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The liquid–liquid coexistence curves of {x dimethyl adipate + (1 − x) n-hexane} and {x dimethyl adipate + (1 − x) n-heptane} in the critical region
J. Chem. Thermodynamics 48 (2012) 229–234
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The liquid–liquid coexistence curves of {x dimethyl adipate + (1 x) n-hexane} and {x dimethyl adipate + (1 x) n-heptane} 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 14 September 2011 Received in revised form 15 December 2011 Accepted 20 December 2011 Available online 27 December 2011 Keywords: Critical behavior Coexistence curve Dimethyl adipate Complete scaling theory
a b s t r a c t The liquid–liquid coexistence curves for (dimethyl adipate + n-hexane), (dimethyl adipate + n-heptane) have been measured, from which the critical amplitudes and the critical exponents are deduced. The critical exponent b corresponding to the coexistence curves are consistent with the 3D-Ising value. The experimental results have also been analyzed to determine the critical amplitudes of Wegner-correction terms when b and D are fixed at their theoretical values, and to examine the asymmetry of the diameters for the coexistence curves. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction A fluid mixture may exhibit a liquid–liquid phase separation with a critical solution point, near which the critical phenomena are observed. It is commonly accepted that the critical behavior of fluids and fluid mixtures belong to the universality class of the three-dimensional Ising model [1]. In the region sufficiently close to the critical point, the differences of the general density variable between the coexisting phases Dq may be expressed by:
Dq ¼ jq2 q1 j ¼ Bsb ;
ð1Þ
where b is the corresponding critical exponent; the subscript 1 or 2 denotes each of the two coexisting phases; s is the reduced temperature (s = |T Tc|/Tc, Tc is the critical temperature) and B is the critical amplitude. The current theoretical value of b is 0.326 from the Renormalization Group calculations [2]. The asymmetric criticality of the coexistence curve has been paid much attention by researchers recently. A few years ago, Fisher and co-workers [3,4] proposed a general formulation of complete scaling for one-component fluids, which showed that the three scaling fields should be the linear mix of all physical fields: not only the chemical potential and the temperature but also the pressure. It was concluded that the strong singularity of the diameter of the coexistence densities was a consequence of the complete scaling. Recently, Anisimov and co-workers [5–7] have extended the complete scaling to binary mixtures for both incompressible and weakly compress⇑ Corresponding author at: School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China. Tel.: +86 21 64250047/931 64253966; fax: +86 21 64252510. E-mail address: [email protected] (W. Shen). 0021-9614/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2011.12.024
ible 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 are highly required. Dimethyl adipate has excellent properties for industrial applications, which is environmentally friendly, low cost, low toxicity, stable, and has rather high boiling temperature with their viscosity and density being close to those of water. The small size of its molecules combined with the high polarity makes it as an effective and novel solvent. It is already widely used in the paint, coating, and polymer industry and as industrial cleaners [8–10]. For better applications of dimethyl adipate the knowledge of the thermophysical properties including the behaviors of its binary solutions are essential. In this paper, we report the liquid–liquid coexistence curves of (dimethyl adipate + n-hexane) and (dimethyl adipate + n-heptane) determined by measurements of the refractive index n. The experimental results are analyzed to determine the critical exponent b and the critical amplitude B. The experimental results of the coexistence curves have also been analyzed to examine the Wegner correction terms and asymmetric behavior of the diameters of the coexistence curves. Further more, the asymmetries of the diameters for the coexistence curves are discussed according to the complete scaling theory proposed by Anisimov et al. [5–7]. 2. Experimental 2.1. Chemicals The dimethyl adipate ((CH3O2C)2C4H8, mass fraction 0.99), n-hexane (C6H14, mass fraction 0.99) and n-heptane (C7H16, mass
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
fraction 0.99) used in the experiments were purchased from Alfa Aesar Co. All the chemicals were dried and stored over 0.4 nm molecular sieves. 2.2. Apparatus and procedure The critical compositions were determined by adjusting the proportion of the two components to achieve ‘‘equal volume’’ of the two phases at the phase-separation point [11]. The samples with the critical concentrations were prepared in glass tubes provided with Acethread connections, and the tubes were placed into the water bath and refractive indexes in the two coexisting phases were measured using ‘‘minimum deviation angle’’ technique described previously [12]. The temperature in the water bath was measured with a platinum resistance thermometer and a Keithley 2700 digital multimeter with an uncertainty of ±0.001 K. During measurements of the refractive indexes, the temperature in water bath 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 at k ¼ 632:8 nm in each coexisting phase. A vibrating-tube densimeter (Anton Paar model DMA 5000 M) with automatic viscosity correction was used to measure the densities of dimethyl adipate. The temperature in the cell was regulated to ±0.001 K by a Peltier unit and measured by the built-in platinum resistance thermometers with an accuracy of ±0.01 K and a repeatability of ±0.001 K. The accuracy and the repeatability in the density measurements were stated by the manufacturer to be (±5 106 and ±1 106) g cm3, respectively. 3. Results and discussion The critical mole fractions and the critical temperatures were determined to be xc = (0.353 ± 0.002) and Tc = (289.3 ± 0.1) K for {x dimethyl adipate + (1 x) n-hexane} and xc = (0.393 ± 0.002) and Tc = (294.2 ± 0.1) K for {x dimethyl adipate + (1 x) n-heptane}, respectively. It was observed that the critical temperature was influenced by the impurities introduced in the preparation of the samples, nominally of the same composition, had different values of the critical temperatures, differing by as much as 0.1 K. However, the final results are not affected because only one sample was used in the measurement of the whole coexistence curve and only the temperature difference (T Tc) was important in the data analysis for obtaining 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 of table 1 for {x dimethyl adipate + (1 x) n-hexane} and table 2 for {x dimethyl adipate + (1 x) n-heptane}. 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. In order to convert the refractive index n to the mole fraction x to obtain the coexistence curves of temperature against mole fraction (T, x), a series of binary mixtures of {x dimethyl adipate + (1 x) n-heptane} with known mole fraction were prepared and their refractive indexes were measured in one phase region at various temperatures and are listed in table 3. With the assumption that no significant critical anomaly is present in the refractive index [12], the refractive index n of a liquid or a mixture may be expressed as a linear function of temperature in a certain temperature range [13]:
nðT; xÞ ¼ nðT 0 ; xÞ þ RðxÞ ðT T 0 Þ;
ð2Þ
RðxÞ ¼ x R1 þ ð1 xÞ R2 ;
ð3Þ
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
TABLE 1 Coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) n-hexane}. Subscripts 1 and 2 relate to upper and lower phases, respectively. T/K
n1
n2
x1
x2
/1
/2
289.248 289.246 289.247 289.244 289.235 289.229 289.225 289.213 289.202 289.185 289.167 289.148 289.129 289.110 289.080 289.050 288.996 288.949 288.886 288.821 288.682 288.535 288.381 288.178 287.875 287.514 286.928 286.306 285.498 284.442 283.434 282.449 281.298 280.112
1.3961 1.3960 1.3958 1.3956 1.3955 1.3953 1.3951 1.3947 1.3947 1.3943 1.3940 1.3940 1.3938 1.3936 1.3933 1.3932 1.3928 1.3925 1.3923 1.3920 1.3915 1.3912 1.3908 1.3904 1.3900 1.3896 1.3892 1.3887 1.3883 1.3881 1.3879 1.3879 1.3879 1.3880
1.3984 1.3986 1.3987 1.3989 1.3992 1.3994 1.3995 1.4000 1.4001 1.4004 1.4008 1.4010 1.4011 1.4013 1.4017 1.4018 1.4022 1.4024 1.4027 1.4031 1.4037 1.4043 1.4048 1.4054 1.4063 1.4071 1.4083 1.4093 1.4105 1.4120 1.4133 1.4144 1.4156 1.4169
0.330 0.328 0.325 0.321 0.319 0.316 0.312 0.306 0.306 0.299 0.293 0.293 0.290 0.286 0.281 0.279 0.272 0.266 0.263 0.257 0.248 0.241 0.234 0.225 0.216 0.206 0.195 0.182 0.168 0.156 0.144 0.136 0.126 0.117
0.369 0.373 0.375 0.378 0.383 0.387 0.388 0.397 0.399 0.404 0.411 0.414 0.416 0.419 0.426 0.428 0.434 0.438 0.442 0.449 0.459 0.468 0.476 0.486 0.500 0.512 0.529 0.543 0.559 0.579 0.596 0.609 0.622 0.638
0.382 0.380 0.376 0.373 0.371 0.367 0.363 0.356 0.356 0.348 0.343 0.342 0.339 0.335 0.329 0.327 0.319 0.313 0.309 0.303 0.292 0.285 0.277 0.267 0.257 0.246 0.233 0.218 0.202 0.188 0.174 0.165 0.153 0.143
0.424 0.427 0.429 0.433 0.438 0.442 0.444 0.453 0.454 0.460 0.467 0.470 0.472 0.475 0.482 0.484 0.491 0.494 0.499 0.506 0.515 0.525 0.533 0.542 0.556 0.568 0.585 0.598 0.614 0.633 0.649 0.661 0.674 0.689
x = 0, respectively. Fitting equation (2) to the data listed in table 3 for pure components dimethyl adipate and n-heptane give R1 = 4.07 104 K1 for dimethyl adipate and R2 = 4.90 104 K1 for n-heptane. Substituting the values of R1 and R2 into equation (3), the data in table 3 for n(T, x) then were fitted to equations (2) and (3) to obtain n(T0, x) with a standard deviation of 0.0001. The value of T0 was chosen as 289.468 K, which is the middle temperature of the coexistence curve of {x dimethyl adipate + (1 x) n-heptane}. The small standard deviation indicates that equations (2) and (3) are valid. We fitted the n(T0, x) data to a polynomial form for various x at T0 and obtained the expression:
nðT 0 ¼ 289:468K; xÞ ¼ 1:3884 þ 0:0369x þ 0:0189x2 0:0217x3 þ 0:0063x4
ð4Þ
with a standard deviation of less than 0.0001. The coexistence curves (T, n) then were converted to (T, x) by simultaneously solving equations (2)–(4) by the Newton iteration method. The validness of equations (2) and (3) allowed us to simplify the procedure for determination of the dependence of n on x. We measured the refractive indexes at various temperatures for pure dimethyl adipate and n-hexane, and at various concentrations for {x dimethyl adipate + (1 x) n-hexane} solutions at 289.28 K, which are listed in tables 4 and 5, respectively. Fitting equation (2) to the values listed in table 4 gives R1 = 4.01 104 K1 for dimethyl adipate and R2 = 5.39 104 K1 for n-hexane. Then the values of n at 289.28 K and various compositions for {x dimethyl adipate + (1 x) n-hexane} were used to calculate the values of n at the middle temperature T0 = 284.680 K through equations (2) and (3) and further to obtain the polynomial form:
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
To convert the (T, x) curve to the coexistence curve of temperature against volume fraction (T, /), the volume fraction / of dimethyl adipate was calculated from the mole fraction by
TABLE 2 Coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) nheptane}. Subscripts 1 and 2 relate to upper and lower phases, respectively. T/K
n1
n2
x1
x2
/1
/2
294.216 294.213 294.211 294.208 294.202 294.194 294.184 294.169 294.149 294.131 294.114 294.092 294.073 294.044 294.013 293.963 293.902 293.838 293.753 293.601 293.443 293.275 293.077 292.775 292.363 291.787 291.167 290.399 289.392 288.377 287.174 285.728 284.719
1.4014 1.4012 1.4011 1.4010 1.4009 1.4008 1.4005 1.4003 1.4001 1.3999 1.3998 1.3997 1.3996 1.3994 1.3992 1.3991 1.3988 1.3986 1.3984 1.3980 1.3977 1.3974 1.3972 1.3969 1.3966 1.3963 1.3961 1.3959 1.3959 1.3958 1.3958 1.3960 1.3962
1.4037 1.4038 1.4039 1.4041 1.4042 1.4043 1.4045 1.4048 1.4050 1.4052 1.4054 1.4055 1.4057 1.4059 1.4061 1.4063 1.4067 1.4069 1.4073 1.4077 1.4082 1.4087 1.4091 1.4098 1.4105 1.4114 1.4124 1.4133 1.4145 1.4156 1.4167 1.4181 1.4190
0.369 0.364 0.362 0.359 0.357 0.355 0.348 0.343 0.338 0.333 0.331 0.328 0.326 0.321 0.316 0.313 0.306 0.300 0.295 0.284 0.275 0.266 0.260 0.249 0.238 0.224 0.213 0.199 0.188 0.174 0.160 0.148 0.141
0.421 0.424 0.426 0.43 0.433 0.435 0.439 0.446 0.451 0.455 0.459 0.462 0.466 0.470 0.475 0.479 0.487 0.491 0.500 0.507 0.517 0.527 0.535 0.548 0.560 0.575 0.593 0.606 0.624 0.640 0.655 0.674 0.685
0.395 0.390 0.388 0.386 0.383 0.381 0.374 0.369 0.364 0.359 0.356 0.353 0.351 0.346 0.341 0.338 0.330 0.324 0.319 0.307 0.298 0.289 0.282 0.271 0.259 0.245 0.232 0.218 0.206 0.191 0.176 0.163 0.155
0.449 0.451 0.454 0.458 0.460 0.463 0.467 0.474 0.478 0.483 0.487 0.490 0.494 0.498 0.503 0.507 0.515 0.519 0.528 0.535 0.545 0.555 0.562 0.576 0.588 0.603 0.620 0.633 0.650 0.666 0.679 0.698 0.709
ð6Þ
K ¼ d1 M 2 =d2 M1 ;
ð7Þ
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 were measured at various temperatures by the vibrating-tube densimeter and are listed in table 6. The values of d2 were obtained from reference [14]. The values of x and / of coexisting phases at various temperatures are listed in columns 4 to 7 of tables 1 and 2, and are shown in figures 1b and c and 2b and c, respectively. In the region sufficiently close to the critical point, the coexistence curve can be described by equation (1). The differences of the 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 [15]
jq2 q1 j ¼ B sb þ B1 sbþD þ
ð8Þ
to obtain the parameters B and B1. The results are summarized in table 8, from which it may be seen that the contributions of B1 term for the variables x and / are negligible. It is 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 the complete scaling [16], while the presence of the term s2b was attributed to a wrong choice for the order parameters in the past [17,18]. It is almost impossible to simultaneously obtain the coefficients of the above three terms by fitting the experimental data of the coexistence
nðT 0 ¼ 284:680 K;xÞ ¼ 1:3784 þ 0:0612x þ 0:0048x2 0:0247x3 þ 0:0111x4
1=/ ¼ ð1 KÞ þ K=x;
ð5Þ
with a standard deviation of less than ±0.0001. Equation (5) together with equations (2) and (3) were used to convert refractive indexes to mole fractions.
290
a
c
b
288
T/K
286
284
282
280 1.39
1.40
n
1.41
0.2
0.3
0.4
x
0.5
0.6
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-hexane}. N, Experimental values of diameter qd of the coexisting phases; d, experimental values of density variables q of the coexisting phases; —, density variables qcalc and diameters qd,calc of the coexisting phases calculated from a combination of equation (10) and (11) with coefficients listed in tables 8 and 9.
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
a
294
c
b
292
T/K
290
288
286
284 1.40
1.41
1.42
0.2 0.3 0.4 0.5 0.6 0.7
n
0.2 0.3 0.4 0.5 0.6 0.7
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-heptane}. N, Experimental values of diameter qd of the coexisting phases; d, experimental values of density variables q of the coexisting phases; —, density variables qcalc and diameters qd,calc of the coexisting phases calculated from a combination of equations (10) and (11) with coefficients listed in tables 8 and 9.
TABLE 3 Refractive indexes n at wavelength k = 632.8 nm for {x dimethyl adipate + (1 x) n-heptane} at various compositions and temperatures. x
T/K
n
T/K
n
T/K
n
T/K
n
0
289.623 290.675
1.3883 1.3878
291.710 292.700
1.3873 1.3868
293.739 294.707
1.3863 1.3858
295.773
1.3853
0.1018
292.222 293.269
1.3909 1.3904
295.394 294.374
1.3893 1.3898
296.451 297.447
1.3888 1.3883
0.2007
291.465 292.535
1.3955 1.3950
293.578 294.616
1.3945 1.3940
295.636 296.702
1.3935 1.3930
297.717
1.3925
0.2976
294.958 295.952
1.3980 1.3975
296.908 298.025
1.3971 1.3965
299.113 300.152
1.3960 1.3955
301.211
1.3950
0.3992
295.613 296.649
1.4020 1.4016
297.636 298.661
1.4011 1.4007
299.700 300.670
1.4002 1.3997
301.743
1.3992
0.4976
295.798 296.900
1.4062 1.4056
297.952 298.924
1.4052 1.4048
299.927 300.929
1.4043 1.4039
301.978
1.4034
0.5999
292.852 293.869
1.4120 1.4115
294.863 295.831
1.4111 1.4107
296.810 297.865
1.4103 1.4098
0.7009
293.075 294.117
1.4159 1.4155
295.151 296.148
1.4150 1.4146
297.230 298.153
1.4142 1.4138
0.8048
293.283 294.218
1.4200 1.4196
295.121 296.094
1.4192 1.4188
297.081 298.016
1.4184 1.4180
0.8979
293.203 294.114
1.4236 1.4232
295.008 295.944
1.4228 1.4224
296.895 297.805
1.4220 1.4216
1
289.503 290.608
1.4287 1.4283
291.652 292.731
1.4279 1.4274
294.899 295.943
1.4265 1.4261
296.944
1.4257
curves to a corresponding equation; however the diameter qd may be fitted to the form: TABLE 4 Refractive indexes n at wavelength k = 632.8 nm for dimethyl adipate, and n-hexane at various temperatures. T/K
n
T/K
n
283.482 284.370
1.4312 1.4309
285.276 286.097
283.721 284.520
1.3789 1.3785
285.347 286.214
T/K
n
T/K
n
Dimethyl adipate 1.4305 286.922 1.4302 287.648
1.4298 1.4296
288.488
1.4292
n-Hexane 1.3781 287.099 1.3776 287.941
1.3771 1.3767
288.712
1.3762
qd ¼ qc þ D sz
ð9Þ
with an apparent exponent Z being fixed at the values 1, 1 a = 0.89 and 2b = 0.652 in separate fitting procedures. The results are compared in table 9. In table 9, the experimental values of nc were obtained by extrapolating the refractive indexes against temperatures in the one-phase region to the critical temperature. The experimental values of xc and /c were determined by the technique ‘‘equal volume’’ and calculated by using equations (6) and (7), respectively. The goodness of the fit to equation (9) may be indicated by the reduced
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234 TABLE 5 Refractive indexes n at wavelength k = 632.8 nm for {x dimethyl adipate + (1 x) nhexane} at 289.28 K. x
n
x
n
x
n
0.101 0.202 0.301
1.3821 1.3884 1.3944
0.394 0.500 0.599
1.3998 1.4056 1.4107
0.706 0.799 0.882
1.416 1.4203 1.4239
chi-squared value v2/N (N is the number of degree of freedom of the fitting) [19], which are also listed in table 8. It shows that the fit with Z = 2b for x is the best among the three variables, while the fits with Z = 1 and Z = 1 a for / are better than that for the other two variables. Our experimental results implied that the term s2b is dominant for variable x, while it is not so important as compared with terms s1a and s for variable /. 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 [20]. The values of this ratio calculated from the values of D and B listed in tables 9 and 7 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 (8) and (9) yields:
q1 ¼ qc þ Dsz ð1=2ÞBsb ð1=2ÞB1 sbþD ;
ð10Þ
q2 ¼ qc þ Dsz þ ð1=2ÞBsb þ ð1=2ÞB1 sbþD :
ð11Þ
Fixing Z, b, D at 0.89, 0.326, 0.5 respectively and taking the values of D, qc, B and B1 from tables 9 and 8, the values of qd, q1 and q2 for two mixture systems were calculated from equations (10) and (11). 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 and coworkers [7], the width of the phase boundary Dxcxc and the deviation function of the diameter Dxd of a coexistence curve can be expressed by:
^x jsjb ð1 þ B ^ x jsjD þ B ^ x jsj2b Þ; Dxcxc jx1 x2 j=2 B 0 1 2
ð12Þ
^ jsj1a =ð1 aÞ þ B ^ x jsj2b þ D ^ x ðA ^ cr jsjÞ; Dxd ðx1 þ x2 Þ=2 xc D 2 1 0 ð13Þ where x is the mole fraction of dimethyl adipate, the subscripts 1 ^x , B ^x , B ^x , D ^ x and D ^ x are and 2 relate to upper and lower phases; B 0 1 2 2 1 ^ is a reduced variable of the the system dependent amplitudes; A 0 critical amplitude corresponding to the heat capacity in two-phase ^ cr ¼ Bcr V c =R is a reduced variable with Bcr being the critical region; B fluctuation-induced contribution to the background of heat capacity and R being the gas constant. As discussed above the contribution of the sb term in equation (12) is dominant. Perez-Sanchez and co-workers [7] pointed that the contributions of |s|1a and |s| in
TABLE 6 Densities of dimethyl adipate at various temperatures. T/K
q/(g cm3)
T/K
q/(g cm3)
T/K
q/(g cm3)
T/K
q/(g cm3)
281.15 282.15 283.15 284.15 285.15 286.15 287.15
1.073093 1.072116 1.071142 1.070167 1.069191 1.068215 1.067240
288.15 289.15 290.15 291.15 292.15 293.15 294.15
1.066264 1.065290 1.064317 1.063342 1.062368 1.061393 1.060420
295.15 296.15 297.15 298.15 299.15 300.15 301.15
1.059446 1.058471 1.057499 1.056524 1.055549 1.054576 1.053601
302.15 303.15 304.15 305.15 306.15 307.15 308.15
1.052628 1.051654 1.050677 1.049703 1.048727 1.047752 1.046777
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-hexane} and {x dimethyl adipate + (1 x) nheptane}. Order parameter
TcT < 1 K
TcT < 10 K
B
b
n x /
0.092 ± 0.004 1.60 ± 0.07 1.69 ± 0.07
x Dimethyl adipate + (1 x) n-hexane 0.325 ± 0.005 0.326 ± 0.005 0.325 ± 0.002
n x /
0.072 ± 0.002 1.67 ± 0.03 1.71 ± 0.04
x Dimethyl adipate + (1 x) n-heptane 0.325 ± 0.003 0.326 ± 0.003 0.326 ± 0.003
B
b
0.090 ± 0.002 1.61 ± 0.03 1.69 ± 0.02
0.322 ± 0.003 0.326 ± 0.003 0.326 ± 0.001
0.070 ± 0.001 1.68 ± 0.01 1.71 ± 0.02
0.322 ± 0.001 0.326 ± 0.001 0.326 ± 0.001
TABLE 8 Parameters of equation (8) for coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) n-hexane} and {x dimethyl adipate + (1 x) n-heptane}. Order parameter
B
B1
B
B1
{x Dimethyl adipate + (1 x) n-hexane}
x Dimethyl adipate + (1 x) n-heptane
n
0.0912 ± 0.0003 0.0941 ± 0.0004
0.026 ± 0.003
0.0714 ± 0.0002 0.0736 ± 0.0002
0.019 ± 0.002
x
1.609 ± 0.002 1.610 ± 0.006
0.01 ± 0.05
1.672 ± 0.002 1.677 ± 0.004
0.047 ± 0.03
/
1.693 ± 0.003 1.705 ± 0.006
0.11 ± 0.05
1.704 ± 0.002 1.714 ± 0.004
0.083 ± 0.04
234
Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
TABLE 9 Parameters of equation (9) and the reduced chi-squared value v2/N in qd for diameters of coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) n-hexane} and {x dimethyl adipate + (1 x) n-heptane}. (T, n)
qc,expta
(T, x)
x Dimethyl adipate + (1 x) n-hexane 1.3974 ± 0.0001 0.353 ± 0.002
(T, /) 0.406 ± 0.002
Z=1
qc D
v2/N
qc D 2
v /N
1.3973 ± 0.0001 0.162 ± 0.001 0.258
0.352 ± 0.001 0.87 ± 0.03 0.43
0.404 ± 0.001 0.37 ± 0.02 0.204
1.3971 ± 0.0001 0.046 ± 0.001 5.453
Z = 2b = 0.652 0.350 ± 0.001 0.25 ± 0.01 0.198
0.404 ± 0.001 0.11 ± 0.01 0.225
v2/N
Z = 1a 1.3973 ± 0.0001 0.109 ± 0.001 0.53
qc,expta
x Dimethyl adipate +(1 x) n-heptane 1.4026 ± 0.0001 0.393 ± 0.002
qc D
= 0.89 0.352 ± 0.001 0.59 ± 0.02 0.28
0.404 ± 0.001 0.25 ± 0.01 0.2 0.420 ± 0.002
^ 0 ¼ V 0 =V c , V ^ 0 ¼ V 0 =V c ; where V 0 and V 0 are molar volwith V 1;c 2;c 1;c 2;c 1;c 2;c umes of alkane and dimethyl adipate at the corresponding critical solution temperatures, respectively; Vc is molar volume of the solution at the critical temperature and V c ¼ ð1 xc ÞV 01;c ðT c Þ þ xc V 02;c ðT c Þ with xc being the critical mole fraction of dimethyl adipate. The value of V 01;c and V 02;c were obtain 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 solution temperature, respectively. The densities of dimethyl adipate at various temperatures listed in table 6 were used to obtain a polynomial form: q/(g cm3) = 1.119 108(T/ K)2 9.810 104(T/K) + 1.348021. The densities of n-hexane and n-heptane were taken form the database of Cibulka [14]. The theoretical values of a1 were calculated by equation (15) are 0.26 for {x dimethyl adipate + (1 x) n-hexane} and 0.12 for {x dimethyl adipate + (1 x) n-heptane}, which significantly departure from that calculated from equation (14). One possible way to solve this incon^ jsj1a = ^ x ðA sistency is to take the heat capacity term D 1 0 ^ cr jsjÞ in equation (13) into account [21], which requires ð1 aÞ þ B the precise measurements of the capacities of the corresponding binary solutions both in the critical and non-critical regions.
Z=1
qc D 2
v /N
qc D
v2/N
qc D
v2/N a
1.4025 ± 0.0001 0.157 ± 0.001 0.121
0.395 ± 0.001 0.59 ± 0.02 0.174
0.421 ± 0.001 0.32 ± 0.01 0.118
1.4022 ± 0.0001 0.045 ± 0.001 5.135
Z = 2b = 0.652 0.394 ± 0.001 0.17 ± 0.01 0.155
0.421 ± 0.001 0.09 ± 0.01 0.165
Z = 1a = 0.89 1.4025 ± 0.0001 0.395 ± 0.001 0.106 ± 0.001 0.40 ± 0.01 0.325 0.127
0.421 ± 0.001 0.22 ± 0.01 0.118
qc,expt is the critical value of the order parameter determined by the techniques
described in the text.
equation (13) have opposite sign and the net effect of the contributions appears to be minor significance to Dxd. Following their fitting ^ x to be zero and neglected the procedure, we sat the parameter D 1 terms of |s|D and |s|2b in equation (12), and fitted the experimental data of (T, x) coexistence curves within Tc T < 10 K with b being fixed at the theoretical value 0.326. We obtained the amplitudes ^ x and B ^ x for {x dimethyl adipate + (1 x) n-hexane} and {x diD 2 0 methyl adipate + (1 x) n-heptane}. The asymmetric criticality of ^ x and ðB ^ x Þ2 according coexistence curve was related to the ratio of D 2 0 to the complete scaling theory we have [7]:
Acknowledgements This work was supported by the National Natural Science Foundation of China (Projects 20973061, 21073063 and 21173080), the Chinese Ministry of Education (Key project 105074). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
^ x =ðB ^ x Þ2 ¼ a1 =ð1 a1 xc Þ; D 2 0
ð14Þ
where a1 is the asymmetric coefficient of the coexistence curve. The values of a1 were calculated by equation (14) to be (0.36 ± 0.02) for {x dimethyl adipate + (1 x) n-hexane} and (0.29 ± 0.01) for {x dimethyl adipate + (1 x) n-heptane}, respectively. With an assumption that the volume change upon mixing may be neglected and in a nearly incompressible limit of a binary solution, it has been shown [7] that the asymmetric coefficient a1 of the coexistence curve may be deduced as a simple linear function of solute/solvent ^ 0 =V ^0 : reduced molar volume ratio V 2;c 1;c
a1 ¼ 1 V^ 02;c =V^ 01;c
ð15Þ
[14] [15] [16] [17] [18] [19] [20] [21]
J.V. Sengers, J.M.H. Levelt Sengers, Annu. Rev. Phys. Chem. 37 (1986) 189–222. J.V. Sengers, J.G. Shanks, J. Stat. Phys. 137 (2009) 857–877. M.E. Fisher, G. Orkoulas, Phys. Rev. Lett. 85 (2000) 696–699. Y.C. Kim, M.E. Fisher, G. Orkoulas, Phys. Rev. E 67 (2003) 061506. C.A. Cerdeiriña, M.A. Anisimov, J.V. Sengers, Chem. Phys. Lett. 424 (2006) 414– 419. J.T. Wang, C.A. Cerdeiriña, M.A. Anisimov, J.V. Sengers, Phys. Rev. E 77 (2008) 031127. G. Perez-Sanchez, P. Losada-Perez, C.A. Cerdeiriña, J.V. Sengers, M.A. Anisimov, J. Chem. Phys. 132 (2010) 154502. M.J.P. Comuñas, J-P. Bazile, L. Lugo, A. Baylaucq, J. Fernández, C. Boned, J. Chem. Eng. Data 55 (2010) 3697–3703. E. Ince, J. Chem. Eng. Data 52 (2007) 1889–1893. U.-P. Marketta, J.R. Robert, B. Paul, A.T. Béatrice, A.B. Kris, J. Chem. Eng. Data 41 (1996) 235–238. W.G. Shen, G.R. Smith, C.M. Knobler, R.L. Scott, J. Phys. Chem. 95 (1991) 3376– 3379. X.Q. An, W.G. Shen, H.J. Wang, G.K. Zheng, J. Chem. Thermodyn. 25 (1993) 1373–1783. X.Q. An, H.H. Zhao, F.G. Jiang, C.F. Mao, W.G. Shen, J. Chem. Thermodyn. 29 (1997) 1047–1054. I. Cibulka, Fluid Phase Equilib. 89 (1993) 1–18. F.J. Wenger, Phys. Rev. B 5 (1972) 4529–4536. C.A. Cerdeiriña, M.A. Anisimov, J.V. Sengers, Chem. Phys. Lett. 424 (2006) 414– 419. S.C. Greer, M.R. Moldover, Ann. Rev. Phys. Chem. 32 (1981) 233–265. A. Oleinikova, H. Weingärtner, Chem. Phys. Lett. 319 (2000) 119–124. P.R. Bevington, Data Reduction and Error Analysis for Physical Sciences, McGraw-Hill, New York, 1969. T.X. Yin, Y.T. Lei, M.J. Huang, Z.Y. Chen, C.F. Mao, X.Q. An, W.G. Shen, J. Chem. Thermodyn. 43 (2011) 656–663. 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.
JCT-11-409
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J. Chem. Thermodynamics journal homepage: www.elsevier.com/locate/jct
The liquid–liquid coexistence curves of {x dimethyl adipate + (1 x) n-hexane} and {x dimethyl adipate + (1 x) n-heptane} 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 14 September 2011 Received in revised form 15 December 2011 Accepted 20 December 2011 Available online 27 December 2011 Keywords: Critical behavior Coexistence curve Dimethyl adipate Complete scaling theory
a b s t r a c t The liquid–liquid coexistence curves for (dimethyl adipate + n-hexane), (dimethyl adipate + n-heptane) have been measured, from which the critical amplitudes and the critical exponents are deduced. The critical exponent b corresponding to the coexistence curves are consistent with the 3D-Ising value. The experimental results have also been analyzed to determine the critical amplitudes of Wegner-correction terms when b and D are fixed at their theoretical values, and to examine the asymmetry of the diameters for the coexistence curves. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction A fluid mixture may exhibit a liquid–liquid phase separation with a critical solution point, near which the critical phenomena are observed. It is commonly accepted that the critical behavior of fluids and fluid mixtures belong to the universality class of the three-dimensional Ising model [1]. In the region sufficiently close to the critical point, the differences of the general density variable between the coexisting phases Dq may be expressed by:
Dq ¼ jq2 q1 j ¼ Bsb ;
ð1Þ
where b is the corresponding critical exponent; the subscript 1 or 2 denotes each of the two coexisting phases; s is the reduced temperature (s = |T Tc|/Tc, Tc is the critical temperature) and B is the critical amplitude. The current theoretical value of b is 0.326 from the Renormalization Group calculations [2]. The asymmetric criticality of the coexistence curve has been paid much attention by researchers recently. A few years ago, Fisher and co-workers [3,4] proposed a general formulation of complete scaling for one-component fluids, which showed that the three scaling fields should be the linear mix of all physical fields: not only the chemical potential and the temperature but also the pressure. It was concluded that the strong singularity of the diameter of the coexistence densities was a consequence of the complete scaling. Recently, Anisimov and co-workers [5–7] have extended the complete scaling to binary mixtures for both incompressible and weakly compress⇑ Corresponding author at: School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China. Tel.: +86 21 64250047/931 64253966; fax: +86 21 64252510. E-mail address: [email protected] (W. Shen). 0021-9614/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2011.12.024
ible 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 are highly required. Dimethyl adipate has excellent properties for industrial applications, which is environmentally friendly, low cost, low toxicity, stable, and has rather high boiling temperature with their viscosity and density being close to those of water. The small size of its molecules combined with the high polarity makes it as an effective and novel solvent. It is already widely used in the paint, coating, and polymer industry and as industrial cleaners [8–10]. For better applications of dimethyl adipate the knowledge of the thermophysical properties including the behaviors of its binary solutions are essential. In this paper, we report the liquid–liquid coexistence curves of (dimethyl adipate + n-hexane) and (dimethyl adipate + n-heptane) determined by measurements of the refractive index n. The experimental results are analyzed to determine the critical exponent b and the critical amplitude B. The experimental results of the coexistence curves have also been analyzed to examine the Wegner correction terms and asymmetric behavior of the diameters of the coexistence curves. Further more, the asymmetries of the diameters for the coexistence curves are discussed according to the complete scaling theory proposed by Anisimov et al. [5–7]. 2. Experimental 2.1. Chemicals The dimethyl adipate ((CH3O2C)2C4H8, mass fraction 0.99), n-hexane (C6H14, mass fraction 0.99) and n-heptane (C7H16, mass
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
fraction 0.99) used in the experiments were purchased from Alfa Aesar Co. All the chemicals were dried and stored over 0.4 nm molecular sieves. 2.2. Apparatus and procedure The critical compositions were determined by adjusting the proportion of the two components to achieve ‘‘equal volume’’ of the two phases at the phase-separation point [11]. The samples with the critical concentrations were prepared in glass tubes provided with Acethread connections, and the tubes were placed into the water bath and refractive indexes in the two coexisting phases were measured using ‘‘minimum deviation angle’’ technique described previously [12]. The temperature in the water bath was measured with a platinum resistance thermometer and a Keithley 2700 digital multimeter with an uncertainty of ±0.001 K. During measurements of the refractive indexes, the temperature in water bath 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 at k ¼ 632:8 nm in each coexisting phase. A vibrating-tube densimeter (Anton Paar model DMA 5000 M) with automatic viscosity correction was used to measure the densities of dimethyl adipate. The temperature in the cell was regulated to ±0.001 K by a Peltier unit and measured by the built-in platinum resistance thermometers with an accuracy of ±0.01 K and a repeatability of ±0.001 K. The accuracy and the repeatability in the density measurements were stated by the manufacturer to be (±5 106 and ±1 106) g cm3, respectively. 3. Results and discussion The critical mole fractions and the critical temperatures were determined to be xc = (0.353 ± 0.002) and Tc = (289.3 ± 0.1) K for {x dimethyl adipate + (1 x) n-hexane} and xc = (0.393 ± 0.002) and Tc = (294.2 ± 0.1) K for {x dimethyl adipate + (1 x) n-heptane}, respectively. It was observed that the critical temperature was influenced by the impurities introduced in the preparation of the samples, nominally of the same composition, had different values of the critical temperatures, differing by as much as 0.1 K. However, the final results are not affected because only one sample was used in the measurement of the whole coexistence curve and only the temperature difference (T Tc) was important in the data analysis for obtaining 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 of table 1 for {x dimethyl adipate + (1 x) n-hexane} and table 2 for {x dimethyl adipate + (1 x) n-heptane}. 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. In order to convert the refractive index n to the mole fraction x to obtain the coexistence curves of temperature against mole fraction (T, x), a series of binary mixtures of {x dimethyl adipate + (1 x) n-heptane} with known mole fraction were prepared and their refractive indexes were measured in one phase region at various temperatures and are listed in table 3. With the assumption that no significant critical anomaly is present in the refractive index [12], the refractive index n of a liquid or a mixture may be expressed as a linear function of temperature in a certain temperature range [13]:
nðT; xÞ ¼ nðT 0 ; xÞ þ RðxÞ ðT T 0 Þ;
ð2Þ
RðxÞ ¼ x R1 þ ð1 xÞ R2 ;
ð3Þ
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
TABLE 1 Coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) n-hexane}. Subscripts 1 and 2 relate to upper and lower phases, respectively. T/K
n1
n2
x1
x2
/1
/2
289.248 289.246 289.247 289.244 289.235 289.229 289.225 289.213 289.202 289.185 289.167 289.148 289.129 289.110 289.080 289.050 288.996 288.949 288.886 288.821 288.682 288.535 288.381 288.178 287.875 287.514 286.928 286.306 285.498 284.442 283.434 282.449 281.298 280.112
1.3961 1.3960 1.3958 1.3956 1.3955 1.3953 1.3951 1.3947 1.3947 1.3943 1.3940 1.3940 1.3938 1.3936 1.3933 1.3932 1.3928 1.3925 1.3923 1.3920 1.3915 1.3912 1.3908 1.3904 1.3900 1.3896 1.3892 1.3887 1.3883 1.3881 1.3879 1.3879 1.3879 1.3880
1.3984 1.3986 1.3987 1.3989 1.3992 1.3994 1.3995 1.4000 1.4001 1.4004 1.4008 1.4010 1.4011 1.4013 1.4017 1.4018 1.4022 1.4024 1.4027 1.4031 1.4037 1.4043 1.4048 1.4054 1.4063 1.4071 1.4083 1.4093 1.4105 1.4120 1.4133 1.4144 1.4156 1.4169
0.330 0.328 0.325 0.321 0.319 0.316 0.312 0.306 0.306 0.299 0.293 0.293 0.290 0.286 0.281 0.279 0.272 0.266 0.263 0.257 0.248 0.241 0.234 0.225 0.216 0.206 0.195 0.182 0.168 0.156 0.144 0.136 0.126 0.117
0.369 0.373 0.375 0.378 0.383 0.387 0.388 0.397 0.399 0.404 0.411 0.414 0.416 0.419 0.426 0.428 0.434 0.438 0.442 0.449 0.459 0.468 0.476 0.486 0.500 0.512 0.529 0.543 0.559 0.579 0.596 0.609 0.622 0.638
0.382 0.380 0.376 0.373 0.371 0.367 0.363 0.356 0.356 0.348 0.343 0.342 0.339 0.335 0.329 0.327 0.319 0.313 0.309 0.303 0.292 0.285 0.277 0.267 0.257 0.246 0.233 0.218 0.202 0.188 0.174 0.165 0.153 0.143
0.424 0.427 0.429 0.433 0.438 0.442 0.444 0.453 0.454 0.460 0.467 0.470 0.472 0.475 0.482 0.484 0.491 0.494 0.499 0.506 0.515 0.525 0.533 0.542 0.556 0.568 0.585 0.598 0.614 0.633 0.649 0.661 0.674 0.689
x = 0, respectively. Fitting equation (2) to the data listed in table 3 for pure components dimethyl adipate and n-heptane give R1 = 4.07 104 K1 for dimethyl adipate and R2 = 4.90 104 K1 for n-heptane. Substituting the values of R1 and R2 into equation (3), the data in table 3 for n(T, x) then were fitted to equations (2) and (3) to obtain n(T0, x) with a standard deviation of 0.0001. The value of T0 was chosen as 289.468 K, which is the middle temperature of the coexistence curve of {x dimethyl adipate + (1 x) n-heptane}. The small standard deviation indicates that equations (2) and (3) are valid. We fitted the n(T0, x) data to a polynomial form for various x at T0 and obtained the expression:
nðT 0 ¼ 289:468K; xÞ ¼ 1:3884 þ 0:0369x þ 0:0189x2 0:0217x3 þ 0:0063x4
ð4Þ
with a standard deviation of less than 0.0001. The coexistence curves (T, n) then were converted to (T, x) by simultaneously solving equations (2)–(4) by the Newton iteration method. The validness of equations (2) and (3) allowed us to simplify the procedure for determination of the dependence of n on x. We measured the refractive indexes at various temperatures for pure dimethyl adipate and n-hexane, and at various concentrations for {x dimethyl adipate + (1 x) n-hexane} solutions at 289.28 K, which are listed in tables 4 and 5, respectively. Fitting equation (2) to the values listed in table 4 gives R1 = 4.01 104 K1 for dimethyl adipate and R2 = 5.39 104 K1 for n-hexane. Then the values of n at 289.28 K and various compositions for {x dimethyl adipate + (1 x) n-hexane} were used to calculate the values of n at the middle temperature T0 = 284.680 K through equations (2) and (3) and further to obtain the polynomial form:
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
To convert the (T, x) curve to the coexistence curve of temperature against volume fraction (T, /), the volume fraction / of dimethyl adipate was calculated from the mole fraction by
TABLE 2 Coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) nheptane}. Subscripts 1 and 2 relate to upper and lower phases, respectively. T/K
n1
n2
x1
x2
/1
/2
294.216 294.213 294.211 294.208 294.202 294.194 294.184 294.169 294.149 294.131 294.114 294.092 294.073 294.044 294.013 293.963 293.902 293.838 293.753 293.601 293.443 293.275 293.077 292.775 292.363 291.787 291.167 290.399 289.392 288.377 287.174 285.728 284.719
1.4014 1.4012 1.4011 1.4010 1.4009 1.4008 1.4005 1.4003 1.4001 1.3999 1.3998 1.3997 1.3996 1.3994 1.3992 1.3991 1.3988 1.3986 1.3984 1.3980 1.3977 1.3974 1.3972 1.3969 1.3966 1.3963 1.3961 1.3959 1.3959 1.3958 1.3958 1.3960 1.3962
1.4037 1.4038 1.4039 1.4041 1.4042 1.4043 1.4045 1.4048 1.4050 1.4052 1.4054 1.4055 1.4057 1.4059 1.4061 1.4063 1.4067 1.4069 1.4073 1.4077 1.4082 1.4087 1.4091 1.4098 1.4105 1.4114 1.4124 1.4133 1.4145 1.4156 1.4167 1.4181 1.4190
0.369 0.364 0.362 0.359 0.357 0.355 0.348 0.343 0.338 0.333 0.331 0.328 0.326 0.321 0.316 0.313 0.306 0.300 0.295 0.284 0.275 0.266 0.260 0.249 0.238 0.224 0.213 0.199 0.188 0.174 0.160 0.148 0.141
0.421 0.424 0.426 0.43 0.433 0.435 0.439 0.446 0.451 0.455 0.459 0.462 0.466 0.470 0.475 0.479 0.487 0.491 0.500 0.507 0.517 0.527 0.535 0.548 0.560 0.575 0.593 0.606 0.624 0.640 0.655 0.674 0.685
0.395 0.390 0.388 0.386 0.383 0.381 0.374 0.369 0.364 0.359 0.356 0.353 0.351 0.346 0.341 0.338 0.330 0.324 0.319 0.307 0.298 0.289 0.282 0.271 0.259 0.245 0.232 0.218 0.206 0.191 0.176 0.163 0.155
0.449 0.451 0.454 0.458 0.460 0.463 0.467 0.474 0.478 0.483 0.487 0.490 0.494 0.498 0.503 0.507 0.515 0.519 0.528 0.535 0.545 0.555 0.562 0.576 0.588 0.603 0.620 0.633 0.650 0.666 0.679 0.698 0.709
ð6Þ
K ¼ d1 M 2 =d2 M1 ;
ð7Þ
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 were measured at various temperatures by the vibrating-tube densimeter and are listed in table 6. The values of d2 were obtained from reference [14]. The values of x and / of coexisting phases at various temperatures are listed in columns 4 to 7 of tables 1 and 2, and are shown in figures 1b and c and 2b and c, respectively. In the region sufficiently close to the critical point, the coexistence curve can be described by equation (1). The differences of the 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 [15]
jq2 q1 j ¼ B sb þ B1 sbþD þ
ð8Þ
to obtain the parameters B and B1. The results are summarized in table 8, from which it may be seen that the contributions of B1 term for the variables x and / are negligible. It is 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 the complete scaling [16], while the presence of the term s2b was attributed to a wrong choice for the order parameters in the past [17,18]. It is almost impossible to simultaneously obtain the coefficients of the above three terms by fitting the experimental data of the coexistence
nðT 0 ¼ 284:680 K;xÞ ¼ 1:3784 þ 0:0612x þ 0:0048x2 0:0247x3 þ 0:0111x4
1=/ ¼ ð1 KÞ þ K=x;
ð5Þ
with a standard deviation of less than ±0.0001. Equation (5) together with equations (2) and (3) were used to convert refractive indexes to mole fractions.
290
a
c
b
288
T/K
286
284
282
280 1.39
1.40
n
1.41
0.2
0.3
0.4
x
0.5
0.6
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-hexane}. N, Experimental values of diameter qd of the coexisting phases; d, experimental values of density variables q of the coexisting phases; —, density variables qcalc and diameters qd,calc of the coexisting phases calculated from a combination of equation (10) and (11) with coefficients listed in tables 8 and 9.
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Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
a
294
c
b
292
T/K
290
288
286
284 1.40
1.41
1.42
0.2 0.3 0.4 0.5 0.6 0.7
n
0.2 0.3 0.4 0.5 0.6 0.7
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-heptane}. N, Experimental values of diameter qd of the coexisting phases; d, experimental values of density variables q of the coexisting phases; —, density variables qcalc and diameters qd,calc of the coexisting phases calculated from a combination of equations (10) and (11) with coefficients listed in tables 8 and 9.
TABLE 3 Refractive indexes n at wavelength k = 632.8 nm for {x dimethyl adipate + (1 x) n-heptane} at various compositions and temperatures. x
T/K
n
T/K
n
T/K
n
T/K
n
0
289.623 290.675
1.3883 1.3878
291.710 292.700
1.3873 1.3868
293.739 294.707
1.3863 1.3858
295.773
1.3853
0.1018
292.222 293.269
1.3909 1.3904
295.394 294.374
1.3893 1.3898
296.451 297.447
1.3888 1.3883
0.2007
291.465 292.535
1.3955 1.3950
293.578 294.616
1.3945 1.3940
295.636 296.702
1.3935 1.3930
297.717
1.3925
0.2976
294.958 295.952
1.3980 1.3975
296.908 298.025
1.3971 1.3965
299.113 300.152
1.3960 1.3955
301.211
1.3950
0.3992
295.613 296.649
1.4020 1.4016
297.636 298.661
1.4011 1.4007
299.700 300.670
1.4002 1.3997
301.743
1.3992
0.4976
295.798 296.900
1.4062 1.4056
297.952 298.924
1.4052 1.4048
299.927 300.929
1.4043 1.4039
301.978
1.4034
0.5999
292.852 293.869
1.4120 1.4115
294.863 295.831
1.4111 1.4107
296.810 297.865
1.4103 1.4098
0.7009
293.075 294.117
1.4159 1.4155
295.151 296.148
1.4150 1.4146
297.230 298.153
1.4142 1.4138
0.8048
293.283 294.218
1.4200 1.4196
295.121 296.094
1.4192 1.4188
297.081 298.016
1.4184 1.4180
0.8979
293.203 294.114
1.4236 1.4232
295.008 295.944
1.4228 1.4224
296.895 297.805
1.4220 1.4216
1
289.503 290.608
1.4287 1.4283
291.652 292.731
1.4279 1.4274
294.899 295.943
1.4265 1.4261
296.944
1.4257
curves to a corresponding equation; however the diameter qd may be fitted to the form: TABLE 4 Refractive indexes n at wavelength k = 632.8 nm for dimethyl adipate, and n-hexane at various temperatures. T/K
n
T/K
n
283.482 284.370
1.4312 1.4309
285.276 286.097
283.721 284.520
1.3789 1.3785
285.347 286.214
T/K
n
T/K
n
Dimethyl adipate 1.4305 286.922 1.4302 287.648
1.4298 1.4296
288.488
1.4292
n-Hexane 1.3781 287.099 1.3776 287.941
1.3771 1.3767
288.712
1.3762
qd ¼ qc þ D sz
ð9Þ
with an apparent exponent Z being fixed at the values 1, 1 a = 0.89 and 2b = 0.652 in separate fitting procedures. The results are compared in table 9. In table 9, the experimental values of nc were obtained by extrapolating the refractive indexes against temperatures in the one-phase region to the critical temperature. The experimental values of xc and /c were determined by the technique ‘‘equal volume’’ and calculated by using equations (6) and (7), respectively. The goodness of the fit to equation (9) may be indicated by the reduced
233
Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234 TABLE 5 Refractive indexes n at wavelength k = 632.8 nm for {x dimethyl adipate + (1 x) nhexane} at 289.28 K. x
n
x
n
x
n
0.101 0.202 0.301
1.3821 1.3884 1.3944
0.394 0.500 0.599
1.3998 1.4056 1.4107
0.706 0.799 0.882
1.416 1.4203 1.4239
chi-squared value v2/N (N is the number of degree of freedom of the fitting) [19], which are also listed in table 8. It shows that the fit with Z = 2b for x is the best among the three variables, while the fits with Z = 1 and Z = 1 a for / are better than that for the other two variables. Our experimental results implied that the term s2b is dominant for variable x, while it is not so important as compared with terms s1a and s for variable /. 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 [20]. The values of this ratio calculated from the values of D and B listed in tables 9 and 7 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 (8) and (9) yields:
q1 ¼ qc þ Dsz ð1=2ÞBsb ð1=2ÞB1 sbþD ;
ð10Þ
q2 ¼ qc þ Dsz þ ð1=2ÞBsb þ ð1=2ÞB1 sbþD :
ð11Þ
Fixing Z, b, D at 0.89, 0.326, 0.5 respectively and taking the values of D, qc, B and B1 from tables 9 and 8, the values of qd, q1 and q2 for two mixture systems were calculated from equations (10) and (11). 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 and coworkers [7], the width of the phase boundary Dxcxc and the deviation function of the diameter Dxd of a coexistence curve can be expressed by:
^x jsjb ð1 þ B ^ x jsjD þ B ^ x jsj2b Þ; Dxcxc jx1 x2 j=2 B 0 1 2
ð12Þ
^ jsj1a =ð1 aÞ þ B ^ x jsj2b þ D ^ x ðA ^ cr jsjÞ; Dxd ðx1 þ x2 Þ=2 xc D 2 1 0 ð13Þ where x is the mole fraction of dimethyl adipate, the subscripts 1 ^x , B ^x , B ^x , D ^ x and D ^ x are and 2 relate to upper and lower phases; B 0 1 2 2 1 ^ is a reduced variable of the the system dependent amplitudes; A 0 critical amplitude corresponding to the heat capacity in two-phase ^ cr ¼ Bcr V c =R is a reduced variable with Bcr being the critical region; B fluctuation-induced contribution to the background of heat capacity and R being the gas constant. As discussed above the contribution of the sb term in equation (12) is dominant. Perez-Sanchez and co-workers [7] pointed that the contributions of |s|1a and |s| in
TABLE 6 Densities of dimethyl adipate at various temperatures. T/K
q/(g cm3)
T/K
q/(g cm3)
T/K
q/(g cm3)
T/K
q/(g cm3)
281.15 282.15 283.15 284.15 285.15 286.15 287.15
1.073093 1.072116 1.071142 1.070167 1.069191 1.068215 1.067240
288.15 289.15 290.15 291.15 292.15 293.15 294.15
1.066264 1.065290 1.064317 1.063342 1.062368 1.061393 1.060420
295.15 296.15 297.15 298.15 299.15 300.15 301.15
1.059446 1.058471 1.057499 1.056524 1.055549 1.054576 1.053601
302.15 303.15 304.15 305.15 306.15 307.15 308.15
1.052628 1.051654 1.050677 1.049703 1.048727 1.047752 1.046777
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-hexane} and {x dimethyl adipate + (1 x) nheptane}. Order parameter
TcT < 1 K
TcT < 10 K
B
b
n x /
0.092 ± 0.004 1.60 ± 0.07 1.69 ± 0.07
x Dimethyl adipate + (1 x) n-hexane 0.325 ± 0.005 0.326 ± 0.005 0.325 ± 0.002
n x /
0.072 ± 0.002 1.67 ± 0.03 1.71 ± 0.04
x Dimethyl adipate + (1 x) n-heptane 0.325 ± 0.003 0.326 ± 0.003 0.326 ± 0.003
B
b
0.090 ± 0.002 1.61 ± 0.03 1.69 ± 0.02
0.322 ± 0.003 0.326 ± 0.003 0.326 ± 0.001
0.070 ± 0.001 1.68 ± 0.01 1.71 ± 0.02
0.322 ± 0.001 0.326 ± 0.001 0.326 ± 0.001
TABLE 8 Parameters of equation (8) for coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) n-hexane} and {x dimethyl adipate + (1 x) n-heptane}. Order parameter
B
B1
B
B1
{x Dimethyl adipate + (1 x) n-hexane}
x Dimethyl adipate + (1 x) n-heptane
n
0.0912 ± 0.0003 0.0941 ± 0.0004
0.026 ± 0.003
0.0714 ± 0.0002 0.0736 ± 0.0002
0.019 ± 0.002
x
1.609 ± 0.002 1.610 ± 0.006
0.01 ± 0.05
1.672 ± 0.002 1.677 ± 0.004
0.047 ± 0.03
/
1.693 ± 0.003 1.705 ± 0.006
0.11 ± 0.05
1.704 ± 0.002 1.714 ± 0.004
0.083 ± 0.04
234
Z. Chen et al. / J. Chem. Thermodynamics 48 (2012) 229–234
TABLE 9 Parameters of equation (9) and the reduced chi-squared value v2/N in qd for diameters of coexistence curves of (T, n), (T, x) and (T, /) for {x dimethyl adipate + (1 x) n-hexane} and {x dimethyl adipate + (1 x) n-heptane}. (T, n)
qc,expta
(T, x)
x Dimethyl adipate + (1 x) n-hexane 1.3974 ± 0.0001 0.353 ± 0.002
(T, /) 0.406 ± 0.002
Z=1
qc D
v2/N
qc D 2
v /N
1.3973 ± 0.0001 0.162 ± 0.001 0.258
0.352 ± 0.001 0.87 ± 0.03 0.43
0.404 ± 0.001 0.37 ± 0.02 0.204
1.3971 ± 0.0001 0.046 ± 0.001 5.453
Z = 2b = 0.652 0.350 ± 0.001 0.25 ± 0.01 0.198
0.404 ± 0.001 0.11 ± 0.01 0.225
v2/N
Z = 1a 1.3973 ± 0.0001 0.109 ± 0.001 0.53
qc,expta
x Dimethyl adipate +(1 x) n-heptane 1.4026 ± 0.0001 0.393 ± 0.002
qc D
= 0.89 0.352 ± 0.001 0.59 ± 0.02 0.28
0.404 ± 0.001 0.25 ± 0.01 0.2 0.420 ± 0.002
^ 0 ¼ V 0 =V c , V ^ 0 ¼ V 0 =V c ; where V 0 and V 0 are molar volwith V 1;c 2;c 1;c 2;c 1;c 2;c umes of alkane and dimethyl adipate at the corresponding critical solution temperatures, respectively; Vc is molar volume of the solution at the critical temperature and V c ¼ ð1 xc ÞV 01;c ðT c Þ þ xc V 02;c ðT c Þ with xc being the critical mole fraction of dimethyl adipate. The value of V 01;c and V 02;c were obtain 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 solution temperature, respectively. The densities of dimethyl adipate at various temperatures listed in table 6 were used to obtain a polynomial form: q/(g cm3) = 1.119 108(T/ K)2 9.810 104(T/K) + 1.348021. The densities of n-hexane and n-heptane were taken form the database of Cibulka [14]. The theoretical values of a1 were calculated by equation (15) are 0.26 for {x dimethyl adipate + (1 x) n-hexane} and 0.12 for {x dimethyl adipate + (1 x) n-heptane}, which significantly departure from that calculated from equation (14). One possible way to solve this incon^ jsj1a = ^ x ðA sistency is to take the heat capacity term D 1 0 ^ cr jsjÞ in equation (13) into account [21], which requires ð1 aÞ þ B the precise measurements of the capacities of the corresponding binary solutions both in the critical and non-critical regions.
Z=1
qc D 2
v /N
qc D
v2/N
qc D
v2/N a
1.4025 ± 0.0001 0.157 ± 0.001 0.121
0.395 ± 0.001 0.59 ± 0.02 0.174
0.421 ± 0.001 0.32 ± 0.01 0.118
1.4022 ± 0.0001 0.045 ± 0.001 5.135
Z = 2b = 0.652 0.394 ± 0.001 0.17 ± 0.01 0.155
0.421 ± 0.001 0.09 ± 0.01 0.165
Z = 1a = 0.89 1.4025 ± 0.0001 0.395 ± 0.001 0.106 ± 0.001 0.40 ± 0.01 0.325 0.127
0.421 ± 0.001 0.22 ± 0.01 0.118
qc,expt is the critical value of the order parameter determined by the techniques
described in the text.
equation (13) have opposite sign and the net effect of the contributions appears to be minor significance to Dxd. Following their fitting ^ x to be zero and neglected the procedure, we sat the parameter D 1 terms of |s|D and |s|2b in equation (12), and fitted the experimental data of (T, x) coexistence curves within Tc T < 10 K with b being fixed at the theoretical value 0.326. We obtained the amplitudes ^ x and B ^ x for {x dimethyl adipate + (1 x) n-hexane} and {x diD 2 0 methyl adipate + (1 x) n-heptane}. The asymmetric criticality of ^ x and ðB ^ x Þ2 according coexistence curve was related to the ratio of D 2 0 to the complete scaling theory we have [7]:
Acknowledgements This work was supported by the National Natural Science Foundation of China (Projects 20973061, 21073063 and 21173080), the Chinese Ministry of Education (Key project 105074). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
^ x =ðB ^ x Þ2 ¼ a1 =ð1 a1 xc Þ; D 2 0
ð14Þ
where a1 is the asymmetric coefficient of the coexistence curve. The values of a1 were calculated by equation (14) to be (0.36 ± 0.02) for {x dimethyl adipate + (1 x) n-hexane} and (0.29 ± 0.01) for {x dimethyl adipate + (1 x) n-heptane}, respectively. With an assumption that the volume change upon mixing may be neglected and in a nearly incompressible limit of a binary solution, it has been shown [7] that the asymmetric coefficient a1 of the coexistence curve may be deduced as a simple linear function of solute/solvent ^ 0 =V ^0 : reduced molar volume ratio V 2;c 1;c
a1 ¼ 1 V^ 02;c =V^ 01;c
ð15Þ
[14] [15] [16] [17] [18] [19] [20] [21]
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JCT-11-409