LiquidâLiquid Coexistence Curves and Turbidity Measurements of

Article pubs.acs.org/jced

Liquid−Liquid Coexistence Curves and Turbidity Measurements of Benzonitrile + Tridecane in the Critical Region Tianxiang Yin,† Yongliang Bai,† Jingjing Xie,† 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: Liquid−liquid coexistence curves and turbidity measurements for the binary fluid mixture of {benzonitrile + tridecane} were conducted, from which the corresponding critical exponents and critical amplitudes were obtained. It was found that the critical exponents were in accordance with the theoretic predictions and the critical amplitudes supported the well-known two-scale-factor universality. Moreover, the coexistence curve plotted with temperature against mole fraction was found to be well-represented by the formulism proposed by Gutkowski et al. in the frame of the crossover theory, and the asymmetry of its diameter confirmed the complete scaling theory.

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is also used as an additive in fuels and fibers. Benzonitrile and alkanes can form partially miscible solutions with moderate upper critical temperatures; thus the precise experimental determinations of the (liquid + liquid) coexistence curves of mixtures of {benzonitrile + alkane} in the critical region are easily accessible. As one of the series of studies on the critical phenomena of the binary solutions of {benzonitrile + alkane}, in this work, we report the (liquid + liquid) equilibrium data and the turbidities in one-phase region for the critical binary mixture of {benzonitrile + tridecane}. From the experimental results, the values of the critical amplitudes B, ξ0, and χ0 are deduced and used to test the two-scale-factor universality. The coexistence curve of the temperature against the mole fraction is analyzed with the crossover model and the complete scaling theory.

INTRODUCTION Large fluctuations appear when a system approaches the critical point, which makes the system-dependent microscopic details insignificant. The behaviors of anomalous thermodynamic properties for different systems may be expressed by the same scaling functions based on the universality class. As widely accepted, binary solutions belong to the 3D-Ising universality class,1,2 where the asymptotical behaviors of the difference of the general density variables in two coexisting phases Δρ, the correlation length ξ, and the osmotic compressibility χ can be described by simple power laws: Δρ = |ρ+ − ρ− | = Bt β ξ = ξ0t

(1)

−ν

χ = χ0 t

(2)

−γ

■

(3)

EXPERIMENTAL SECTION Chemicals. Chemicals used in this work are summarized in Table 1.

where t is the reduced temperature (t = |Tc − T|/Tc, with Tc being the critical temperature); ρ is the general density variable and the superscripts “+” and “−”denote the upper and the lower phase, respectively; β, ν, and γ are the universal critical exponents with the values of 3D-Ising universality class, which were well-confirmed by theoretical calculations and experiments;1−5 B, ξ0, and χ0 are the critical amplitudes. The asymmetric criticality has caught much attention in recent decade since the concept of “the complete scaling” was proposed by Fisher and co-workers,6−8 which suggests that the scaling fields should be the linear combination of both dependent and independent physical fields. The complete scaling theory has been well-applied in weekly compressible liquid mixtures by Anisimov et al.,9 which showed that the asymmetrical diameter of the coexistence curve can be interpreted by the complete scaling theory. Moreover, as we proposed recently,10 the contribution of the heat capacity plays an significant role in describing the asymmetric criticality by the complete scaling theory. Benzonitrile is widely used as a useful solvent and a versatile precursor to many derivatives, such as resins and coatings, and © 2012 American Chemical Society

Table 1. Purities and Suppliers of the Chemicals chemical

supplier

purity, mass fraction

dried and stored method

benzonitrile tridecane

Alfa Aesar Merck Co.

0.99 0.99

0.4 nm molecular sieves 0.4 nm molecular sieves

Apparatus and Experimental Procedure. We have described the details of the measurements of critical mole fraction, critical temperature, and the coexistence curve previously. “The critical mole fraction of the binary mixture of {benzonitrile + tridecane} was determined by the technique of “equal volume method” described previously”.11 “The phase separation temperature of the sample with critical mole fraction Received: April 29, 2012 Accepted: August 15, 2012 Published: August 23, 2012 2479

dx.doi.org/10.1021/je300630u | J. Chem. Eng. Data 2012, 57, 2479−2485

Journal of Chemical & Engineering Data

Article

Table 2. Coexistence Curves of (T, n), (T, x), and (T, ϕ) for {x C6H5CN + (1 − x) CH3(CH2)11CH3}. Refractive Indexes n Were Measured at Wavelength λ = 632.8 nm, Pressure p = 0.1 MPa. Mole Fraction and Volume Fraction are Denoted by x and ϕ, Separately. Superscripts “+” and “−” Denote the Up and the Low Phase, Respectivelya (Tc = 295.488 K)

was carefully measured and taken as the critical temperature. The coexistence curve was obtained by measuring the refractive indexes n in two coexisting phases by the method of “minimum deviations”.11,12 “A rectangular fluorescence cell provided with an Ace-thread connection was served as sample cell. A water bath with a temperature stability of ± 0.002 K measured by a platinum resistance thermometer and a Keithley 2700 digital multimeter was used for temperature control. A He−Ne laser with the wavelength being 632.8 nm was supplied as a light source for the refractive index measurement”.11,12 “The accuracy in measurements of critical mole fraction, refractive index, the temperature, and the temperature difference (Tc − T) were ± 0.001, ± 0.0001, ± 0.02 K, and ± 0.002 K, respectively”.12,13 A brief description of the experimental method for turbidity measurements is presented here. “The sample with the critical composition was prepared in a sample cell purchased from Aceglass Co. The cell was set in the sample holder located in a wellstirred 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. A low-power He−Ne laser with the wavelength of 632.8 nm was used as a light source. The intensities of incident and transmitted light were measured by using a light-power meter (model 1918C) purchased 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. After each of temperature changes, we waited for at least one hour for thermal equilibrium and the new equilibrium was indicated by the stable light intensity detected from the light-power meter.”13

(Tc − T)/K

n−

n+

x−

x+

ϕ−

ϕ+

0.005 0.012 0.016 0.022 0.029 0.037 0.047 0.063 0.077 0.092 0.117 0.144 0.177 0.217 0.260 0.318 0.385 0.460 0.545 0.665 0.814 0.976 1.182 1.481 1.885 2.460 3.087 3.869 4.870 6.085 7.389 9.826

1.4653 1.4661 1.4664 1.4667 1.4671 1.4675 1.4678 1.4683 1.4687 1.4690 1.4695 1.4700 1.4703 1.4709 1.4713 1.4721 1.4724 1.4732 1.4736 1.4744 1.4755 1.4761 1.4773 1.4784 1.4799 1.4817 1.4833 1.4853 1.4875 1.4899 1.4923 1.4962

1.4610 1.4603 1.4600 1.4597 1.4593 1.4590 1.4587 1.4582 1.4579 1.4576 1.4572 1.4568 1.4565 1.4559 1.4554 1.4549 1.4546 1.4539 1.4537 1.4530 1.4521 1.4517 1.4508 1.4500 1.4492 1.4481 1.4471 1.4463 1.4453 1.4444 1.4435 1.4426

0.633 0.640 0.643 0.646 0.649 0.653 0.656 0.660 0.663 0.666 0.670 0.674 0.677 0.682 0.685 0.692 0.694 0.700 0.703 0.709 0.717 0.721 0.730 0.737 0.747 0.758 0.768 0.779 0.791 0.803 0.815 0.832

0.591 0.584 0.581 0.578 0.574 0.571 0.567 0.562 0.559 0.555 0.551 0.546 0.543 0.536 0.530 0.524 0.521 0.512 0.509 0.500 0.489 0.483 0.471 0.459 0.446 0.429 0.411 0.395 0.374 0.353 0.331 0.299

0.422 0.429 0.432 0.435 0.439 0.443 0.446 0.451 0.454 0.457 0.462 0.467 0.470 0.475 0.479 0.487 0.489 0.497 0.500 0.507 0.517 0.522 0.533 0.542 0.555 0.570 0.583 0.598 0.615 0.633 0.650 0.676

0.379 0.372 0.370 0.367 0.363 0.360 0.357 0.352 0.349 0.346 0.341 0.337 0.334 0.328 0.323 0.318 0.315 0.307 0.305 0.297 0.288 0.283 0.273 0.264 0.254 0.241 0.228 0.216 0.202 0.187 0.173 0.153

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RESULTS AND DISCUSSION The critical mole fraction xc (the mole fraction of benzonitrile) and the critical temperature Tc of {x benzontrile + (1 − x) tridecane} were determined, which were 0.613 ± 0.001 and 295.5 ± 0.2 K, respectively. We measured the refractive indexes n of two coexisting phases at various temperatures for the critical solution {benzontrile + tridecane}, which are summarized in columns 2 and 3 of Table 2. Figure 1a shows the plot of temperature against refractive index for this coexistence curve. In a certain temperature range, the relation between refractive index and temperature was proposed to be described by:11−13 n(T , x) = n(T 0 , x) + R(x) ·(T − T 0)

(4)

R(x) = x·R1 + (1 − x) ·R 2

(5)

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, and u(ϕ) = 0.003. a

eqs 4 and 5, which were then fitted into a polynomial form for {x benzonitrile + (1 − x) tridecane} to obtain: n(290.573 K, x) = 1.4255 + 0.0363x + 0.0521x 2 − 0.0402x 3 + 0.0527x 4

(6)

The standard deviation of the fit was 0.0001. The measured (T, n) coexistence curve was then converted to the (T, x) coexistence curve (plot of temperature against mole fraction) by simultaneously solving eqs 4 to 6. The volume fraction of benzonitrile was then calculated by 1 K = (1 − K ) + ϕ x (7)

where R(x), R1, and R2 are the derivatives of n with respect to T for a solution with a certain mole fraction x, pure benzonitrile, and pure tridecane, respectively. Table 3 lists the refractive indexes of pure benzonitrile and pure tridecane at various temperatures, from which we obtained R1 = −0.00049 K−1 and R2 = −0.00044 K−1, respectively. A series of binary mixtures with a certain mole fraction x were prepared and their refractive indexes were measured in one phase region at 295.84 K, which are listed in Table 4. With the help of the values of R1 and R2, the values of n for various compositions at 295.84 K listed in Table 4 were converted to the corresponding values at the middle temperature T0 (290.573 K) of the coexistence curve by

with

K=

d1M 2 d 2M1

(8)

where d1 and d2 are the mass densities of benzonitrile and tridecane, which were taken from the literature;14 M1 and M2 2480



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Figure 1. Coexistence curves of (a) temperature against refractive index (T, n), (b) temperature against mole fraction (T, x), and (c) temperature against volume fraction (T, ϕ) for {x C6H5CN + (1 − x) CH3(CH2)11CH3}. ●, experimental values of general density variables ρ of the coexisting phases; ▲, experimental values of diameter ρd of the coexisting phases; the solid line is calculated from eqs 14 to 16 with coefficients listed in Tables 6 and 7.

Table 3. Refractive Indexes n at Wavelength λ = 632.8 nm for C6H5CN and CH3(CH2)11CH3 at Pressure p = 0.1 MPa and Various Temperatures Ta T/K

n

293.369 295.089 296.798 298.196 299.262 300.308 301.349 302.420 305.644

1.5252 1.5243 1.5236 1.5228 1.5223 1.5218 1.5213 1.5208 1.5192

293.200 296.326 299.424

1.4244 1.4230 1.4216

T/K

n

C6H5CN 293.987 1.5249 295.661 1.5241 297.155 1.5233 298.803 1.5225 300.062 1.5219 301.126 1.5214 302.265 1.5209 303.409 1.5203 306.894 1.5186 CH3(CH2)11CH3 294.257 1.4239 297.354 1.4225 300.387 1.4212

T/K

n

294.531 296.114 297.911 298.833 300.124 301.233 302.311 304.490 307.877

1.5247 1.5238 1.5230 1.5226 1.5219 1.5214 1.5209 1.5198 1.5181

295.293 298.374 301.470

1.4234 1.4221 1.4207

are the corresponding molar masses. Columns 4 to 7 of Table 2 summarize the values of x and ϕ for each of the coexisting phases at different temperatures, and Figure 1 parts b and c show the (T, x) and (T, ϕ) coexistence curves, respectively. Equation 1 was used to fit Δρ for n, x, and ϕ to obtain the optimal values of β and B, which are listed in Table 5. The values of β show somewhat small changes with different choices of temperature range. However, in (Tc − T) < 1 K, they are accordance with the theoretical prediction (0.326) and the experimental results for other similar binary solutions.12,13,15 A more precise critical amplitude B was suggested to be obtained by fitting the experimental data in temperature region of (Tc − T) < 1 K to eq 1 with β = 0.326,13 which is listed in column 4 of Table 5. In a wide temperature range, eq 9 instead of eq 1 should be used to analyze the experimental data: Δρ = |ρ+ − ρ− | = Bt β + B1t β +Δ + ···

a

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

where Δ is a universal exponent. Fixing β = 0.326 and Δ = 0.50 (i.e., their theoretical values), eq 9 was used to fit the experimental results and give the values of B and B1, which are summarized in Table 6. The difference of the mole fractions of the two coexisting phases, Δx, may be described by eq 10 according to the crossover theory proposed by Gutkowski et al.16 in a wide temperature range:

Table 4. Refractive Indexes n at Wavelength λ = 632.8 nm for {x C6H5CN + (1 − x) CH3 (CH2)11CH3} at Pressure p = 0.1 MPa and T = 295.84 Ka x

n

x

n

x

n

0 0.3192 0.6113 0.9009

1.4231 1.4390 1.4629 1.5032

0.1148 0.3935 0.7078 1.0000

1.4280 1.4443 1.4738 1.5239

0.2007 0.5000 0.8007

1.4323 1.4525 1.4865

(9)

Δx = B0 (tY (2β− 1)/ Δ)1/2 + at 2xc

a

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

(10)

where a and B0 are constant coefficients; Y is the crossover function expressed as: 2481



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Table 5. Values of Critical Amplitude B and Critical Exponent β in eq 1 for Coexistence Curves of (T, n), (T, x), and (T, ϕ) for {x C6H5CN + (1 − x) CH3(CH2)11CH3}. The Numbers in the Bracket Were Fixed in the Fits (Tc − T) < 1 K

(Tc − T) < 1 K

(Tc − T) < 10 K

order parameter

B

β

B

β

B

β

n x ϕ

0.157 ± 0.003 1.55 ± 0.03 1.54 ± 0.02

0.326 ± 0.002 0.328 ± 0.002 0.325 ± 0.002

0.1575 ± 0.0004 1.533 ± 0.004 1.545 ± 0.004

(0.326) (0.326) (0.326)

0.166 ± 0.001 1.68 ± 0.01 1.62 ± 0.01

0.334 ± 0.001 0.339 ± 0.002 0.333 ± 0.001

Table 6. Values of Critical Amplitudes B and B1 in eq 9 for Coexistence Curves of (T, n), (T, x), and (T, ϕ) for {x C6H5CN + (1− x) CH3(CH2)11CH3} order parameter

B

B1

n x ϕ

0.1566 ± 0.0004 1.516 ± 0.004 1.538 ± 0.004

0.034 ± 0.003 0.57 ± 0.03 0.29 ± 0.03

1/2 ⎡ ⎛ Λ ⎞2 ⎤ 1 − (1 − u ̅ )Y = u ̅ ⎢1 + ⎜ ⎟ ⎥ Y ν / Δ ⎝κ⎠ ⎦ ⎣

(11)

κ 2 = cttY (2ν− 1)/ Δ

(12)

Figure 2. Plot of log(x1 − x2)/(2xc) against log(t) for the order parameter x of {x C6H5CN + (1 − x) CH3(CH2)11CH3}. The points are from experimental values. The line is calculated from eq 10.

with u̅ and Λ being the crossover parameters. The terms on the right side of eq 10 are the singular term and the regular term in sequence. It has been pointed out16 that the regular term should be considered in the temperature region t > 10−2. Additionally, it should be noted that the term t3β should be also considered for Δx in the distance far away from the critical point in the complete scaling theory;9 however, since it is indistinguishable from the linear term in eq 10 in practice, we ignored the t3β term in the analysis. In principle, the values of four parameters u,̅ c0.5 t /Λ, B0, and a can be obtained simultaneously by fitting the experimental data with eq 10; however, the correlations among these parameters were so strong that large errors in determination of them were resulted from the fits. Following the way we used previously,13 the value of coupling constant u̅ was set between 0 and 3 with an increment of 0.1, and at each setting value the experimental data of Δx were fitted to eq 10 with c0.5 t /Λ, B0, and a as adjusting variables. The value of u̅ with the smallest standard deviation of the fit was adopted as the optimal one, and then the corresponding values of c0.5 t /Λ, B0, and a were obtained, which were u̅ = 0.6, c0.5 t /Λ = 2.74, B0 = 2.09, and a = −0.73. Figure 2 shows the good agreement between the experimental results and the calculated values of Δx/(2xc) through eq 10, which indicates that the crossover model can well represent the coexistence curve (T, x) of {benzonitrile + tridecane}. The diameter of the coexistence curve can be described by: +

ρd =

with Z being an apparent exponent. Taking D and ρc as adjustable parameters and fixing Z at 1, or 1 − α = 0.89, or 2β = 0.652, we fitted the experimental results of ρd to eq 14 separately. The fitting results including the standard deviations S of fits are listed and compared in Table 7. The experimental values listed in line 2 of Table 7 were obtained from extrapolation of refractive indexes in the one-phase region to the critical temperature for nc, determined with the method of “equal volume” for xc, and calculated by eqs 7 and 8 for ϕc, respectively. The uncertainties of the parameters listed in Table 7 do not include the systematic ones resulted from conversions Table 7. Parameters of eq 14 and the Standard Deviations S in ρd for Diameters of Coexistence Curves of (T, n), (T, x), and (T, ϕ) for {x C6H5CN + (1 − x) CH3(CH2)11CH3}. ρc(expt.) Is the Experimental Critical Value of the Order Parameter; nc Were Obtained by Extrapolating Refractive Indexes against Temperatures in the One-Phase Region to the Critical Temperature; The Experimental Values of xc and ϕc Were Determined by the Technique “Equal Volume” and Calculated by Using Equations 7 and 8 (T, n) ρc(expt.)

−

ρ +ρ = ρc + D1t + D2t 1 − α + D3t 2β + ··· 2

(13)

ρc D S

where ρc is the value of the diameter at the critical point; D1, D2, and D3 are the coefficients. The singular term t2β is a direct inference from the complete scaling theory,8−10,17 while it was attributed to an incorrect choice of the order parameter before.1,18 It is difficult to obtain the coefficients D1, D2, and D3 simultaneously by fit of the experimental data with eq 13; thus the diameter ρd was fitted to an alternative expression: ρ + + ρ− = ρc + Dt Z ρd = 2

ρc D S ρc D S

(14) 2482

1.4633 ± 0.0001

(T, x)

0.613 ± 0.001 Z=1 1.4633 ± 0.0001 0.610 ± 0.001 0.188 ± 0.001 −1.51 ± 0.06 5.5·10−5 2.5·10−3 Z = 2β = 0.652 1.4629 ± 0.0001 0.613 ± 0.001 0.053 ± 0.002 −0.441 ± 0.003 2.5·10−4 4.0·10−4 Z = 1 − α = 0.89 1.4632 ± 0.0001 0.611 ± 0.001 0.126 ± 0.001 −1.03 ± 0.03 5.5·10−5 1.9·10−3

(T, ϕ) 0.401 ± 0.002 0.401 ± 0.001 0.41 ± 0.01 4.2·10−4 0.401 ± 0.001 0.118 ± 0.003 5.2·10−4 0.401 ± 0.001 0.279 ± 0.005 3.5·10−4



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of n to x and x to ϕ. It was estimated that these uncertainties were about ± 0.002 for x and ϕ. Thus the critical values of x and ϕ given by fits of eq 14 are in reasonable agreement with the experimental observations. The values of S indicate that the fit with Z = 2β is better than that with Z = 1 − α for the order parameters x, while the goodness of fits with Z = 2β, Z = 1− α, and Z = 1 for order parameter ϕ are nearly equal. These results show that the presence of term t2β is the consequence of the formulizm of the complete scaling; however, the significance of the 1 − α term somewhat depends on the choice of the order parameter. The combination of eqs 9 and 14 yields: ρ− = ρc + Dt Z + (1/2)Bt β + (1/2)B1t β +Δ

(15)

ρ+ = ρc + Dt Z − (1/2)Bt β − (1/2)B1t β +Δ

(16)

x D̂2 2 a1 x2 2 = − ̂ 1 − a1x 2, c (B0 )

with a1 being the asymmetric coefficient. The value of a1 was calculated through eq 21 and found to be −0.97 ± 0.02. As pointed previously,10,19 the asymmetric coefficient a1 may also be estimated from a simple relation in the incompressible liquid mixtures: a1 ≈ 1 −

V01,0

x 2+ + x 2− 2

− ⎞ x x ⎛ Â ≈ x 2,c + D̂2 2 t 2β + D1̂ 2 ⎜ 0 t 1 − α + Bcr̂ t ⎟ ⎝1 − α ⎠

Δxcxc

(17)

where x2 is the mole fraction of the component with larger molar volume, that is, tridecane in this work; D̂ x22, D̂ x12, B̂ x02, B̂ x12, and B̂ x22 are the amplitudes; B̂ cr = BcrVc/R is the dimensionless critical background of the heat capacity and Â −0 = A−0 Vc/αR is the dimensionless critical amplitude of the heat capacity in the two-phase region, where Vc and R are the critical volume of the solution and the gas constant, respectively. As it was pointed out previously,10,19 the contribution of term B̂ x02tβ was dominant in eq 18, and the assumption of B̂ x12 = B̂ x22 = 0 only caused little changes in B̂ x02. Perez-Sanchez et al.9 pointed that the net effect of the contributions proportional to terms t1−α and t in eq 17, which have opposite sign, appears to be negligible for x2,d. Thus, we neglected the term D̂ x12[(Â −0 /(1 − α))t1−α + B̂ crt] in eq 17 and terms B̂ x12tΔ and B̂ x22t2β in eq 18. Equations 17 and 18 may be simplified as:

Δxcxc =

x 2+

− 2

x 2−

(23)

Table 8 lists the values of τ calculated by eq 24. 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:20 τ=

(19)

2 π 3 ⎛ ∂n2 ⎞ kBTχf (c) ⎜ ⎟ λ 04 ⎝ ∂ϕ ⎠

f (c ) =

x

≈ B0̂ 2 t β

(22)

V02,0

for each temperature, where the constant L = 0.680 is the length of the light path. In this way, the influences of the light reflected and absorbed by the water bath were eliminated, but the reflection, the scattering, and the adsorption from the cell and the solution in the cell were not corrected. Therefore the so-called background contribution τb was introduced to account for these effects. The parameter τb was determined at 310.0 K, 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 values of total turbidity τT: τ = τT − τb (24)

(18)

x 2,d

0 V1,c

1 ⎛I ⎞ τT = − ln⎜ 1 ⎟ L ⎝ I2 ⎠

x + − x 2− x x x = 2 ≈ B0̂ 2 t β(1 + B1̂ 2 t Δ + B2̂ 2 t 2β + ···) 2

x + + x 2− x = 2 ≈ x 2,c + D̂2 2 t 2β 2

0 V 2,c

where and are the molar volumes of benzonitrile and tridecane in the consolute point. The calculated value of a1 by eq 22 was −1.37. The value of a1 obtained from eq 21 with the values of D̂ x22 and B̂ x02 from fitting the simplified eqs 19 and 20 is significantly different from that calculated from eq 22. One reasonable explanation may be that the heat capacity related term D̂ x12[(Â −0 / (1 − α))t1−α + B̂ crt] should be considered in eq 17, which calls for precise heat capacity data in both the critical region and the region far away from the critical point. A critical binary solution of {benzonitrile + tridecane} was prepared, and the critical temperature was remeasured, which was 295.6 ± 0.2 K; the transmitted light intensities I1 and I2 corresponding to the sample cell in and out of the laser beam at various temperatures in one-phase region were then measured. The total turbidities τT were calculated by:

Fixing Z, β, and Δ at 0.652, 0.326, and 0.5, and taking the values of D, ρc, B, and B1 listed in Tables 6 and 7, ρ+, ρ−, and ρd were calculated through eqs 14 to 16. Figure 1 shows the accordance of the calculated results represented by the solid lines with the experimental ones. The “width” and “diameter” of coexistence curve (T, x) can be described by the following expressions derived from the “complete scaling theory”: x 2,d =

(21)

(20)

(2c 2 + 2c + 1)ln(1 + 2c) 2(1 + c) − 3 c c2

(25)

(26)

with c = 2(2nπξ/λ0) . In eqs 25 and 26, λ0 is the wavelength of light in vacuum; ϕ is the volume fraction of benzonitrile; kB is the Boltzmann’s constant; the values of the refractive index n and (∂n2/∂ϕ) of the solution were computed from the relation of the refractive index with the composition and temperature obtained in the coexistence curve measurement. 2

The experimental coexistence curve (T, x) data were fitted to x x eqs 19 and 20 to obtain the values of x2,c, D̂ 22, and B̂ 02, which were 0.387 ± 0.001, 0.440 ± 0.003, and 0.789 ± 0.003, x x respectively. The values of B̂ 02 and D̂ 22 can be related by eq 21 according to the complete scaling theory: 2483



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Table 8. Turbidity τ of the Critical Solution {x C6H5CN + (1 − x) CH3(CH2)11CH3} with the Critical Composition x = 0.613 at the Wavelength λ = 632.8 nm and at Various Temperatures and p = 0.1 MPa. Tc = 295.601 Ka (T − Tc)/K

τ/cm−1

(T − Tc)/K

τ/cm−1

0.025 0.042 0.056 0.066 0.077 0.092 0.106 0.127 0.153 0.180

0.672 0.497 0.416 0.368 0.333 0.292 0.260 0.220 0.186 0.160

0.211 0.246 0.289 0.337 0.388 0.453 0.522 0.622 0.753 0.955

0.138 0.118 0.101 0.086 0.077 0.066 0.054 0.045 0.033 0.024

Figure 4. Plot of τ against t for {x C6H5CN + (1 − x) CH3(CH2)11CH3}. The points are the experimental data. The solid line is calculated by eq 25.

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

a

The well-known two-scale-factor universality relating Bϕ, ξ0, and χ0 has the form:

To avoid the large uncertainties in simultaneous determination of the values of ξ0, χ0, ν, and γ due to their strong coupling, we first determined the critical exponent γ by analyzing the turbidity data in the temperature range of t > 1·10−3 where the simple power law τ = τ0t−γ is valid with τ0 = (8/3)(π3/λ40)(∂n2/∂ϕ)2kBTχ0. Thus the plot of log(τ) − log(t) should yield a straight line as shown in Figure 3. A linear least-

⎛ (B /2)2 ⎞1/3 ϕ ⎟⎟ R = ξ0⎜⎜ k ⎝ BTcχ0 ⎠

(27)

The value of Bϕ listed in row 5 of Table 5 and the values of ξ0 and χ0 reported above were used to calculate R by eq 27, which was 0.66 ± 0.01 and well agreed with 0.65 from series expansions21 or 0.67 from ε expansions in a renormalizationgroup setting.22

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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 and 21173080). Notes

The authors declare no competing financial interest.

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Figure 3. Plot of log(τ) against log(t) for {x C6H5CN + (1 − x) CH3(CH2)11CH3} in the temperature range of 1·10−3 < t < 4·10−3. The points are the experimental data. The solid line represents a linear fit of the experimental data.

REFERENCES

(1) Greer, S. C.; Moldover, M. R. Thermodynamic Anomalies at Critical Points of Fluids. Annu. Rev. Phys. Chem. 1981, 32, 233−265. (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) Guida, R.; Zinn-Justin, J. Critical Exponents of the N-Vector Model. J. Phys. A 1998, 31, 8103−8121. (4) Pelissetto, A.; Vicari, E. Critical Phenomena and Renormolization-Group Theory. Phys. Rep. 2002, 368, 549−727. (5) Sengers, J. V.; Shanks, J. G. Experimental Critical-Exponent Values for Fluids. J. Stat. Phys. 2009, 137, 857−877. (6) Fisher, M. E.; Orkoulas, G. The Yang-Yang Anomaly in Fluid Criticality: Experiment and Scaling Theory. Phys. Rev. Lett. 2000, 85, 696−699. (7) Orkoulas, G.; Fisher, M. E.; Ustun, C. The Yang-Yang Relation and the Specific Heats of Propane and Carbon Dioxide. J. Chem. Phys. 2000, 113, 7530−7545. (8) Kim, Y. C.; Fisher, M. E.; Orkoulas, G. Asymmetric Fluid Criticality. I. Scaling with Pressure Mixing. Phys. Rev. E 2003, 67, 061506. (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.

squares fit gave the value of critical exponent γ, which was 1.25 ± 0.05, in good agreement with 1.239 of 3D-Ising value. Subsequently, all of the turbidity data were fitted to eq 25 with γ = 1.239 to determine ν, which was 0.62 ± 0.01 and in accordance with the value of 0.630 predicted for 3D-Ising universality class. To test the two-scale-factor universality, a precise determination of χ0 and ξ0 is required. Therefore, the fit of the turbidity data to eq 25 was carried out again with γ = 1.239 and ν = 0.63 to determine the values of χ0 and ξ0, which were (1.21 ± 0.01)·10−8 m3·J−1 and (0.288 ± 0.003) nm, respectively. The turbidity values calculated from eq 25 using the optimal values of the critical amplitudes χ0 and ξ0 are shown in Figure 4 as solid line, where the points represent the experimental results. It can be seen from Figure 4 that the experimental values and the calculated ones are in good agreement. 2484



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(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) 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. (12) Huang, M. J.; Xiang, Y. S.; Chen, Z. Y.; Cai, L.; An, X. Q.; Shen, W. G. Liquid-Liquid Coexistence Curves of Dimethyl Carbonate + nDecane and Dimethyl Carbonate + n-Tetradecane in the Critical Region. J. Chem. Eng. Data 2009, 54, 1477−1482. (13) Yin, T. X.; Lei, Y. T.; Huang, M. J.; Chen, Z, Y; An, X. Q.; Shen, W. G. Critical Behavior of Binary Mixture of {x C6H5CN + (1−x) CH3(CH2)12CH3}: Measurements of Coexistence Curves, Turbidity, and Heat Capacity. J. Chem. Thermodyn. 2011, 43, 656−663. (14) Dean, J. A. Lange’s Handbook of Chemistry, 15th ed.; McGrawHill: New York, 1998. (15) Mao, C. F.; Wang, N.; Peng, X. H.; An, X. Q.; Shen, W. G. The Measurements of Coexistence Curves and Light Scattering for {x C6H5CN + (1−x) CH3(CH2)10CH3} in the Critical Region. J. Chem. Thermodyn. 2008, 40, 424−430. (16) Gutkowski, K. I.; Anisimov, M. A.; Sengers, J. V. Crossover Criticality in Ionic Solutions. J. Chem. Phys. 2001, 114, 3133−3148. (17) Wang, J. T.; Cerdeirina, C. A.; Anisimov, M. A.; Sengers, J. V. Principle of Isomorphism and Complete Scaling for Binary-Fluid Criticality. Phys. Rev. E 2008, 77, 031127. (18) 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. (19) Huang, M. J.; Chen, Z. Y.; Yin, T. X.; An, X. Q.; Shen, W. G. Critical Behavior of Bianry Mixtures of Dimethyl Carbonate + Nonane and Dimethyl Carbonate + Dodecane: Measurements of the Coexistence Curves. J. Chem. Eng. Data 2011, 56, 2349−2355. (20) Puglielli, G.; Ford, N. C., Jr. Turbidity Measurements in SF6 Near Its Critical Point. Phys. Rev. Lett. 1970, 25, 143−147. (21) 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. (22) Hohenberg, C.; Aharony, A.; Halperin, B. I.; Sigia, E. D. TwoScale-Factor Universality and the Renormalization Group. Phys. Rev. B 1979, 13, 2986−2996.

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