Peng-Robinson EOS와 회합이론에 의한 용해도 거동의 열역학적 해석

성환도, 이광희, 심재진

영남대학교 응용화학공학부

Thermodynamic Analysis of Solubility Behavior

by Peng-Robinson EOS and Association Theory

Hwan-Do Sung, Kwang-Hee Lee, Jae-Jin Shim

School of Chemical Engineering and Technology, Yeungnam University

Introduction

Chemical theorem suggests that several molecules associate with each other to form a large conglomerate, which behaves like a single molecule. Chemical theorem has been known to represent well the VLE data of the associating molecules. Anderko (1989) proposed the Association Equation of State (AEOS) for polar substances such as ethanol, propanol, and their mixtures, which can associate with each other[1]. Alsten et al. (1993) suggested a chemical model to represent cosolvent effects on the solid solubility in supercritical fluids[2]. But,  their models can hardly estimate the fugacity coefficients of the associating molecules. Based on the chemical theorem we derived a new equation representing the effects of association on dye solubility in supercritical fluids. We correlated the experimental data to estimate the cosolvent effect on dye solubility.

Theory

Solid solubility in pure supercritical fluids has, in general, a linear relationship with density when it was plotted in a logarithmic or semilogarithmic plot[3]. This linear relationship between the logarithm of solubility and the logarithm of density (or density itself) can be modified to include the temperature effect on the solubility as in the following equation.

[eqn: ln { y}_{ i}{} ={}a{} ln□+{}b{}+{} { c} over {T }]                (1)

[eqn: ln { y}_{ i}{} ={}a' {}□+{}b'{}+ {}{ c'} over {T }]                 (2)

Both equations fitted the experimental solubility data well[3]. The slope of the logarithm of solid solubility versus logarithm of density may be expressed thermodynamically as follows:

[eqn: LEFT({partial ln y_i } over { partial ln rho }RIGHT)_T,y_i = {v_i ^s - bar{v_i } } over {RT K_T }]                  (3)

where yi is the solid solubility, [eqn: v_i ^s] is the solid molar volume, [eqn: bar{v_i }] is the partial molar volume of the solid, KT is the compressibility factor. The partial molar volume can be expressed as a pressure-dependent property.

[eqn: bar{v_i} {}={} K_T V{} LEFT({partial P}over{partialn_i}RIGHT)_T,n_j]                  (4)

The cosolvent effect results from the physical interaction between  the solid and the solvent and the associative interaction between the solid molecules due to the specific forces by hydrogen bonding. We define these contributions as the "physical interaction" and the "quasi-chemical interaction". It would be better to express the cosolvent effect by the physical interaction term. The fugacity coefficient,[eqn: psi_i], derived from the Helmholtz energy may be separated directly as a physical interaction term and a quasi-chemical interaction term as follows:

[eqn: ln psi_i {} ={} ln psi_i^phy + lnpsi_i^chm]                 (5)

To derive an equation for the enhancement of solid solubility by with the quasi-chemical term for the system of CO2(1)-dye(2)-cosolvent(3), we assumed an equilibrium between the solute and cosolvent molecules and between cosolvent molecules. another assumption is that carbon dioxide does not take part in the reaction.

[eqn: A + iB {} LRARROW {}AB_i]                      (6)

where A is a solute and B is a cosolvent molecule. The equilibrium constant at each state may be defined as

[eqn: K_23^i {} ={} {epsilon_AB_i barrho }over {( epsilon_AB_i-1 bar rho ) (epsilon_B_1 bar rho ) }]                 (7)

where the subscript "2" is solute; "3" is cosolvent; "i" is the number of association; K23 is the equilibrium constant between the solute and cosolvent; [eqn: epsilon _A_1] is the fraction of solute that exists as a monomer A; [eqn: epsilon _B_1] is the fraction of cosolvent that exists as monomer B; and [eqn: bar rho] is the true molecular density ([eqn: n_T]/N). The enhanced solubility in supercritical fluids is then expressed as follows:

[eqn: y_2 {}={} y_2^phy left[1+ SUM from { {i }=1} to i (K_23 y_B_1 rho)^i right]]               (8)

Results and Discussion

In our previous study[4], we measured the solubilities of C.I. Disperse Red 60 and C.I. Disperse Blue 60 in supercritical carbon dioxide[3] and the enhanced solubility of C.I. Disperse Red 60 in supercritical carbon dioxide using either ethanol or acetone as a cosolvent. In this study we measured the enhanced solubility of C.I. Disperse Blue 60 at the same condition. While the solubility of Red 60 enhanced about 200% by adding 4.3 mol% ethanol, that of Blue 60 enhanced about 150%, as shown in Figures 1 and 2. For Blue 60 the solubility enhancement was described well with the physical interaction term(Figure 1). The solubility enhancement for Red 60, however, could not be explained only with the physical term. With both the physical and quasi-chemical interaction terms, the solubility enhancement fitted the experimental data well(Figure 2).

Figure 1. Solubility of C.I. Disperse Blue 60 in supercritical carbon dioxide with 4.3 mol % ethanol. -- solubility calculated with the physical interaction term, … solubility in pure carbon dioxide.		Figure 2. Solubility of C.I. Disperse Red 60 in supercritical carbon dioxide with 4.3 mol % ethanol. ― solubility calculated with both physical and quasi- chemical terms, -- solubility with the physical interaction term, … solubility in pure carbon dioxide.

Figures 3 and 4 show the enhanced solubility by adding 4.3 mol% acetone as a cosolvent. The solubility enhancement with either acetone or ethanol shows similar behavior for each disperse dye. Red 60 forms strong hydrogen bonding with cosolvents, enhancing the dye solubility, while Blue 60 does not. This is due to the difference in the molecular structure of disperse dyes. Red 60 has a hydroxyl group and an amine group that can form the strong hydrogen bonding with ethanol and acetone. Blue 60 does not have any functional group leading to hydrogen bonding.

Figure 3. Solubility of C.I. Disperse Blue 60 in supercritical carbon dioxide with 4.3 mol % acetone. -- solubility calculated with the physical interaction term, … solubility in pure carbon dioxide.		Figure 4. Solubility of C.I. Disperse Red 60 in supercritical carbon dioxide with 4.3 mol % acetone. ― solubility calculated with both physical and quasi- chemical terms, -- solubility with the physical interaction term, … solubility in pure carbon dioxide.

Acknowledgments

This research was supported by R&D Management Center for Energy and Resources (1999-Save-96). We thank the Dyestuffs Research Center of LG Chemical for the disperse dye samples.

References

[1] Anderko, A.  "A  Simple Equation of State incorporating Association", Fluid Phase Equil., 45, 39-67 (1989).

[2] Alsten, V.; Eckert, C. A.  "Effect of Entrainers and of Solute Size and Polarity in Supercritical Fluid Solutions", J. Chem. Eng. Data, 38, 605-610 (1993).

[3] Sung, H.-D.; Shim, J.-J. "Solubility of C.I. Disperse Red 60 and C.I. Disperse Blue 60 in Supercritical Carbon Dioxide", J. Chem. Eng. Data, 44(5), 985-989 (1999).

[4] Muthukumaran, P.; Gupta, R. B.; Sung, H.-D.; Shim, J.-J., Bae, H. K. "Dye Solubility in Supercritical Carbon Dioxide. Effect of Hydrogen Bonding with Cosolvents", Korean J. Chem. Eng., 16, 111-117 (1999).