9), Zd is at least the sum of the three terms proportional to ||2ˇ , ||1-˛ and ||. These terms were considered as a direct consequence of the complete scaling [16]. Because of the small values of Zd and the strong correlations among the three terms it is almost impossible to obtain the reliable coefficients of the three terms simultaneously by fitting the experimental data of the coexistence curves with Eq. (9). So we fitted each of them in ˆ Z, D ˆ Z and D ˆ Z with separate procedures and obtain the values of D 2 1 0 the standard deviations s of the fits, which are listed in Table 8. The difference of s between the fit with the 2ˇ term and the fit with the (1 − ˛) term or the fit with the liner term judges the significance of the 2ˇ term, the larger of the difference, the more significant of the 2ˇ term is. It can be seen from Table 8 that this difference for  ˆ m and  ˆ m x are significantly smaller than that for x, indicating the 2ˇ term is more significant for x than the other two variables. It is consistent with the hypothesis of the incompressible or weakly compressible liquids and liquid mixtures, which results in a3 ≈ 0, ˆ ˆ m ≈ 0 and D ˆ ˆ m x ≈ 0 in Eq. (12). hence D 2 2

Table 6 Values of the critical amplitude B and the critical exponent ˇ in Eq. (1) for coexistence curves of (T, n), (T, x) and (T, ) of {x [C8 mim][PF6 ] + (1 − x) 1-butanol} and {x [C8 mim][PF6 ] + (1 − x) 2-butanol}. Order parameter

Tc − T < 1 K

Tc − T < 15 K

B

ˇ

B

ˇ

x [C8 mim][PF6 ] + (1 − x) 1-butanol n x 

0.058 ± 0.002 0.894 ± 0.012 1.967 ± 0.051

0.326 ± 0.004 0.327 ± 0.002 0.326 ± 0.004

0.044 ± 0.001 0.948 ± 0.006 1.545 ± 0.012

0.289 ± 0.003 0.336 ± 0.001 0.301 ± 0.002

x [C8 mim][PF6 ] + (1 − x) 2-butanol n x 

0.059 ± 0.003 0.765 ± 0.020 1.644 ± 0.031

0.327 ± 0.005 0.328 ± 0.003 0.325 ± 0.003

0.048 ± 0.001 0.814 ± 0.004 1.463 ± 0.011

0.299 ± 0.002 0.336 ± 0.001 0.310 ± 0.002

Z. Cui et al. / Fluid Phase Equilibria 348 (2013) 52–59

57

ˆ m ); (b) temperature against dimensionless partial molar density (T,  ˆ m x) for {x Fig. 3. Coexistence curves of (a) temperature against dimensionless molar density (T,  [C8 mim][PF6 ] + (1 − x) 1-butanol}., Experimental values of the diameter Zd of the coexisting phases;䊉 experimental values of the density variables Z of the coexisting phase; the lines represent the vales of the density variables Zcalc and the diameters Zd,calc of the coexisting phases calculated from Eqs. (16) and (17) with coefficients listed in Tables 7 and 8.

Combination of Eq. (8) neglecting the 3ˇ term and Eq. (9) with only 2ˇ term yields ˆ Z ||2ˇ + Bˆ Z ||ˇ + Bˆ Z Bˆ Z ||ˇ+1 Z + = Zc + D 2 0 0 1

(16)

shown as lines in Figs. 1–4, which are in well agreement with the experimental results. The volume fraction  may be calculated by

ˆ Z ||2ˇ − Bˆ Z ||ˇ − Bˆ Z Bˆ Z ||ˇ+1 Z − = Zc + D 2 0 0 1

(17)

= ˆ mx ·

ˆ Z, Fixing ˇ and 1 at 0.327 and 0.5 and taking the values of D 2 Bˆ 0Z and Bˆ 1Z from the corresponding values listed in Tables 7 and 8, the values of Z+ and Z− for x,  ˆ m and  ˆ m x were calculated and are

V20 Vc

(18)

where V20 is the molar volume of pure [C8 mim][PF6 ]. The calculated values of  are shown as lines in Figs. 1c and 2c, which also well agree with the experimental results.

Fig. 4. Coexistence curves of (a) temperature against dimensionless molar density (T,  ˆ m ); (b) temperature against dimensionless partial molar density (T,  ˆ m x) for {x [C8 mim][PF6 ] + (1 − x) 2-butanol} , Experimental values of the diameter Zd of the coexisting phases;䊉 experimental values of the density variables Z of the coexisting phase; the lines represent the vales of the density variables Zcalc and the diameters Zd,calc of the coexisting phases calculated from Eqs. (16) and (17) with coefficients listed in Tables 7 and 8.

58

Z. Cui et al. / Fluid Phase Equilibria 348 (2013) 52–59

Table 7 Parameters of Eq. (8) and the standard deviation s of coexistence curves of ˆ m ) and (T,  ˆ m x) for {x [C8 mim][PF6 ] + (1 − x) 1-butanol} and {x (T, x), (T,  [C8 mim][PF6 ] + (1 − x) 2-butanol}. Order parameter

(T, x)

x [C8 mim][PF6 ] + (1 − x) 1-butanol Zcxc ≈ Bˆ 0Z ||ˇ Bˆ 0Z 0.454 ± 0.001 s 1.2 × 10−3

 ˇ

Zcxc ≈ Bˆ 0Z || Bˆ 0Z Bˆ 1Z s Zcxc

1+Bˆ 1Z ||1



0.441 ± 0.001 0.27 ± 0.03 6.3 × 10−4



≈ Bˆ 0Z ||ˇ 1 + Bˆ 1Z ||1 + Bˆ 2Z ||2ˇ



Bˆ 0Z Bˆ 1Z Bˆ Z

0.451 ± 0.003 −0.9 ± 0.3 1.8 ± 0.4 2 5.3 × 10−4 s x [C8 mim][PF6 ] + (1 − x) 2-butanol Zcxc ≈ Bˆ 0Z ||ˇ 0.389 ± 0.001 Bˆ 0Z s 9.2 × 10−4



Zcxc ≈ Bˆ 0Z ||ˇ 1+Bˆ 1Z ||1 Bˆ 0Z Bˆ 1Z s

 ˇ

Zcxc ≈ Bˆ 0Z ||



0.378 ± 0.001 0.28 ± 0.03 4.5 × 10−4

 2ˇ

(T,  ˆ m)

(T,  ˆ m x)

−0.692 ± 0.003 3.8 × 10−3

0.352 ± 0.002 2.0 × 10−3

−0.739 ± 0.002 −0.56 ± 0.02 8.1 × 10−4

0.377 ± 0.001 −0.59 ± 0.02 4.1 × 10−4

−0.746 ± 0.004 −1.1 ± 0.3 0.8 ± 0.4 7.8 × 10−4

0.380 ± 0.002 −1.0 ± 0.3 0.7 ± 0.4 4.0 × 10−4

−0.614 ± 0.002 2.2 × 10−3

0.314 ± 0.001 1.2 × 10−3

−0.642 ± 0.002 −0.43 ± 0.02 7.9 × 10−4

0.329 ± 0.001 −0.45 ± 0.02 4.1 × 10−4

−0.637 ± 0.004 0.1 ± 0.3 −0.8 ± 0.5 7.6 × 10−4

0.326 ± 0.002 0.1 ± 0.3 −0.9 ± 0.5 3.9 × 10−4

1 + Bˆ 1Z ||1 + Bˆ 2Z ||

Bˆ 0Z Bˆ 1Z Bˆ Z

0.382 ± 0.002 −0.2 ± 0.3 0.8 ± 0.5 4.4 × 10−4

2

s

According to Eq. (12), the asymmetric coefficient a1 of the coexˆ x to (Bˆ x )2 by istence curve may be related to the ratio of D 2 0 ˆx D a1 = −  22 1 − a1 xc Bˆ 0x

(19)

Table 8 Parameters of Eq. (9) and the standard deviation s of coexistence curves of ˆ m ) and (T,  ˆ m x) for {x [C8 mim][PF6 ] + (1 − x) 1-butanol} and {x (T, x), (T,  [C8 mim][PF6 ] + (1 − x) 2-butanol}. Order parameter

(T, x)

x [C8 mim][PF6 ] + (1 − x) 1-butanol ˆ Z ||2ˇ Zd ≈ D 2 ˆZ D 0.593 ± 0.005 2 1.2 × 10−3 s

(T,  ˆ m)

(T,  ˆ m x)

−0.365 ± 0.005 1.2 × 10−3

0.230 ± 0.003 8.6 × 10−4

−0.92 ± 0.02 2.0 × 10−3

0.58 ± 0.01 9.6 × 10−4

−1.39 ± 0.04 2.8 × 10−3

0.88 ± 0.02 1.5 × 10−3

−0.352 ± 0.004 9.4 × 10−4

0.219 ± 0.002 5.4 × 10−4

ˆ− A

ˆZ 0 D 1 1−˛ s

1.49 ± 0.03 3.0 × 10−3

ˆ Z Bˆ cr || Zd ≈ −D 1 ˆ Z Bˆ cr -D 2.26 ± 0.06 1 4.4 × 10−3 s x [C8 mim][PF6 ] + (1 − x) 2-butanol ˆ Z ||2ˇ Zd ≈ D 2 ˆZ D 0.501 ± 0.003 2 7.3 × 10−4 s ˆ− A

ˆ Z 0 ||1−˛ Zd ≈ D 1 1−˛ ˆ− A



xcxc ≈ ±Bˆ 0Z ||ˇ 1 + Bˆ 1Z ||1



ˆ x ||2ˇ xd ≈ D 2

(20) (21)

ˆ x and Bˆ x were obtained by fitting the data The values of D 2 0 of xcxc and xd to Eqs. (20) and (21), respectively. The asymmetry coefficient a1 were then calculated through Eq. (19) from ˆ x and Bˆ x , which were (−4.35 ± 0.03) the obtained values of D 2 0 for {x [C8 mim][PF6 ] + (1 − x) 1-butanol} and (−4.97 ± 0.02) for {x [C8 mim][PF6 ] + (1 − x) 2-butanol}, respectively. It has been reported that with an assumption of the ideal mixing the asymmetric coefficient a1 of the coexistence curve in a nearly incompressible limit of a binary solution may be deduced as a simple linear func0 /V ˆ 0 [18]. tion of solute/solvent reduced molar volume ratio Vˆ 2,c 1,c

a1 = 1 −

0 Vˆ 2,c 0 Vˆ 1,c

(22)

0 = V 0 /V , V 0 0 0 0 with Vˆ 1,c c ˆ 2,c = V2,c /Vc ; where V1,c and V2,c are molar 1,c volumes of butanol and [C8 mim][PF6 ] at the corresponding criti0 and V 0 cal solution temperatures, respectively. The values of V1,c 2,c 0 = M /d , where M and d were obtained by Vi,c i i,c are the molar i i,c mass and the density of the pure component i (i = 1 or 2) at the critical solution temperatures, respectively. The value of di,c was calculated by the linear temperature dependence of the density reported in the references [22,23]. The values of a1 calculated by Eq. (22) are (−1.96 ± 0.03) for {x [C8 mim][PF6 ] + (1 − x) 1-butanol} and (−1.95 ± 0.03) for {x [C8 mim][PF6 ] + (1 − x) 2-butanol}, which significantly departure from that calculated Eq. (19). It demon − by  ˆ ||1−˛ / (1 − ˛) + Bˆ cr || ˆx A strates that the heat capacity term D 0 1 is important in describing the asymmetry criticality of the coexistence curve. One possible way solve this inconsistency  to  is to ˆ − ||1−˛ / (1 − ˛) + Bˆ cr || in data ˆx A keep the heat capacity term D 1 0 analysis [24], which requires the precise measurements of the capacities of the corresponding binary solutions in both the critical and non-critical regions.

4. Conclusion

ˆ Z 0 ||1−˛ Zd ≈ D 1 1−˛ ˆ− A

ˆ − ||1−˛ / (1 − ˛) + ˆ x (A Perez-Sanchez [18] neglected the term D 1 0 ˆBcr ||) for Z = x in Eq. (9) when they tested the complete scaling theory because of the absence of heat capacity data. As we mentioned above the first two terms in Eq. (8) are sufficient for fitting the experimental data of the width of the coexistence curves. Following Perez-Sanchez’s approximation, Eqs. (8) and (9) could be simplified as

ˆZ 0 D 1 1−˛ s

1.29 ± 0.03 2.6 × 10−3

−0.90 ± 0.03 2.3 × 10−3

0.57 ± 0.01 1.1 × 10−3

ˆ Z Bˆ cr || Zd ≈ −D 1 ˆ Z Bˆ cr -D 1 s

1.97 ± 0.06 3.6 × 10−3

−1.37 ± 0.05 2.9 × 10−3

0.86 ± 0.03 1.5 × 10−3

We measured the coexistence curves for systems (x [C8 mim][PF6 ] + (1 − x) 1-butanol) and (x [C8 mim][PF6 ] + (1 − x) 2-butanol), from which the values of the critical exponent ˇ were determined and found to be consistent with the 3D-Ising value. The experimental data were used to study the asymmetry of diameter of the coexistence curves in terms of the complete scaling theory proposed by Anisimov under incompressible conditions and neglecting the excess volume. The results indicated that the heat capacity possibly makes important contribution in describing the asymmetric behavior of the coexistences curves. Acknowledgements This work was supported by the National Natural Science Foundation of China (Projects 20973061 and 21173080). References [1] R.D. Rogers, K.R. Seddon, Science 302 (2003) 792–793.

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