Critical behavior of binary mixture of {(1 − x) C6H5CN + x CH3(CH2)9CH3}: Measurements of coexistence curves, turbidity, and heat capacity

doi:10.1016/j.jct.2012.04.018

The Journal of Chemical Thermodynamics

Volume 53, October 2012, Pages 42-51

https://doi.org/10.1016/j.jct.2012.04.018 Get rights and content

Abstract

(Liquid + liquid) coexistence curve, turbidity, and isobaric heat capacity per unit volume for the critical solution of {benzonitrile + n-undecane} have been measured. The critical exponents β, ν, γ, and α have been deduced, which were found to be consistent with the theoretic predictions. Meanwhile, the experimental data have also been analyzed to obtain the system-dependent critical amplitudes B, ξ₀, χ₀, A^±, and D corresponding to the difference of the general density variable of two coexisting phases Δρ, the correlation length ξ, the osmotic compressibility χ, the isobaric heat capacity per unit volume C_pV⁻¹, and the first term of correction-to-scaling for the isobaric heat capacity per unit volume, which were used to test some universal ratios. It was found that the coexistence curve may be well described by the crossover model proposed by Gutkowski et al. The critical-fluctuation induced contribution to the background heat capacity B_cr was obtained and used to analyze the asymmetric behavior of the diameter of the coexistence curve. The result indicated that the asymmetry of the coexistence curve can be well described by the complete scaling theory proposed by Anisimov et al., and the heat capacity does make a significant contribution to this asymmetric behavior.

Highlights

► Coexistence curve, isobaric heat capacity and turbidity measurements have been reported. ► Asymmetry of the coexistence curves has been analyzed by the complete scaling theory. ► Heat capacity has been shown to be important in describing the asymmetric criticality. ► Universal amplitude ratios have been tested.

Introduction

The presence of large critical fluctuations in the close vicinity of critical point makes the system-dependent microscopic details insignificant; and the anomalous thermodynamic properties of different systems may be described with the same scaling functions depended on the universality class. As commonly accepted, the fluids, fluid mixtures and uniaxial ferromagnetic materials belong to the universality class of 3D-Ising-like systems [1], [2], i.e., systems with short-range interactions and a scalar order parameter.

In the immediate region of critical point, various thermodynamic properties such as the difference of the general density variables of two coexisting phases Δρ, the correlation length ξ, the osmotic compressibility χ, and the isobaric heat capacity per unit volume C_pV⁻¹ show power-law dependences described as follows [1], [2]: $Δ ρ = | ρ_{1} - ρ_{2} | = {Bt}^{β},$ $ξ = ξ_{0} t^{- ν},$ $χ = χ_{0} t^{- γ},$ $C_{p} V^{- 1} = (A^{\pm} / α) | t |^{- α},$ where t is the reduced temperature ( $t = | T - T_{c} | / T_{c},$ T_c is the critical temperature); ρ is the general density variable and the subscripts 1 and 2 denote each of the two coexisting phases; α, β, ν, and γ are the critical exponents for 3D-Ising universality class which were well developed from both theoretical calculations and experiments [1], [2], [3], [4], [5]; B, ξ₀, χ₀, and A^± are the critical amplitudes corresponding to the difference of the general density variables of two coexisting phases, the correlation length, the osmotic compressibility, and the isobaric heat capacity per unit volume with “+” or “−” denoting the one-phase or the two-phase region.

However, in a wide temperature range, corrections of simple power-law dependences should be taken and equations (1), (4) may be rewritten as: $| ρ_{1} - ρ_{2} | = {Bt}^{β} + B_{1} t^{β + Δ} + \dots,$ $C_{p} V^{- 1} = C_{p 0} + Et + (A^{\pm} / α) | t |^{- α} (1 + D | t |^{Δ} + \dots),$ where Δ is the universal correction-to-scaling exponent; B₁ and D are the amplitudes of the first terms of correction-to-scaling; the background term $C_{p 0}$ is the sum of the regular background heat capacity at the non-critical state B_bg and the critical-fluctuation induced contribution to the background heat capacity B_cr, $C_{p 0} = B_{bg} + B_{cr}$ ; the value of D and the value of C_p₀ were taken to be the same in the one-phase and the two-phase region [6]; Et is the non-critical linear term arising from the regular part of the free energy, which should be the same in both sides of the critical point.

B, ξ₀, χ₀, A^±, D, and B_cr are all system-dependent amplitudes, however, some ratios of them were predicted to be universal [1], [2], [6], [7], [8], [9] and required further verification by more precise experimental results.

The asymmetric criticality has been a long question and attracted much attention in recent decade accompanied by the raise of the concept “the complete scaling” proposed by Fisher and co-workers [10], [11], [12], which implies that the scaling fields should be the mix of all dependent and independent physical fields. It was concluded that the singularity of the diameter of the coexistence curve was a consequence of the complete scaling. Anisimov et al. showed that the complete scaling theory could be well applied in weekly compressible liquid mixtures [13]. Moreover, as we indicated recently [14], the contribution of the heat capacity does play an important role in describing the asymmetric criticality within the frame of the complete scaling theory.

As a part of the continuous investigations on the critical behavior of the binary mixtures, in this paper, we report the measurements of the (liquid + liquid) coexistence curve, turbidity, and isobaric heat capacity per unit volume for the binary solution of {benzonitrile + n-undecane}. The experimental results have been analyzed to obtain the values of the amplitudes B, ξ₀, χ₀, A^±, D, and B_cr to test some universal ratios. The asymmetry of the coexistence curve in the critical region has also been analyzed through the complete scaling theory, which indicates the significance of heat capacity in description of the asymmetry of the coexistence curve and verification of the complete scaling theory.

Section snippets

Chemicals

Table 1 lists the purities and suppliers of the chemicals used in this work.

Coexistence curve

The critical mole fraction of the binary solution of (benzonitrile + n-undecane) was determined by adjusting the proportion of the two components to achieve “equal volume” of the two phases at the phase-separation temperature [15], [16], [17]. A solution with the critical composition was carefully prepared in a rectangular fluorescence cell provided with an Ace-thread connection. The sample cell was then placed in a water

Coexistence curves

The critical mole fraction and the critical temperature of {(1 − x) benzontrile + x n-undecane} were determined to be x_c = 0.441 ± 0.001 and T_c = 290.1 ± 0.2 K, respectively, where x is the mole fraction of n-undecane and the subscript c denotes the critical value. The measured refractive indexes n for each coexisting phases are listed in columns 2 and 3 of table 2. They are also shown in figure 1a as the plot of temperature against refractive index, and denoted as the (T, n) coexistence curve.

The refractive

Analysis of the asymmetric behavior of the coexistence curve with complete scaling theory

The complete scaling theory has been successfully applied to explain the singularity of the diameter of the coexistence curves both in one-component fluids [33] and binary fluid systems [13], [14], [23]. The ‘width’ and ‘diameter’ of coexistence curve could be expressed by the following equations derived from the “complete scaling theory” as: $Z_{d} \equiv \frac{Z^{+} + Z^{-}}{2} \approx Z_{c} + {\hat{D}}_{2}^{Z} | t |^{2 β} + {\hat{D}}_{1}^{Z} (\frac{{\hat{A}}_{0}^{-}}{1 - α} | t |^{1 - α} + {\hat{B}}_{cr} | t |),$ $Δ Z_{cxc} \equiv \frac{Z^{+} - Z^{-}}{2} \approx {\hat{B}}_{0}^{Z} | t |^{β} (1 + {\hat{B}}_{1}^{z} | t |^{Δ} + {\hat{B}}_{2}^{z} | t |^{2 β}),$ ${\hat{B}}_{0}^{x} = (1 - a_{1} x_{c}) B_{0} | τ_{0} |^{β}, {\hat{B}}_{0}^{\hat{ρ}} = (a_{1} + a_{3}) B_{0} | τ_{0} |^{β}, {\hat{B}}_{0}^{\hat{ρ} x} = (1 + a_{3} x_{c}) B_{0} | τ$

Acknowledgements

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

References (46)

A. Pelissetto et al.
Phys. Rep.
(2002)
A.J. Liu et al.
Physica A
(1989)
T. Yin et al.
J. Chem. Thermodyn.
(2011)
Y. Lei et al.
J. Chem. Thermodyn.
(2010)
C.A. Cerdeirina et al.
Thermochim. Acta
(2000)
C. Bagnuls et al.
Phys. Lett. A
(1985)
M.A. Anisimov et al.
Physica A
(1992)
S.C. Greer et al.
Ann. Rev. Phys. Chem.
(1981)
J.V. Sengers et al.
Sengers
Ann. Rev. Phys. Chem.
(1986)
R. Guida et al.
J. Phys. A
(1998)

J.V. Sengers et al.

J. Stat. Phys.

(2009)

C. Bagnuls et al.

Phys. Rev. B

(1987)

M.E. Fisher et al.

J. Phys. A

(1998)

M. Campostrini et al.

Phys. Rev. E

(1999)

M.E. Fisher et al.

Phys. Rev. Lett.

(2000)

G. Orkoulas et al.

J. Chem. Phys.

(2000)

Y.C. Kim et al.

Phys. Rev. E

(2003)

G. Perez-Sanchez et al.

J. Chem. Phys.

(2010)

M. Huang et al.

J. Phys. Chem. B

(2011)

W. Shen et al.

J. Phys. Chem.

(1991)

X. An et al.

J. Chem. Thermodyn.

(1993)

M. Huang et al.

J. Chem. Eng. Data

(2011)

J.A. Dean

Lange’s Handbook of Chemistry

(1998)

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¹: Present address: Department of Basic Science, China Pharmaceutical University, Nanjing, Jiangsu 210009, China.

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Critical behavior of binary mixture of {(1 − x) C6H5CN + x CH3(CH2)9CH3}: Measurements of coexistence curves, turbidity, and heat capacity

Abstract

Highlights

Introduction

Section snippets

Chemicals

Coexistence curve

Coexistence curves

Analysis of the asymmetric behavior of the coexistence curve with complete scaling theory

Acknowledgements

Phys. Rep.

Physica A

J. Chem. Thermodyn.

J. Chem. Thermodyn.

Thermochim. Acta

Phys. Lett. A

Physica A

Ann. Rev. Phys. Chem.

Sengers

Ann. Rev. Phys. Chem.

J. Phys. A

J. Stat. Phys.

Phys. Rev. B

J. Phys. A

Phys. Rev. E

Phys. Rev. Lett.

J. Chem. Phys.

Phys. Rev. E

J. Chem. Phys.

J. Phys. Chem. B

J. Phys. Chem.

J. Chem. Thermodyn.

J. Chem. Eng. Data

Lange’s Handbook of Chemistry

Critical behavior of binary mixture of {(1 − x) C₆H₅CN + x CH₃(CH₂)₉CH₃}: Measurements of coexistence curves, turbidity, and heat capacity