LiquidâLiquid Phase Equilibria and Critical Phenomena of the Binary

Article pubs.acs.org/jced

Liquid−Liquid Phase Equilibria and Critical Phenomena of the Binary Mixture Nitrobenzene + n‑Nonane Tianxiang Yin,† Aiqin Shi,† Jingjing Xie,† Mingjie Wang,† Zhiyun Chen,† Xueqin An,† and Weiguo Shen†,‡,* †

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

‡

ABSTRACT: The liquid−liquid coexistence curve for nitrobenzene + n-nonane has been measured in a wide temperature range. The values of the critical exponent β obtained from the liquid−liquid equilibrium data in the critical region were found to be in good agreement with that of the 3D Ising universality. The monotonous crossover behavior from Ising criticality to mean-field one was observed. The isobaric heat capacities per unit volume were measured, with which the asymmetric behavior of the diameter of the coexistence curve was welldescribed by the complete scaling theory. Moreover, the turbidity data near the critical point were obtained to deduce the critical amplitudes of the correlation length and the osmotic compressibility, which together with the amplitudes related to the coexistence curve and the heat capacity were used to calculate the universal scaling ratios and to confirm their theoretical predictions.

■

INTRODUCTION The measurement of liquid−liquid equilibrium data is an important way to know the interactions between the solvent and solute and also a powerful tool for designing the separation processes. Nitrobenzene is an important basic organic intermediate, which can form partially miscible mixtures with alkanes in an atmospheric pressure. The refractive index of nitrobenzene is significantly larger than that of alkanes, which made it possible for us to precisely determine the liquid−liquid equilibrium behavior for the binary solutions of nitrobenzene + alkanes by measuring the refractive index in each of the coexisting phases. Moreover, the binary solutions of nitrobenzene + alkanes also have appropriate critical temperatures1 to be easily experimentally approached; thus the critical phenomena corresponding to the coexistence curves may be precisely examined in these systems. It is commonly accepted that the critical behaviors of fluids and fluid mixtures belong to the universality class of the 3DIsing model.2 Sufficiently closing to the critical point, various thermodynamic properties can be described by universal power-law of the reduced temperature ΔT̂ (ΔT̂ = (|T − Tc|/ Tc), with Tc being the critical temperature): F = A |ΔT̂ |λ

recent years, the asymmetric behavior of the diameter of the coexistence curve in the critical region has been discussed by the newly developed “complete scaling theory”,7−9 which mixed both the dependent and independent physical fields into the scaling fields. Our previous investigations on binary systems of benzonitrile + n-alkane, dimethyl carbonate + n-alkane, and nitrobenzene + n-alkane showed the complete scaling theory can well represent the experimental asymmetric behaviors of the diameters of the coexistence curves.1,10,11 In this work, we report the liquid−liquid coexistence curve, turbidity, and isobaric heat capacity for the binary solution of {nitrobenzene + n-nonane}. The critical exponents and the universal ratios of critical amplitudes were deduced to test the theoretical predictions. The asymmetric behaviors of the diameter of the coexistence curve are discussed in the frame of the complete scaling theory.

■

EXPERIMENTAL SECTION Chemicals. Table 1 lists the purities and the suppliers of nitrobenzene and n-nonane used in this work. Measurement of Coexistence Curve. The experimental apparatus was a home-built one.12 The critical composition of the binary mixture was determined by adjusting the proportion of the two components to achieve the “equal volume” of the two phases at the phase-separation point. The critical mole fraction xc was able to be determined within ± 0.001 by visual

(1)

where F represents a thermodynamic property, λ is the universal critical exponent,3 and A is the corresponding critical amplitude. The critical amplitudes are system-dependent; however some combinations of the critical amplitudes for different thermodynamic properties show universal values as predicted by theories and confirmed by experiments.4−6 In © 2014 American Chemical Society

Received: December 19, 2013 Accepted: February 27, 2014 Published: March 7, 2014 1312

dx.doi.org/10.1021/je401097f | J. Chem. Eng. Data 2014, 59, 1312−1319

Journal of Chemical & Engineering Data

Article

herein. The sample with the critical composition was prepared in a sample cell purchased from Ace-glass Co. The cell was set in the sample holder located in a water bath with the temperature stability better than ± 0.003 K, which was adjusted in multiple dimensions of translation and rotation to ensure the incident light being at the center of the cell and normal to it. The light source was a low-power He−Ne laser with the wavelength of 632.8 nm. The intensities of incident and transmitted light beams were measured by a light-power meter (Model 1918C) from Newport Co. The total uncertainties in measurements of the light intensity and temperature difference (T − Tc) were about ± 2 % and ± 0.005 K, respectively. All measurements were carried out in the one-phase region. Measurement of Isobaric Heat Capacity Per Unit Volume. The isobaric heat capacities per unit volume of the mixture with the critical composition for various temperatures were measured by a differential scanning calorimeter, Micro DSC III (Setaram, France) based on the Tian-Calvet’s principle with two calibration liquids of n-heptane and 1-butanol. The detailed description of the apparatus and the measurement procedure can be found in previous articles.10,11 The background noise of Micro DSC III was less than ± 0.2 μW, and the

Table 1. Purities and Suppliers of the Chemicals chemical

supplier

purity, mass fraction

dried and stored method

nitrobenzene n-nonane

Alfa Aesar Alfa Aesar

0.99 0.99

0.4 nm molecular sieves 0.4 nm molecular sieves

observations. Thereafter, the sample with the critical composition was prepared, and the sample cell was sealed and set on a sample holder in a water bath. The temperature in the bath was controlled within ± 0.002 K. A platinum resistance thermometer connected with the Keithley 2700 digital multimeter was used to measure the temperature with the uncertainties of ± 0.02 K for T and of ± 0.002 K for (Tc − T). The refractive indexes were measured according to the method of “minimum deviation”.12 A He−Ne laser with wavelength λ being 632.8 nm was used as a light source. The uncertainty in measurement of refractive index in each coexisting phase was estimated to be ± 0.0001. Measurement of Turbidity. The apparatus and the method for the turbidity measurement have been described in the previous work.13 A brief introduction was presented

Table 2. Coexistence Curves of (T, n), (T, x), (T, φ), (T, ρ̂), and (T, ρ̂x) for the Binary Solution of {Nitrobenzene + nNonane}a (Tc-T)/K

n+

n−

x+

x−

φ+

φ−

ρ̂+

ρ̂−

ρ̂x+

ρ̂x−

0.007 0.010 0.013 0.017 0.022 0.029 0.038 0.050 0.068 0.094 0.124 0.159 0.196 0.243 0.301 0.366 0.437 0.530 0.638 0.800 0.991 1.229 1.533 2.000 2.621 3.327 4.151 5.182 6.176 7.243 8.298 9.474

1.4593 1.4590 1.4585 1.4581 1.4577 1.4573 1.4568 1.4563 1.4555 1.4548 1.4540 1.4534 1.4529 1.4520 1.4514 1.4506 1.4499 1.4491 1.4482 1.4472 1.4461 1.4450 1.4437 1.4421 1.4405 1.4390 1.4376 1.4361 1.4349 1.4339 1.4330 1.4322

1.4662 1.4666 1.4671 1.4675 1.4679 1.4684 1.4689 1.4694 1.4703 1.4711 1.4719 1.4725 1.4731 1.4740 1.4747 1.4756 1.4765 1.4774 1.4786 1.4797 1.4811 1.4826 1.4843 1.4866 1.4891 1.4916 1.4942 1.4971 1.4996 1.5022 1.5044 1.5066

0.520 0.518 0.514 0.511 0.508 0.505 0.501 0.498 0.492 0.486 0.480 0.475 0.472 0.465 0.460 0.453 0.448 0.441 0.433 0.425 0.415 0.405 0.394 0.378 0.362 0.347 0.331 0.313 0.298 0.284 0.271 0.258

0.568 0.571 0.574 0.577 0.580 0.583 0.586 0.590 0.596 0.601 0.606 0.610 0.614 0.620 0.624 0.629 0.635 0.640 0.648 0.654 0.662 0.671 0.680 0.693 0.706 0.718 0.731 0.745 0.756 0.767 0.776 0.785

0.382 0.380 0.377 0.374 0.371 0.369 0.365 0.362 0.356 0.351 0.346 0.342 0.338 0.332 0.327 0.322 0.317 0.311 0.305 0.297 0.289 0.281 0.271 0.258 0.246 0.233 0.221 0.207 0.196 0.185 0.176 0.167

0.430 0.432 0.436 0.439 0.441 0.445 0.448 0.451 0.457 0.463 0.468 0.472 0.476 0.482 0.487 0.493 0.499 0.505 0.513 0.520 0.529 0.538 0.549 0.563 0.579 0.593 0.609 0.625 0.640 0.654 0.666 0.677

0.987 0.985 0.983 0.982 0.980 0.979 0.977 0.975 0.972 0.969 0.966 0.963 0.961 0.958 0.955 0.952 0.949 0.946 0.943 0.938 0.934 0.929 0.924 0.917 0.911 0.904 0.898 0.891 0.885 0.880 0.875 0.871

1.014 1.015 1.017 1.019 1.020 1.022 1.024 1.026 1.030 1.033 1.036 1.038 1.041 1.044 1.047 1.050 1.054 1.057 1.062 1.066 1.071 1.077 1.084 1.092 1.102 1.111 1.121 1.131 1.140 1.150 1.157 1.165

0.513 0.510 0.505 0.502 0.498 0.494 0.490 0.485 0.478 0.471 0.464 0.458 0.453 0.445 0.439 0.432 0.425 0.417 0.408 0.399 0.388 0.377 0.364 0.347 0.330 0.313 0.297 0.279 0.264 0.250 0.237 0.225

0.576 0.580 0.584 0.588 0.592 0.596 0.601 0.605 0.613 0.621 0.628 0.633 0.639 0.647 0.653 0.661 0.669 0.677 0.688 0.697 0.709 0.722 0.737 0.757 0.777 0.798 0.819 0.842 0.862 0.882 0.899 0.915

Refractive indexes n were measured at wavelength λ = 632.8 nm, pressure p = 0.1 MPa. x, φ, ρ̂, and ρ̂x refer to mole fraction, volume fraction, dimensionless density, and dimensionless partial mole density. Superscripts “+” and “−” denote the up and the low phases, respectively (Tc = 294.169 K). Standard uncertainties u are u(p) = 10 kPa, u(T) = 0.02 K, u(Tc − T) = 0.002 K, u(n) = 0.0001, u(x) = 0.003, u(φ) = 0.003, u(ρ̂) = 0.001, and u(ρ̂x) = 0.002. a

1313



■

temperature stability was better than ± 0.002 K. The measurements were carried out in the down scanning model with the scanning rate being selected as 0.1 K·min−1. The uncertainty of the isobaric heat capacity per unit volume was estimated to be ± 0.0005 J·K−1·cm−3.

■

RESULTS

Coexistence Curves. The critical mole fraction of nitrobenzene and the critical temperature were determined to be xc = 0.544 and Tc = 294.169 K for binary solution of {nitrobenzene + n-nonane}, which were in good agreement with those reported by Latos et al.,14 namely, xc = 0.54 and Tc = 294.05 K. Latos et al.14 determined the coexistence curve for the same system by the visual method, and thus the concentrations of the conjugate phases were unable to be directly obtained simultaneously and precisely enough to deduce the corresponding critical exponent and amplitude, to analysis the asymmetry behavior of the diameter, and to test the universal ratios of the critical amplitudes, which are the aim of this work. The measured refractive indexes n for each coexisting phase at various temperatures are listed in columns 2 and 3 of Table 2. They are also shown in Figure 1a as a plot of temperature against refractive index, denoted as the (T, n) coexistence curve.

THEORETICAL BACKGROUND

According to the concept of the complete scaling theory,7 a scaling field is the mixture of both dependent and independent physical fields. As a consequence,9 the term proportional to |ΔT̂ |2β has been shown to be presented in the description of physical densities Z such as the density ρ̂ (ρ̂ = Vc/V with V and Vc being the volume of the solution and its critical value respectively), the mole fraction x, the volume fraction φ, and the partial mole density ρ̂x (density multiplied by mole fraction), and so forth, for each of the two coexisting phases: Z Z Z Z ± ≈ Zc ± B0̂ |ΔT̂ | β (1 + B1̂ |ΔT |Δ ) + D̂2 |ΔT̂ |2β − ⎞ Z ⎛ Â + D1̂ ⎜ 0 |ΔT̂ |1 − α + Bcr̂ |ΔT̂ |⎟ ⎠ ⎝1 − α

Article

(2)

where β, α, and Δ are the universal critical exponents3 corresponding to the coexistence curve, the isobaric heat capacity per unit volume, and the correction-to-scaling term; Â −0 = A−0 Vc/αR is the dimensionless critical amplitude corresponding to the heat capacity in the two-phase region and B̂ cr = BcrVc/R is the dimensionless critical background of the heat capacity. The scaling amplitudes B̂ Z0 , D̂ Z2 , and D̂ Z1 are given as: ρ̂

x

B0̂ = (1 − a1xc)B0 |τ0| β

B0̂ = (a1 + a3)B0 |τ0| β

ρ x̂

B0̂ = (1 + a3xc)B0 |τ0| β x D̂2 = −

x a1(B0̂ )2 1 − a1xc

Figure 1. Coexistence curves of (a) the temperature against the refractive index (T, n), (b) the temperature against the volume fraction (T, φ), (c) the temperature against the mole fraction (T, x), (d) the temperature against the dimensionless density (T, ρ̂), and (e) the temperature against the dimensionless partial mole density (T, ρ̂x) for binary solution of {nitrobenzene + n-nonane}. ●, experimental values of the general density variables ρ of the coexisting phases; ▲, experimental values of the diameter ρd of the coexisting phases.

(3)

ρ̂ D̂2 =

ρ̂ a3(B0̂ )2 a1 + a3

ρ x̂

ρ x̂ D̂2 =

a3(B0̂ )2 1 + a3xc

x D1̂ = (b2xc − b4)|τ0|−1

(4)

Since only weakly critical anomaly has been observed in the refractive index,12,15 the refractive index can be transformed to the mole fraction with the assumption that the refractive index n of the binary solution of {x nitrobenzene + (1 − x) n-alkane} may be expressed as a linear function of temperature in a certain temperature range:12

ρ̂ D1̂ = −(b2 + b3)|τ0|−1

ρ x̂ D1̂ = −(b4 + b3xc)|τ0|−1

(5)

for the mole fraction x, the density ρ̂, and the partial mole density ρ̂x, respectively, where B0 is the scaling amplitude; ai and bi are coefficients in the relations between physical fields and scaling fields;9 τ0 = 1 − b2(a2/a1); xc is the critical mole fraction. Therefore, the width ΔZcxc and the diameter Zd of the coexistence curves may be represented by: ΔZcxc ≡

Z− − Z+ Z Z ≈ B0̂ |ΔT̂ | β (1 + B1̂ |ΔT̂ |Δ ) 2

Z− + Z+ Z ≈ Zc + D̂2 |ΔT̂ |2β 2 − ⎞ Z ⎛ Â 0 ̂ + D1 ⎜ |ΔT̂ |1 − α + Bcr̂ |ΔT̂ |⎟ ⎠ ⎝1 − α

R(x) = xR1 + (1 − x)R 2

(8)

n(T , x) = n(T 0 , x) + R(x) (T − T 0)

(9)

where R(x) is the derivative of n with respect to T for a particular composition x; and R1 and R2 are the values of R(x) for x = 1 and x = 0, respectively. We measured the refractive indexes of nitrobenzene and n-nonane at various temperatures and of {x nitrobenzene + (1 − x) n-nonane} solutions for various concentrations at a fixed temperature T = 296.36 K, which was higher than the critical temperature. The results are listed in Table 3. Fitting the refractive indexes of two pure components at various temperatures gave R1 = 4.86·10−4 K−1 for nitrobenzene and R2 = 4.76·10−4 K−1 for n-nonane. The values of n(T0, x) at various compositions and at the middle temperature T0 =289.429 K of the coexistence curve were then

(6)

Zd ≡

(7) 1314



Article

Table 3. Refractive Indexes n at Wavelength λ = 632.8 nm for Nitrobenzene, n-Nonane, and a Binary Solution of {Nitrobenzene + n-Nonane} at Pressure p = 0.1 MPaa

φ=

T

n 1.5501 1.5497 1.5493 1.5439 1.5433 1.5428

T

n

290.885 291.939 292.727

1.4053 1.4048 1.4044 x nitrobenzene

T

n

291.448 1.5488 292.364 1.5484 293.381 1.5479 304.976 1.5423 306.084 1.5418 307.202 1.5412 n-nonane T

n

T

n

294.402 295.501

1.5474 1.5469

308.254 309.343

1.5407 1.5402

T

293.554 1.4040 296.475 294.500 1.4036 297.421 295.507 1.4031 + (1 − x) n-nonane at T = 296.36 K

where is the molar volume of pure nitrobenzene. The calculated values of x, φ, ρ̂, and ρ̂x for each of coexistence phases are summarized in Table 2 and also shown in Figure 1. Turbidity. A sample of the binary solution of {nitrobenzene + n-nonane} with the critical composition was prepared, and its critical temperature was remeasured and found to be 294.154 K, which was used in turbidity data treatment described below. The little difference in the critical temperature determined in turbidity and coexistence curve measurements may be ascribed to the uncontrollable impurity and moisture in the sample preparation.18 The total turbidity τT was calculated from the transmitted light intensities I1 and I2 corresponding to the sample cell in and out of the laser beam by the equation:

n 1.4026 1.4022

τT = −

x

n

x

n

x

n

0 0.153 0.200 0.275

1.4029 1.4162 1.4207 1.4284

0.379 0.541 0.594 0.688

1.4401 1.4612 1.4691 1.4838

0.799 0.896 1

1.5035 1.5231 1.5467

calculated by using eqs 8 and 9, which were further fitted into a polynomial form to obtain: n(T 0 = 289.429 K, x) = 1.4062 + 0.0785x + 0.052x 2

with a standard deviation of less than ± 0.0001. The coexistence curve (T, n) then was converted to (T, x) by simultaneously solving eqs 8 to 10 by the Newton iteration method. With the assumption of ideal mixing, the molar volumes V of the solution in each of the coexistence phases can be expressed by (11)

M1 M + (1 − x−) 2 ρ1 ρ2

(12)

V − = x−

Table 4. Turbidities τ of the Critical Solution of {Nitrobenzene + n-Nonane} at Various Temperatures and a Constant Pressure p = 0.1 MPa Measured by Using a Light Source with the Wavelength λ = 632.8 nma

and the molar volume Vc at the critical point can be calculated by: Vc = xc

M1 M + (1 − xc) 2 ρ1,c ρ2,c

(15)

which are listed in Table 4.

(10)

M M V + = x + 1 + (1 − x +) 2 ρ1 ρ2

1 ⎛ I1 ⎞ ln⎜ ⎟ L ⎝ I2 ⎠

for each temperature, where the constant L = 0.680 is the length of the light path. The background contribution τb was introduced to account for the reflection, the scattering, and the adsorption from the cell and the solution in the cell, which was determined at 15 K away from the critical temperature where the turbidity did not vary with the temperature noticeably and the contribution resulted from the critical concentration fluctuation could be omitted. The turbidity τ originated from the critical concentration fluctuation then was deduced by subtraction of τb from the value of total turbidity τT: τ = τT − τb (16)

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

− 0.0067x 3 + 0.0199x 4

(14)

V01

nitrobenzene 288.974 289.624 290.505 301.664 302.719 303.850

ρ ̂xV10 Vc

(13)

(T − Tc)/K

τ/cm−1

(T − Tc)/K

τ/cm−1

0.010 0.016 0.024 0.033 0.042 0.052 0.064 0.080 0.097 0.118 0.145 0.174

2.015 1.644 1.335 1.106 0.948 0.823 0.707 0.598 0.509 0.430 0.361 0.301

0.209 0.248 0.293 0.346 0.413 0.486 0.569 0.664 0.777 0.932 1.129 1.424

0.253 0.210 0.175 0.147 0.121 0.101 0.085 0.070 0.058 0.046 0.035 0.026

Standard uncertainties u are u(τ) = 0.005 cm−1, u(T − Tc) = 0.005 K, u(x) = 0.001, and u(p) = 10 kPa.

a

where the subscripts 1 and 2 refer to nitrobenzene and nnonane, respectively; the values of ρi under each temperature and ρi,c at the critical temperature of the binary mixture were calculated by the temperature dependence of the density reported in the references.16,17 We calculated the values of ρ̂+ and ρ̂− in each coexisting phases by ρ̂ = Vc/V. Furthermore, the dimensionless partial mole density ρ̂x+ and ρ̂x− were obtained by multiplying ρ̂ by x. Thus, we converted the (T, x) coexistence curve to (T,ρ̂) and (T, ρ̂x) coexistence curves. The volume fraction φ for each of coexistence phases may be calculated by:

The concentration-fluctuation induced part of the turbidity τ in the critical region can be expressed by the integration form of the Ornstein−Zernike equation:19 τ = (π 3/λ 04)(∂n2 /∂φ)2 kBTχf (c) 2

(17) 3

f (c) = {(2c + 2c + 1)ln(1 + 2c)}/c − 2(1 + c)/c

2

(18) 1315



Article

with c = 2(2nπξ/λ0)2, where the osmotic compressibility χ and the correlation length ξ are expressed as χ = χ0|ΔT̂ |−γ and ξ = ξ0|ΔT̂ |−υ with χ0 and ξ0 being the critical amplitudes and ν and γ being the universal critical exponents;3 λ0 is the wavelength of light in vacuum; φ is the volume fraction of nitrobenzene; kB is the Boltzmann’s constant; the values of the refractive index n and (∂n2/∂φ) of the solution at various compositions and temperatures were computed from the relation of the refractive index with the composition and temperature obtained in the coexistence curve measurement. To avoid the large uncertainties in simultaneous determination of the values of ξ0, χ0, ν, and γ due to their strong coupling, we determined these parameters in three separated routings: (1) we first determined the critical exponent γ by analyzing the turbidity data in the temperature range of |ΔT̂ | > 1·10−3 where the simple power law τ = τ0|ΔT̂ |−γ is valid with τ0 = 8/3(π3/λ04)(∂n2/∂φ)2kBTχ0. A plot of log10(τ) against log10|ΔT̂ | yielded a straight line as shown in the inset of Figure 2 with the slope being critical exponent γ, which was 1.23 ±

Figure 3. Plot of CpV−1 against |ΔT̂ | in the temperature region far away from the critical point for the binary solution of {nitrobenzene + n-nonane} with the critical composition.

and 0.138 ± 0.005 J·K−1·cm−3. As we have proposed previously,10 the total background of heat capacity Cp0 contains both noncritical background term Bbg and the critical background term Bcr: Cp0 = Bbg + Bcr. From the values of Cp0 determined by Perez-Sanchez et al.,20 we calculated the critical background term Bcr, which was −0.21 ± 0.01 J·K−1·cm−3 for binary solution of {nitrobenzene + n-nonane}.

■

DISCUSSION Universal Critical Properties of Coexistence Curves. In the temperature region sufficiently close to the critical point, the width of the coexistence curve expressed by eq 6 can be simplified as: Z− − Z+ Z ≈ B0̂ |ΔT̂ | β (19) 2 which has been shown to well describe the experimental data.10,11 The experimental data of the differences of general density variables n, x, φ, ρ̂, and ρ̂x between the coexisting phases within the temperature range Tc − T < 1 K were fitted to eq 19 with critical exponent β being fixed at its theoretical value or as a free parameter. The results are listed in Table 5, which confirms the theoretical value 0.326 of β. However, to represent all of the experimental data of the width of the coexistence curve in a temperature range of (Tc − T < 10 K), the first correction-to-scaling term should be used as expressed in eq 6. The optimal parameters obtained from fitting the experimental data to eq 6 with β and Δ being fixed at their theoretical values of 0.326 and 0.50 respectively are listed in columns 6 and 7 of Table 5. As proposed by the complete scaling theory, the diameter of the coexistence curves should be a combination of |ΔT̂ |2β, |ΔT̂ |1−α, and |ΔT̂ | terms: ΔZcxc ≡

Figure 2. Plot of the turbidity τ against |ΔT̂ | for binary solution of {nitrobenzene + n-nonane}. The points are the experimental data; the solid line is calculated by eqs 17 and 18. The inset shows the plot of log10(τ) against log10(|ΔT̂ |) in the temperature range of 1·10−3< |ΔT̂ | < 5·10−3, where the circles are the experimental data and the solid line represents a linear fit of the experimental data.

0.02 and in good agreement with 1.239 of the 3D-Ising value. (2) All experimental turbidity data were fitted to eqs 17 and 18 with γ being fixed at 1.239. The obtained critical exponent ν was 0.63 ± 0.01, coinciding well with the value of 0.630 predicted for the 3D-Ising universality class. (3) To determine precise values of χ0 and ξ0, the fit of the turbidity data to eqs 17 and 18 was carried out with both γ and ν being fixed at 1.239 and 0.63 respectively, which gave the values of χ0 and ξ0 to be (1.02 ± 0.04)·10−8 m3·J−1 and (0.258 ± 0.008) nm. The turbidity values calculated from eqs 17 and 18 using the optimal values of the critical parameters ξ0, χ0, ν, and γ obtained above are shown as solid line in Figure 2, which are in good agreement with the experimental points represented by unfilled circles. Isobaric Heat Capacity per Unit Volume. The isobaric heat capacities per unit volume of the binary solution of {nitrobenzene + n-nonane} with the critical composition were measured in a temperature range of (315 to 333) K. The heat capacity far away from the critical point can be described by CpV−1 = Bbg + E|ΔT̂ |, where Bbg is the noncritical background of the heat-capacity. The values of Bbg and E can be obtained from the slope and interception from the plot of CpV−1 against |ΔT̂ | as shown in Figure 3, which were 1.6973 ± 0.0005 J·K−1·cm−3

Z− + Z+ eff eff ≈ Zc + D̂2 |ΔT̂ |2β + D1̂ |ΔT̂ |1 − α 2 eff (20) + D̂ |ΔT̂ | eff eff eff where D̂ 2 , D̂ 1 , and D̂ are effective amplitudes for |ΔT̂ |2β, |ΔT̂ |1−α, and |ΔT̂ | terms, respectively. As we presented above, the term proportional to |ΔT̂ |2β has been a direct consequence of the complete scaling theory, while it was thought to be a result of wrong choice of the order parameter previously.21 Therefore, it is interesting to see how important the term |ΔT̂ |2β is in descriptions of different order parameters n, φ, x, ρ̂, Zd ≡

1316



Article

Table 5. Values of the Critical Amplitudes B̂ Z0 and B̂ Z1 , and the Critical Exponent β in eqs 6 and 19 for Coexistence Curves of (T, n), (T, x), (T, φ), (T, ρ̂), and (T, ρ̂x) for the Binary Solution of {Nitrobenzene + n-Nonane}a Tc − T < 1 K n x φ ρ̂ ρ̂x a

Tc − T < 1 K

B̂ Z0

order parameter 0.111 0.785 0.763 0.438 1.023

± ± ± ± ±

B̂ Z0

β 0.001 0.002 0.002 0.001 0.002

(0.326) (0.326) (0.326) (0.326) (0.326)

0.111 0.785 0.765 0.437 1.02

± ± ± ± ±

Tc − T < 10 K B̂ Z0

β 0.001 0.009 0.009 0.005 0.01

0.326 0.326 0.326 0.326 0.326

± ± ± ± ±

0.002 0.002 0.002 0.002 0.002

0.110 0.781 0.760 0.435 1.016

± ± ± ± ±

B̂ Z1 0.001 0.002 0.002 0.001 0.003

0.18 0.21 0.19 0.22 0.25

± ± ± ± ±

0.02 0.02 0.02 0.02 0.02

The numbers in the bracket were fixed in the fits.

Table 6. Parameters and the Standard Deviations of Fitting Zd of Coexistence Curves of (T, n), (T, x), (T, φ), (T, ρ̂), and (T, ρ̂x) with eq 20 for the Binary Solution of {Nitrobenzene + n-Nonane} (T, φ)

(T, n) Zc D̂ eff 1 D̂ eff S zc D̂ eff 2 S

(T, x) 1−α ̂ Zd ≈ Zc + D̂ eff |ΔT | + D̂ eff|ΔT̂ | 1 0.406 ± 0.001 0.544 ± 0.001 0.72 ± 0.06 −2.07 ± 0.07 −0.56 ± 0.08 2.4 ± 0.1 2.0·10−4 2.0·10−4 ̂ 2β Zd ≈ Zc + D̂ eff 2 |ΔT| 0.406 ± 0.001 0.544 ± 0.001 0.148 ± 0.003 −0.214 ± 0.002 4.0·10−4 3.0·10−4

1.4628 ± 0.0001 0.089 ± 0.008 0.077 ± 0.012 3.0·10−5 1.4625 ± 0.0001 0.060 ± 0.002 3.0·10−4

(T, ρ̂)

(T, ρ̂x)

1.000 ± 0.001 0.41 ± 0.03 −0.04 ± 0.05 1.0·10−4

0.545 ± 0.001 0.96 ± 0.08 −0.60 ± 0.11 3.0·10−4

0.999 ± 0.001 0.164 ± 0.004 6.0·10−4

0.544 ± 0.001 0.237 ± 0.004 7.0·10−4

or ρ̂x. We fitted the experimental data to eq 20 with two separate routines: (1) using both |ΔT̂ |1−α and |ΔT̂ | terms; (2) using |ΔT̂ |2β term only. The results are compared in Table 6, and the standard deviations of the fitting indicates that the |ΔT̂ |2β term is more important for the description of parameter x than the other general density variables. It is not surprising because the binary mixture we studied are incompressible or weakly compressible which results in a3 ≈ 0 and hence D̂ ρ̂2 ≈ 0 and D̂ ρ̂2x ≈ 0 in eq 4. Therefore, the expressions for ρ̂ and ρ̂x in the eq 7 may be simplified as: − ⎞ Â ̂ ρ⎛ ρd̂ = ρĉ + D1̂ ⎜ 0 |ΔT̂ |1 − α + Bcr̂ |ΔT̂ |⎟ ⎝1 − α ⎠

(21)

− ⎞ ρ ̂x ⎛ Â 0 ̂ ρ x̂ d = ρ x̂ c + D1 ⎜ |ΔT̂ |1 − α + Bcr̂ |ΔT̂ |⎟ ⎝1 − α ⎠

(22)

Figure 4. Plot of the diameter xd of the coexistence curve (T, x) against |ΔT̂ | for the binary solution of {nitrobenzene + n-nonane}. The points are experimental data; the line is calculated by eq 7. 0 0 a1 ≈ 1 − V1,c /V 2,c

where the contribution of |ΔT̂ | term is negligible. The experimental data of the diameter for ρ̂ and ρ̂x were fitted to eqs 21 and 22, where Â −0 and B̂ cr calculated with the values of A−0 taken from ref 20 and the values of Bcr and Vc determined above. The obtained values of D̂ ρ̂1 and D̂ ρ̂1x were 0.164 ± 0.002 and 0.236 ± 0.002. From eq 5, we derived the relation of D̂ x1 = D̂ ρ̂1x − D̂ ρ̂1 xc, which gave the value of D̂ x1 to be 0.147. Therefore, we fitted the experimental values of diameter of (T, x) coexistence curve to eq 7 with D̂ x1 being fixed at 0.147 to determine D̂ x2, which was −0.361 ± 0.002. With this fitting procedure, the errors of the optimized parameters resulted from the strong coupling between 2β and 1−α terms were significantly reduced.10 The experimental values of xd and the fitting results from eq 7 are compared in Figure 4, indicating a good agreement with each other. The asymmetric coefficient a1 then was calculated from the values of D̂ x2 and B̂ x0 listed in column 6 of Table 5 by eq 4, which was 0.45 ± 0.01. It has been deduced by Anisimov et al.9 from the complete scaling theory that the value of a1 was related to the ratio of mole volume of two components: 2β

(23)

0 0 where V1,c and V2,c are the molar volume of the pure nitrobenzene and n-nonane at the critical temperature of the binary solution. The value of a1 was calculated to be 0.43 by eq 23, which agreed well with 0.45 we determined above. Crossover Behavior. To correlate the difference of the general density variable of the two coexisting phases in both critical and noncritical region, the crossover theory proposed by Gutkowski et al.22 was adopted:

Δρ = B0 (|ΔT̂ |Y (2β− 1)/ Δ)1/2 + a|ΔT̂ | 2ρc

(24)

where a and B0 are the coefficients; the first and the second terms on the right side of eq 24 are the so-called singular and regular terms, respectively; Y is the crossover function expressed as: 1317


Journal of Chemical & Engineering Data 1/2 ⎡ ⎛ Λ ⎞2 ⎤ 1 − (1 − u ̅ )Y = u ̅ ⎢1 + ⎜ ⎟ ⎥ Y υ / Δ ⎝κ⎠ ⎦ ⎣ 2

κ = ct |ΔT̂ |Y

2υ− 1/ Δ

Article

(25)

φ R = ξ0{(B0̂ )2 /kBTcχ0 }1/3

(28)

X = A+ξ03/kB

(29)

where A+ is the critical amplitude of the heat capacity CpV−1 in the one-phase region. Using the values of B̂ φ0 , ξ0, χ0, and Tc we determined in the coexistence curve and turbidity experiments and taking the values of A+ from ref 20, the values of R and X were calculated to be 0.62 ± 0.03 and 0.018 ± 0.002. These results are in good agreement with the theoretical predictions, that is, 0.65 from series expansions23 for R and 0.01966 ± 0.00017 from a d = 3 expansion,24 or 0.01880 ± 0.00008 from high-temperature series5 for X. They are also very consistent with that of other nitrobenzene + n-alkane systems we investigated previously.1

(26)

with u̅ and Λ being the crossover parameters. The values of u,̅ c0.5 t /Λ, a, and B0 were obtained to be u̅ = 0.6, 0.5 ct /Λ = 4.93, a = −2.87 and B0 = 2.95 by fitting experimental data to eq 24. Figure 5a shows the fitting result and the

■

CONCLUSION In this work, we measured the liquid−liquid coexistence curve for binary solution of {nitrobenzene + n-nonane} in both the critical and noncritical region. The values of the critical exponent β deduced from the liquid−liquid equilibrium data in the critical region agreed well with the 3D Ising one. The monotonous crossover from Ising criticality to mean-field one was confirmed by using the crossover formulizm proposed by Gutkowski et al. The isobaric heat capacities per unit volume were measured, and the asymmetric behavior of the diameter of the coexistence curve was found to be well described by the complete scaling theory with the contribution of the heat capacity being considered. Moreover, the turbidity data near the critical point were obtained and analyzed to deduce the critical amplitudes of correlation length and osmotic compressibility, which together with the amplitudes related to the coexistence curve and the heat capacity were used to calculate two-scale-factor universality ratios and to show the good agreement with the theoretical predictions.

■

Figure 5. Plots of (a) ln(Δx/2xc) against log10(|ΔT̂ |), the dots and solid line refer to the experimental data and values calculated from eq 24, respectively; (b) βeff against log10(|ΔT̂ |) for the binary solution of {nitrobenzene + n-nonane}.

AUTHOR INFORMATION

Corresponding Author

*Tel.: +86 21 64250804. Fax: +86 21 64250804. E-mail: [email protected]. Funding

This work was supported by the National Natural Science Foundation of China (Projects 20973061, 21173080, 21373085, and 21303055).

experimental data for comparison, which coincides well with each other and indicates the validity of the crossover theory for the binary solution of {nitrobenzene + n-nonane}. The critical behavior is determined by the singular term of eq 24, which may be expressed as: B0 (|ΔT̂ |Y (2β − 1)/ Δ)1/2 ∝ B |ΔT̂ | β

Notes

The authors declare no competing financial interest.

■

eff

REFERENCES

(1) Yin, T. X.; Lei, Y. T.; Huang, M. J.; Chen, Z. Y.; An, X. Q.; Shen, W. G. Critical Behavior of Binary Mixture of Nitrobenzene + nUndecane and Nitrobenzene + n-Dodecane. J. Solution Chem. 2012, 41, 1866−1888. (2) Sengers, J. V.; Sengers, J. M. H. L. Thermodynamic Behavior of Fluids near the Critical Point. Annu. Rev. Phys. Chem. 1986, 37, 189− 222. (3) Sengers, J. V.; Shanks, J. G. Experimental Critical-Exponent Values for Fluids. J. Stat. Phys. 2009, 137, 857−877. (4) Fisher, M. E.; Zinn, S. Y. The Shape of the van der Waals Loop and Universal Critical Amplitude Ratios. J. Phys. A 1998, 31, L629− L635. (5) Campostrini, M.; Pelissetto, A.; Rossi, P.; Vicari, E. Improved High-Temperature Expansion and Critical Equation of State of ThreeDimensional Ising-Like Systems. Phys. Rev. E 1999, 60, 3526−3563.

(27)

with β referring to the effective critical exponent, which could be obtained from the numerical derivative of the singular term B0(|ΔT̂ |Y(2β−1)/Δ)1/2. The values of βeff are plotted against log10(|ΔT̂ |) in Figure 5b, indicating a monotonous crossover from the Ising to the mean-field critical behavior. Universal Ratios of Critical Amplitudes. It was pointed that the values of the critical amplitudes are generally dependent on the nature of the system, but some combinations of them are universal.4,5,23,24 The test of such universal ratios has been an important aspect in the research of critical phenomena, among which two-scale-factor universality ratios R and X are expressed by: eff

1318



Article

(6) Yin, T. X.; Xie, J. J.; Liu, S. X.; Lei, Y. T.; Chen, Z. Y.; Shen, W. G. Universal Critical Amplitude Ratios in Binary Mixtures of {Benzonitrile + Alkane}. J. Chem. Thermodyn. 2013, 59, 72−79. (7) Fisher, M. E.; Orkoulas, G. The Yang-Yang Anomaly in Fluid Criticality: Experiment and Scaling Theory. Phys. Rev. Lett. 2000, 85, 696−699. (8) Wang, J. T.; Anisimov, M. A. Nature of Vapor-Liquid Asymmetry in Fluid Criticality. Phys. Rev. E 2007, 75, 051107. (9) Perez-Sanchez, G.; Losada-Perez, P.; Cerdeirina, C. A.; Sengers, J. V.; Anisimov, M. A. Asymmetric Criticality in Weakly Compressible Liquid Mixture. J. Chem. Phys. 2010, 132, 154502. (10) Huang, M. J.; Lei, Y. T.; Yin, T. X.; An, X. Q.; Shen, W. G. Heat Capacities and Asymmetric Criticality of Coexistence Curves for Benzonitrile + Alkanes and Dimethyl Carbonate + Alkanes. J. Phys. Chem. B 2011, 115, 13608−13616. (11) Yin, T. X.; Liu, S. X.; Xie, J. J.; Shen, W. G. Asymmetric Criticality of the Osmotic Compressibility in Binary Mixtures. J. Chem. Phys. 2013, 138, 024504. (12) An, X. Q.; Shen, W. G.; Wang, H. J.; Zheng, G. K. The (Liquid + Liquid) Critical Phenomena of (a Polar Liquid + an n-Alkane) 1. Coexistence Curves of (N, N-Dimethylacetamide + Hexane). J. Chem. Thermodyn. 1993, 25, 1373−1383. (13) Yin, T. X.; Bai, Y. L.; Xie, J. J.; Chen, Z. Y.; An, X. Q.; Shen, W. G. Liquid-liquid Coexistence Curves and Turbidity Measurements of Benzonitrile + Tridecane in the Critical Region. J. Chem. Eng. Data 2012, 57, 2479−2485. (14) Latos, A.; Rosa, E.; Rzoska, S. J.; Chrapec, J.; Zioko, J. Critical Properties of Nitrobenzene + n-Alkanes Solutions. Phase Trans. 1987, 10, 131−142. (15) Losada-Perez, P.; Glorieux, C.; Thoen, J. The Critical Behavior of the Refractive Index near Liquid-Liquid Critical Points. J. Chem. Phys. 2012, 136, 144502. (16) Cibulka, I. Saturated Liquid Densities of 1-Alkanols from C1 to C10 and n-Alkanes from C5 to C16: A Critical Evaluation of Experimental Data. Fluid Phase Equilib. 1993, 89, 1−18. (17) Cibulka, I.; Takagi, T. P-ρ-T Data of Liquids: Summarization and Evaluation. 8. Miscellaneous Compounds. J. Chem. Eng. Data 2002, 47, 1037−1070. (18) Jacobs, D. T. Critical Point Shifts in Binary Fluid Mixtures. J. Chem. Phys. 1989, 91, 560−563. (19) Puglielli, G.; Ford, N. C., Jr. Turbidity Measurements in SF6 Near Its Critical Point. Phys. Rev. Lett. 1970, 25, 143−147. (20) Perez-Sanchez, G.; Losada-Perez, P.; Cerdeirina, C. A.; Thoen, J. Critical Behavior of Static Properties for Nitrobenzene-Alkane Mixtures. J. Chem. Phys. 2010, 132, 214503. (21) Greer, S. C.; Das, B. K.; Kumar, A.; Gopal, E. S. R. Critical Behavior of the Diameters of Liquid-Liquid Coexistence Curves. J. Chem. Phys. 1983, 79, 4545−4552. (22) Gutkowski, K.; Anisimov, M. A.; Sengers, J. V. Crossover Criticality in Ionic Solutions. J. Chem. Phys. 2001, 114, 3133−3148. (23) Stanffer, D.; Ferer, M.; Wortis, M. Universality of Second-Order Phase Transition: The Scale Factor for the Correlation Length. Phys. Rev. Lett. 1972, 29, 345−349. (24) Bervillier, C.; Godreche, C. Universal Combination of Critical Amplitudes from Field Theory. Phys. Rev. B 1980, 21, 5427−5431.

1319


LiquidâLiquid Phase Equilibria and Critical Phenomena of the Binary

Recommend Documents

LiquidâLiquid Phase Equilibria and Critical Phenomena of the Binary