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Correlation of the solubility of low-volatile organic compounds in near and supercritical fluids
Fluid Phase Equilibria 194–197 (2002) 469–482
Correlation of the solubility of low-volatile organic compounds in near and supercritical fluids Part II. Applications to Disperse Red 60 and two disubstituted anthraquinones Thomas Kraska a,∗ , Kai O. Leonhard a , Dirk Tuma b , Gerhard M. Schneider b a
Institut für Physikalische Chemie, Universität zu Köln, Luxemburger Str. 116, D-50939 Köln, Germany b Ruhr-Universität Bochum, Fakultät für Chemie, Physikalische Chemie, D-44780 Bochum, Germany Received 26 June 2001; accepted 29 August 2001
Abstract In this paper, we combine experimental investigations and a newly developed correlation to elucidate the high-pressure solubility behavior in CO2 and the saturation pressure of three anthraquinone-based disperse dyestuffs. For modeling pVT-data of the supercritical solvent CO2 , an accurate semiempirical equation of state has been developed. Furthermore, this equation of state has been extended to dilute solutions of the low-volatile anthraquinone derivatives by adapting a fugacity approach. The required saturation pressure of the pure solute has been described by a Clausius–Clapeyron type equation with two adjustable parameters. We have compared the saturation pressures deduced from the sequence of solubility isotherms with those reported in the literature. Besides, the consequences of the choice of saturation pressure on our calculations, especially on the coordinates of solubility extremes, are discussed. The results from our model demonstrate a substantial agreement between measured and calculated solubility data, even for a large solvent density region in its entirety. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Solid–fluid equilibria; Method of calculation; Critical state; Spectroscopy; Dispersion dyestuff
1. Introduction An important property for the design of several processes with supercritical solvents is the solubility of the substances treated in a supercritical fluid. In principle, this challenge is a two-fold one, namely both to correlate existing data and the prediction of phase equilibria in regions where experimental data are not yet available. There is still a multivariant pattern of correlations existing, often applicable only within drastically limited systems. The correlation of such experimental data is one of the challenging problems ∗
Corresponding author. Tel.: +49-221-470-4553; fax: +49-221-470-4900. E-mail address: [email protected] (T. Kraska). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 6 5 2 - 5
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in chemical engineering sciences. The hurdles which a theroretician has to take mainly result from the anomalous behavior, fluids exhibit at the critical point. Most equations of state are not able to describe the pVT-behavior in the critical region [1]. With some exceptions, equations of state which are developed especially for the pVT-behavior in critical region are not accurate outside this region. Due to the lack of simple and accurate equations of state, empirical relationships are often employed for the correlation of solubility data. One of the most popular approaches is the Chrastil equation [2]. We might categorize this model “most popular”, because at present (June 2001) a citation search in the Science Citation Index ExpandedTM gives 148 hits, even 153 if misspelled page and volume numbers are included. It is derived on the basis of phenomenological properties. The solution process is described by two steps: one is the evaporation of the solute molecules from the solid phase, the second is the dissolution of the solute molecules by the solvent molecules. The vapor pressure is modeled by a Clausius–Clapeyron approach, the dissolution by the mass action law for the association of the solute molecule with k solvent molecules. Then, the temperature dependence of the equilibrium constant of this association process is modeled by a van’t Hoff approach. Finally, the combination of these two processes gives the Chrastil equation for the correlation of solubilities in supercritical fluids. Following the Chrastil equation, the plot of the logarithm of the solubility against the logarithm of the solvent density yields a linear function. Slope and intercept of the linear function are dependent on the enthalpy of sublimation of the solute, the enthalpy of the association process, and additionally on the association number and the temperature. This approach has been applied to the correlation of several solubility data. Due to the particular assumptions, it can only be employed over a limited temperature range and for pressures from 10 MPa to approximately 30 MPa depending on the solute/solvent system. In order to improve the Chrastil equation and to extend the range of applicability, it has been empirically modified by Adachi and Lu [3]. However, such an approach still remains empirical, which can only be used for interpolations, and yet it often gives good results. Furthermore, due to its simplified form, it cannot be used for the determination of derived physical properties, such as the saturation pressures of the solute from solubility data. Chrastil’s approach is limited to that range of the solubility isotherms in which the double logarithmic plot gives a linear function. It is known for many substances that they deviate from this linear behavior with increasing pressure. For example, several substances even exhibit a solubility maximum which cannot be correlated by a simple approach as the Chrastil equation [4]. Since this particular double logarithmic plot implies k as constant, it is obvious that a definite solute–solvent complex must be limited to a restricted density range. In turn, this density range usually covers that of technical interest, nevertheless this simplification becomes insufficient outside where a different solute–solvent complex will be formed.
2. Equation of state A more fundamental—and fastidious—approach is the use of equations of state to correlate solubilities in supercritical solvents. With equations of state the pVT-behavior of fluids can be modeled [5]. The accuracy strongly depends on the equation of state employed for the calculation; another characteristic is that they perform well in definite regions and are less precise in others. Fairly widespread are the cubic equations of state [6–10], which are relatively easy to apply and which can model several properties within a reasonable accuracy.
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Besides the cubic equations of state, a group of physically based molecular model equations exists [11–15]. These equations of state are usually evolved on molecular interactions with the methods of molecular thermodynamics. They are usually more complicated than the cubic equations of state and thus less often applied. Attempts to combine microscopic molecular properties and the mathematical simplicity of the equation of state are currently under way [16–18]. For the accurate modeling of the pVT-behavior of pure substances multiparameter equations have been developed. These equations are purely empirical correlations with 30–60 substance-dependent parameters and must rely on high-precision data. They have been developed for several technically important fluids [19]. While such equations are very useful for the accurate calculation of pure substance properties, their parameters cannot be related to physical properties and hence, the determination of properties of dissolved substances based on binary solubility data is not possible. A problem of cubic equations of state and most molecular model equations is the description of the behavior in the critical region [1]. Cubic equations in general cannot model the critical region because of their simple cubic structure. Non-cubic analytical equations cannot model the critical region accurately, either. However, with increasing mathematical complexity it is possible to improve the description of the pVT-behavior in the critical region very close to the critical point. A fundamental approach to describe the non-analytical behavior at the critical point and the properties away from the critical region is the crossover-approach of Sengers and co-workers [20–22]. It requires a numerical iteration in the evaluation, which makes the application for the calculation of solubility data more complicated. Recently, a semiempirical equation of state has been developed [23] in order to combine a good description of the pVT-behavior in the near-critical region with a mathematical function which neither contains 30–60 substance-dependent parameters nor requires a numerical iteration inside the evaluation. Our approach is based on a classical non-cubic equation of state which is based on molecular interactions. Here, we use the Carnahan–Starling repulsion term [24] combined with a van der Waals-like attraction term [16]. The correction of the deviations which such equations usually show in the critical region is accomplished by a perturbation term which in turn describes the deviation of a local density from the average density. The pertinent mathematical procedure is a convolution of the Taylor expansion of the classical equation of state with a density distribution function. The integration of the expansion terms can be achieved analytically, if they are convoluted by the Gauss distribution function. This convolution generates terms in the power series of which some depend on a new parameter σ describing the width of the density distribution. The resulting terms can be divided into terms which are independent of the σ -parameter and others which depend on the σ -parameter. The latter terms represent the perturbation term, while the first term is the reference equation. p = pref (ρ, T )ρ = pref (ρ, T ) + ppert (ρ, T , σ )
(1)
The perturbation term is a generic expression depending on a parameter σ . In the limit of vanishing σ -value the perturbation term also vanishes. lim ppert (ρ, T , σ ) = 0
σ →0
(2)
The density dependence of σ has been introduced as a semiempirical function [23]. The parameters of this function have been obtained by fitting to experimental and theoretically extrapolated data of the critical isotherm of argon [23]. The noble gas argon has been chosen as reference fluid because of its simple molecular interactions. With scaling parameters this function obtained for argon can be applied to other solvents.
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The resulting expression for the perturbation term contains a double sum over many terms. Unfortunately, this sum cannot be transformed into a closed expression and its evaluation is time-consuming in cases where the calculation of the density for given temperature and pressure is required. Therefore, we have developed a simplified perturbation term in closed form which represents the double sum accurately and so can be handled more easily. The details on this equation of state together with an example of the applicability to real systems is given in part one of this publication series [25]. Ay + By2 + Cy3 RT ppert,rep = (3) Vm (3 − 4y)(1 − c1 y) A = 4.757 × 10−5 σn + 12.22 σn2 − 0.5568 σn3
(4)
B = −0.001692 σn + 47.57 σn2 − 1.886 σn3
(5)
C=
−0.0204 σn + 189.3 σn2 + 434.2 σn3 − 1711 σn4 1 + 2.35 σn
(6)
c1 = 1.424 − 0.1 σn + 8.114 σn2 − 4.916 σn3
(7)
σn ≡ σ Vm /b0
(8)
Here, b0 = 1 dm3 mol−1 is the volume unit. The entire perturbation term corresponding to Eq. (10) in [23] is then ppert = ppert,rep − 8RbT∗ Ψ (T )σ 2
(9)
The remaining unknown function such as the density and temperature dependence of σ or the temperature dependence of the attraction term Ψ (T) can be drawn from the aforementioned paper [23]. At this stage, it is possible to calculate the pVT-data of the supercritical solvent, here carbon dioxide, with these expressions. Referring to the object of our investigations, the extension to mixtures of a solid substance dissolved in the supercritical solvent has been accomplished by a fugacity approach [26]. What we finally want to obtain is the equilibrium solubility of the low-volatile solid in the supercritical solvent as a concentration or mole fraction, respectively. This approach requires two quantities, namely the values for the molar volume of the pure solute, which has been taken from the literature where available or fitted to experimental data instead. The other one, the isothermal compressibility κ T of the solute can be taken from the literature. Alternatively, it has also been used as adjustable. Here, we set it equal to zero due to the lack of experimental data, what is very often the case. For the van der Waals parameters of the equation of state T∗ and b quadratic one-fluid mixing rules have been employed. Throughout this work the Lorentz–Berthelot combining rules are used. Furthermore, the saturation pressure of the pure solute has to be known as the third quantity. However, saturation pressure data have been measured only for a few solids, especially for low-volatile solids. Here, we have chosen an indirect way of estimating the saturation pressures. Based on the correlation of the solubility isotherms for different temperatures, we have adjusted the parameters of a Clausius–Clapeyron type function modeling the saturation pressure of the solute: Asat psat = p0 exp − (10) + Bsat T
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with p0 = 1 MPa. In order to obtain the values for Asat and Bsat from the solubility data we have correlated all isotherms corresponding to different temperatures together. Since Eq. (10) is implemented in the model, the parameters Asat and Bsat can be obtained directly in this way. 3. Experimental section We have applied our model to correlate solubility data of three anthraquinone-based disperse dyestuffs. The particular substances (see Fig. 1) are 1-amino-4-hydroxy-2-phenoxy-9,10-anthraquinone (C20 H13 NO4 , C. I. Disperse Red 60; DR60), 1,4-bis-(propylamino)-9,10-anthraquinone (C20 H22 N2 O2 ; AQ03), and 1,4-bis-(octylamino)-9,10-anthraquinone (C30 H42 N2 O2 ; AQ08); the solvent is CO2 . The particular experimental p,T-conditions for these systems have been the following: 7.2 ≤ p (MPa) ≤ 97.6 and 303.1 ≤ T (K) ≤ 322.5 for DR60, 7.86 ≤ p (MPa) ≤ 18.99 and 305 ≤ T (K) ≤ 340 for AQ03, as well 9.04 ≤ p (MPa) ≤ 19.98 and 310.1 ≤ T (K) ≤ 340.1 for AQ08. For all three substances solubilities have been determined spectrophotometrically, in case of DR60 by a similar static technique, of which a schematic diagram is already published in a paper by Tuma and Schneider [27]. The only difference to the equipment reported there is that a cell with two pairs of sapphire windows (a sectional drawing is given in [28]), thus enlarging the accessible concentration range, has been employed. The technique to investigate the two other substances AQ03 and AQ08 in CO2 has been an adapted supercritical fluid chromatography (SFC) (see Fig. 2). For both methods, equilibrium concentrations, i.e. solubilities s, are calculated from the integral absorbance, being the area below the scanned spectrum.
Fig. 1. Structures of the three disperse dyestuffs investigated: 1-amino-4-hydroxy-2-phenoxy-9,10-anthraquinone (Disperse Red 60, DR60), A; 1,4-bis-(propylamino)-9,10-anthraquinone (AQ03), B; 1,4-bis-(octylamino)-9,10-anthraquinone (AQ08), C.
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Fig. 2. Outline of the modified SFC device for dynamic solubility investigations: A, amplifier; C, column filled with charcoal; CR, cryostat; DAD, diode array detector; DMM, multimeter; E, extraction column; F, flow-through cell; FM, flowmeter; H1, H2, heat exchangers; I, current supply; IN1, IN2, injectors; M, multiplexer; P, pressure gauge; PC1, PC2, computers; PR, printer; PT, platinum resistance thermometer; SP, syringe pump; T1–T3, thermocouples; TH, air thermostat; V1–V7, valves.
Calibration is done both in suitable organic solvents, meaning the integrated molar absorption coefficient Bν˜ being similar to the solvent gas, and in CO2 itself [28]. Prior to all solubility investigations, the dyestuffs have been purified by elution of by-products and impurities with CO2 in the SFC-apparatus [28]. Concerning the static method, an excess amount of solid has to be introduced into the spectroscopic autoclave to ensure saturation at all p,T-conditions investigated; whereas for the dynamic method there the low-volatile solid is brought into the supercritical solvent by continuous elution from a reservoir, being the extraction column E in Fig. 2. Both methods have in common, as they base on spectroscopy, cf. the flow-through cell F in Fig. 2, that equilibrium is achieved when the recorded spectrum remains constant. All data presented have been drawn from the doctoral theses of Tuma for DR60 [29], of Wagner for AQ03 [30], and of Kautz for AQ08 [31], respectively. Concerning a more profound description of experiment, analysis, and pretreatment, the authors refer to these references [4,28,32].
4. Results and discussion 4.1. 1-Amino-4-hydroxy-2-phenoxy-9,10-anthraquinone (DR60) + CO2 In Fig. 3 solubility data obtained experimentally and two correlations are plotted for the system DR60 + CO2 . With increasing temperatures the isotherms at T = 303.1, 314.8, and 322.5 K become less steep, meaning a stronger solubility increase within a given pressure interval. From approximately 40 MPa upwards the increase of solubility diminishes as characterized by the curvature. The mutual intersection of the isotherms at p ≈ 20 MPa does not appear when plotting against density ρ instead. The correlations reproduce the complete course of the measured data satisfactorily. It has become apparent that the correlation is improved by a temperature-dependence of the characteristic temperature
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Fig. 3. Experimental (symbols) and calculated solubilities of DR60 in CO2 . The solid curves correspond to a temperature-dependent parameter set (T∗ , b), whereas the dashed curves result from these two parameters set constant, thus temperature-independent. The parameters are listed in Table 1. The data at T = 393.14 and at 423.15 K are taken from the literature [36].
T∗ and the covolume parameter b for modeling the saturation pressure of DR60. This modification is very useful for the calculation of the solute vapor pressure, however, such empirical extensions should be carried out carefully, especially for extrapolations. All equation of state parameters are listed in Table 1. Several investigations on this particular system have been published by other groups so far [33–37]. Muthukumaran et al. [37] have studied the solubility enhancement upon addition of the polar cosolvents ethanol and acetone at T = 40 and 60 ◦ C and up to p ≈ 34 MPa. Common to all five is that their results have been obtained by a dynamic flow-method. The p,T-range investigated amounts to four isotherms between 15 ≤ p (MPa) ≤ 29 and 333.2 ≤ T (K) ≤ 423.2 for Bae and Her [33], four isotherms covering 10 ≤ p (MPa) ≤ 30 and 323.7 ≤ T (K) ≤ 413.7 for Lee et al. [35], and ultimately five isotherms between 10 ≤ p (MPa) ≤ 34 and 313.15 ≤ T (K) ≤ 423.15 measured by Sung and Shim [36]. Özcan et al. [34], however, have only published one single datapoint for DR60 in CO2 at p = 20 MPa and Table 1 Parameters for the solutes dissolved in supercritical carbon dioxide
DR60 + CO2 ; T = 303.1 K DR60 + CO2 ; T = 314.8 K DR60 + CO2 ; T = 322.5 K DR60 + CO2 ; T = 393.15 K [36] DR60 + CO2 ; T = 423.15 K [36] DR60 + CO2 ; 303.1 ≤ T (K) ≤ 322.5 AQ03 + CO2 ; 305 ≤ T (K) ≤ 340 AQ08 + CO2 ; 310.1 ≤ T (K) ≤ 340.1
T22∗ (K)
b22 (cm3 mol−1 )
Asat (K)
Bsat
Vm,s (cm3 mol−1 )
799.872 790.230 784.632 749.937 794.369 705.714 754.423 758.937
81.7009 84.7281 85.3309 80.5537 80.3058 94.1156 80.5019 127.6910
9200.0 9200.0 9200.0 9200.0 9200.0 7709.73 10423.5 15730.4
5.80788 5.58534 5.83209 5.37428 5.70381 1.25074 11.8918 21.4659
312.0 312.0 312.0 312.0 312.0 364.37 322.4 462.7
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T = 353.15 K, respectively. If we compare these data sets both mutually and to our own results, large discrepancies are noticed, in a manner that our data coincide best with those of Sung and Shim [36], their data being approximately 25% higher than ours (exemplarily our value taken at p = 32.83 MPa and T = 314.8 K). Sung and Shim have applied two solubility-density relations originating from Chrastil’s theory, the s(ρ)-plot in semilog and log–log presentation, to correlate their results, showing good consistency. Additionally, they have optimized their correlation by the introduction of a temperature-dependent parameter. However, correlations of this type prove to be unapplicable for an extended density range, as we have investigated for this system. Due to the significant curvature of the isotherms shown in Fig. 3, both the semilog and the log–log relation between solubility and density do not give straight parallel lines for the solubility isotherms any more [4]. Our equation of state overcomes this shortcoming. It is universally applicable and not restricted to a limited density region where Chrastil’s conceptions are applicable [2]. Nevertheless, Chrastil’s model provides a fast and easy tool for correlating solubilities of low-volatile substances in supercritical fluids in a region roughly between 10 and 30 MPa, where, e.g. supercritical fluid chromatography (SFC), supercritical fluid extraction (SFE), and other applications are normally performed. But any extrapolations into regions especially of high (and also very low) densities must be handled with caution. It follows from experience that just the shape of the s(ρ)-isotherm is the crucial point for the success of an extrapolation. In those density regions where the s(ρ)-isotherm shows an extremum, calculations with Chrastil’s model will fail. The correlation of the solubility by our equation of state [23,25] requires the saturation pressure of the solute. These data can either be taken from literature sources or they have to be estimated indirectly from the solubility isotherms. As shown in Fig. 4, several studies on sublimation and vaporization of DR60 have been published because of its particular use in, e.g. thermal-sublimation printing or textile-dyeing [43–46].
Fig. 4. Comparison of the sublimation and vapor pressure of DR60 obtained from solubility correlations (diamonds, derived from the three isotherms shown in Fig. 3) and from direct measurements. In order to fill the gap between our data at low temperature and the directly measured vapor pressures at high temperature [38–41], we have also calculated the vapor pressure from the solubility data of Sung and Shim [36]. The temperature of fusion T fus = 459 K has been taken from [42].
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Fig. 5. Experimental (symbols) [30] and calculated solubility isotherms for AQ03 in CO2 for the temperatures 305, 310, 315, 320, 330, and 340 K. The corresponding parameters are listed in Table 1, being temperature-independent.
4.2. 1,4-Bis-(propylamino)-9,10-anthraquinone (AQ03) + CO2 Fig. 5 shows the experimental data [30], as well as the calculated curves for the system AQ03 + CO2 . There is a good agreement between both within the investigated p,T-range. In contrast to DR60, also Chrastil’s equation gives remarkably good results [30], mainly since the particular p,T-range (pressure region not exceeding 20 MPa) for this system can comply with the specific requirements concerning validity and applicability. The effect of different saturation pressures on the calculation of solubility isotherms is illustrated in Fig. 6. If a Clausius–Clapeyron type function modeling this quantity is based on fitting the solubility data only (dashed curves in Fig. 6, equivalent to the solid curves in Fig. 5 as the best fit in the low-pressure region and based on a temperature-independent parameter set), measured and calculated isotherms coincide well, whereas our model produces very high solubilities at higher pressure. With the employment of vapor pressure data measured by Hickman et al. [47] extrapolated to much lower temperatures below the melting point (see Fig. 7) the solubility maxima are at realistic solubility values (solid curves in Fig. 6). The difference between both correlations is relatively small in the low pressure region in which the solubility was fitted to experimental data. Moreover, a more accurate correlation for the experimental psat -data is obtained for higher temperatures. The (thermodynamically correct) solubility maxima, which do exist in both cases, are at drastically different solubility and pressure. The p,s-coordinates for the maxima in the first correlation (dashed curves) are located at 150 MPa and 2.0 × 10−3 mol dm−3 for T = 305 K and 220 MPa and 2.2 × 10−2 mol dm−3 for T = 340 K, thus, being unrealistic high. In the second (solid lines) correlation a distinct solubility maximum occurs at p = 20 MPa and s = 2.5 × 10−4 mol dm−3 for T = 305 K and p = 35 MPa and s = 9.7×10−4 mol dm−3 for T = 340 K, respectively. This comparison shows that it is not reliable to use an extrapolation of the vapor pressure to low temperatures instead of using the sublimation pressure representing a solid phase in equilibrium with the solution. The change in the saturation pressure has a significant effect on the solubility at the solubility maximum, however
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Fig. 6. The influence of psat on the calculated solubility isotherms of AQ03 in CO2 for the temperatures 305, 310, 315, 320, 330, and 340 K. Dashed curves: best fit in the low-pressure region with adjusted Clausius–Clapeyron parameters; solid curves: correlation for given experimental vapor pressure data published by Hickman et al. [47] extrapolated to low temperatures below the melting point.
the effect on the low pressure data near the crossing of the isotherms is relatively small. Furthermore, for an overall correlation the data for at least one solubility maximum should be known. If that is not possible, mostly because of experimentally disadvantageous pressure conditions, one should at least be sure of the isotherms’ course, such as eventual changes in their trend, any indications for an extremum, and alternatively an extremum at another temperature. Fig. 7 shows the vapor pressure of AQ03 as obtained by direct measurement [47] and the saturation pressure as obtained here from solubility data at temperatures
Fig. 7. Comparison of the vapor/sublimation pressure of AQ03 obtained from the solubility correlations (dashed line) and direct measurement of Hickman et al. [47] (solid line). Also shown is the sublimation pressure of AQ08 as obtained from the solubility correlation (dot-dashed line). The melting temperatures Tfus of AQ03 and AQ08 are 406 and 349 K, respectively [42].
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Fig. 8. Correlation of solubility isotherms for AQ08 in CO2 for the temperatures 310.1, 320, 330, and 340.1 K [31]. The corresponding parameters are listed in Table 1, likewise temperature-independent.
below Tfus . The general behavior of the coexistence pressure over the complete temperature range is comparable to that of DR60 and several other dyestuffs as reported in the literature [43,44] from direct measurements. Another interesting feature from these calculations is that the maximum is moderately shifted to higher pressures with increasing temperatures. This effect has already been found experimentally for the octadecylamino-substituted member of this series AQ18 + CO2 [48]. 4.3. 1,4-Bis-(octylamino)-9,10-anthraquinone (AQ08) + CO2 The system AQ08+CO2 has been subject to the same procedure as the previous two. The results shown in Fig. 8 are similar to those of AQ03 + CO2 , with the correlations even better. In contrast to AQ03, no vapor pressure data have been published so far for AQ08. That is the reason why only the correlation working with fitted saturation pressures (see Fig. 8) has been executed. For AQ08, however, a considerable amount of solubility data both in different solvents (CO2 , N2 O, CClF3 , SF6 ) and (because of the static technique used) covering a by far broader p,T-region has been published by Tuma and Schneider [27,49]. From using this data fund, we expect a better prediction of the coordinates of the solubility maxima. This work is currently in progress. 5. Conclusions A recently developed equation of state has been successfully applied on the modeling of solubility behavior of low-volatile organic compounds, in this case of three anthraquinone derivatives. Solubilities in a remarkably broad density range of the solvent have been modeled showing congruity between our model and the experiments. It is moreover noteworthy to mention that our model predicts the occurrence
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of solubility extremes regardless whether independently determined or along solubility isotherms fitted saturation pressures are considered. As an additional result, the saturation pressure of the particular solute, i.e. the dyestuff, can be deduced from the solubility isotherms, the results (concretely for DR60) being in good agreement with literature data. List of symbols A, B, C, c1 coefficients in the simplified perturbation term Asat , Bsat equation parameter for the saturation pressure b covolume parameter b0 unit volume Bν˜ integrated molar absorption coefficient based on wavenumbers k association number in Chrastil’s model [2] p pressure p0 unit pressure R ideal gas constant s solubility (concentration of saturated solution) T absolute temperature T∗ characteristic temperature, attraction parameter of the equation of state Tfus melting point at normal pressure V volume y packing fraction Greek letters κT ρ σ Ψ (T)
isothermal compressibility density perturbation parameter of the equation of state temperature dependence of the attraction term
Subscripts m n pert ref rep sat T
molar special notation for the parameter σ perturbation state reference state repulsive state state of saturation temperature dependence
Acknowledgements The authors thank the Deutsche Forschungsgemeinschaft (DFG) and the Fonds der Chemischen Industrie e. V. for financial support. One of us (KOL) wishes to thank the Käthe-Hack-Stiftung for support. Additionally, we thank Professor Jae-Jin Shim from Yeungnam University, Kyongsan, Korea, for providing DR60.
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[44] F. Somm, F. Palacin, Melliand Textilber. 64 (1983) 834–837. [45] N. Harada, T. Suzuki, Jpn. Kokai Tokkyo Koho JP 2000272251 A2, 2000. [46] R.-M. Zhou, Y.-Y. Hu, L. Sun, H.-B. Li, Y.-J. Shen, Huadong Ligong Daxue Xuebao (J. East China Univ. Sci. Tech.) 26 (2000) 438–440. [47] K.C.D. Hickman, J.C. Hecker, N.D. Embree, Ind. Eng. Chem. Anal. Ed. 9 (1937) 264–267. [48] U. Haarhaus, P. Swidersky, G.M. Schneider, J. Supercrit. Fluids 8 (1995) 100–106. [49] D. Tuma, G.M. Schneider, J. Supercrit. Fluids 13 (1998) 37–42.
Correlation of the solubility of low-volatile organic compounds in near and supercritical fluids Part II. Applications to Disperse Red 60 and two disubstituted anthraquinones Thomas Kraska a,∗ , Kai O. Leonhard a , Dirk Tuma b , Gerhard M. Schneider b a
Institut für Physikalische Chemie, Universität zu Köln, Luxemburger Str. 116, D-50939 Köln, Germany b Ruhr-Universität Bochum, Fakultät für Chemie, Physikalische Chemie, D-44780 Bochum, Germany Received 26 June 2001; accepted 29 August 2001
Abstract In this paper, we combine experimental investigations and a newly developed correlation to elucidate the high-pressure solubility behavior in CO2 and the saturation pressure of three anthraquinone-based disperse dyestuffs. For modeling pVT-data of the supercritical solvent CO2 , an accurate semiempirical equation of state has been developed. Furthermore, this equation of state has been extended to dilute solutions of the low-volatile anthraquinone derivatives by adapting a fugacity approach. The required saturation pressure of the pure solute has been described by a Clausius–Clapeyron type equation with two adjustable parameters. We have compared the saturation pressures deduced from the sequence of solubility isotherms with those reported in the literature. Besides, the consequences of the choice of saturation pressure on our calculations, especially on the coordinates of solubility extremes, are discussed. The results from our model demonstrate a substantial agreement between measured and calculated solubility data, even for a large solvent density region in its entirety. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Equation of state; Solid–fluid equilibria; Method of calculation; Critical state; Spectroscopy; Dispersion dyestuff
1. Introduction An important property for the design of several processes with supercritical solvents is the solubility of the substances treated in a supercritical fluid. In principle, this challenge is a two-fold one, namely both to correlate existing data and the prediction of phase equilibria in regions where experimental data are not yet available. There is still a multivariant pattern of correlations existing, often applicable only within drastically limited systems. The correlation of such experimental data is one of the challenging problems ∗
Corresponding author. Tel.: +49-221-470-4553; fax: +49-221-470-4900. E-mail address: [email protected] (T. Kraska). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 6 5 2 - 5
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in chemical engineering sciences. The hurdles which a theroretician has to take mainly result from the anomalous behavior, fluids exhibit at the critical point. Most equations of state are not able to describe the pVT-behavior in the critical region [1]. With some exceptions, equations of state which are developed especially for the pVT-behavior in critical region are not accurate outside this region. Due to the lack of simple and accurate equations of state, empirical relationships are often employed for the correlation of solubility data. One of the most popular approaches is the Chrastil equation [2]. We might categorize this model “most popular”, because at present (June 2001) a citation search in the Science Citation Index ExpandedTM gives 148 hits, even 153 if misspelled page and volume numbers are included. It is derived on the basis of phenomenological properties. The solution process is described by two steps: one is the evaporation of the solute molecules from the solid phase, the second is the dissolution of the solute molecules by the solvent molecules. The vapor pressure is modeled by a Clausius–Clapeyron approach, the dissolution by the mass action law for the association of the solute molecule with k solvent molecules. Then, the temperature dependence of the equilibrium constant of this association process is modeled by a van’t Hoff approach. Finally, the combination of these two processes gives the Chrastil equation for the correlation of solubilities in supercritical fluids. Following the Chrastil equation, the plot of the logarithm of the solubility against the logarithm of the solvent density yields a linear function. Slope and intercept of the linear function are dependent on the enthalpy of sublimation of the solute, the enthalpy of the association process, and additionally on the association number and the temperature. This approach has been applied to the correlation of several solubility data. Due to the particular assumptions, it can only be employed over a limited temperature range and for pressures from 10 MPa to approximately 30 MPa depending on the solute/solvent system. In order to improve the Chrastil equation and to extend the range of applicability, it has been empirically modified by Adachi and Lu [3]. However, such an approach still remains empirical, which can only be used for interpolations, and yet it often gives good results. Furthermore, due to its simplified form, it cannot be used for the determination of derived physical properties, such as the saturation pressures of the solute from solubility data. Chrastil’s approach is limited to that range of the solubility isotherms in which the double logarithmic plot gives a linear function. It is known for many substances that they deviate from this linear behavior with increasing pressure. For example, several substances even exhibit a solubility maximum which cannot be correlated by a simple approach as the Chrastil equation [4]. Since this particular double logarithmic plot implies k as constant, it is obvious that a definite solute–solvent complex must be limited to a restricted density range. In turn, this density range usually covers that of technical interest, nevertheless this simplification becomes insufficient outside where a different solute–solvent complex will be formed.
2. Equation of state A more fundamental—and fastidious—approach is the use of equations of state to correlate solubilities in supercritical solvents. With equations of state the pVT-behavior of fluids can be modeled [5]. The accuracy strongly depends on the equation of state employed for the calculation; another characteristic is that they perform well in definite regions and are less precise in others. Fairly widespread are the cubic equations of state [6–10], which are relatively easy to apply and which can model several properties within a reasonable accuracy.
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Besides the cubic equations of state, a group of physically based molecular model equations exists [11–15]. These equations of state are usually evolved on molecular interactions with the methods of molecular thermodynamics. They are usually more complicated than the cubic equations of state and thus less often applied. Attempts to combine microscopic molecular properties and the mathematical simplicity of the equation of state are currently under way [16–18]. For the accurate modeling of the pVT-behavior of pure substances multiparameter equations have been developed. These equations are purely empirical correlations with 30–60 substance-dependent parameters and must rely on high-precision data. They have been developed for several technically important fluids [19]. While such equations are very useful for the accurate calculation of pure substance properties, their parameters cannot be related to physical properties and hence, the determination of properties of dissolved substances based on binary solubility data is not possible. A problem of cubic equations of state and most molecular model equations is the description of the behavior in the critical region [1]. Cubic equations in general cannot model the critical region because of their simple cubic structure. Non-cubic analytical equations cannot model the critical region accurately, either. However, with increasing mathematical complexity it is possible to improve the description of the pVT-behavior in the critical region very close to the critical point. A fundamental approach to describe the non-analytical behavior at the critical point and the properties away from the critical region is the crossover-approach of Sengers and co-workers [20–22]. It requires a numerical iteration in the evaluation, which makes the application for the calculation of solubility data more complicated. Recently, a semiempirical equation of state has been developed [23] in order to combine a good description of the pVT-behavior in the near-critical region with a mathematical function which neither contains 30–60 substance-dependent parameters nor requires a numerical iteration inside the evaluation. Our approach is based on a classical non-cubic equation of state which is based on molecular interactions. Here, we use the Carnahan–Starling repulsion term [24] combined with a van der Waals-like attraction term [16]. The correction of the deviations which such equations usually show in the critical region is accomplished by a perturbation term which in turn describes the deviation of a local density from the average density. The pertinent mathematical procedure is a convolution of the Taylor expansion of the classical equation of state with a density distribution function. The integration of the expansion terms can be achieved analytically, if they are convoluted by the Gauss distribution function. This convolution generates terms in the power series of which some depend on a new parameter σ describing the width of the density distribution. The resulting terms can be divided into terms which are independent of the σ -parameter and others which depend on the σ -parameter. The latter terms represent the perturbation term, while the first term is the reference equation. p = pref (ρ, T )ρ = pref (ρ, T ) + ppert (ρ, T , σ )
(1)
The perturbation term is a generic expression depending on a parameter σ . In the limit of vanishing σ -value the perturbation term also vanishes. lim ppert (ρ, T , σ ) = 0
σ →0
(2)
The density dependence of σ has been introduced as a semiempirical function [23]. The parameters of this function have been obtained by fitting to experimental and theoretically extrapolated data of the critical isotherm of argon [23]. The noble gas argon has been chosen as reference fluid because of its simple molecular interactions. With scaling parameters this function obtained for argon can be applied to other solvents.
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The resulting expression for the perturbation term contains a double sum over many terms. Unfortunately, this sum cannot be transformed into a closed expression and its evaluation is time-consuming in cases where the calculation of the density for given temperature and pressure is required. Therefore, we have developed a simplified perturbation term in closed form which represents the double sum accurately and so can be handled more easily. The details on this equation of state together with an example of the applicability to real systems is given in part one of this publication series [25]. Ay + By2 + Cy3 RT ppert,rep = (3) Vm (3 − 4y)(1 − c1 y) A = 4.757 × 10−5 σn + 12.22 σn2 − 0.5568 σn3
(4)
B = −0.001692 σn + 47.57 σn2 − 1.886 σn3
(5)
C=
−0.0204 σn + 189.3 σn2 + 434.2 σn3 − 1711 σn4 1 + 2.35 σn
(6)
c1 = 1.424 − 0.1 σn + 8.114 σn2 − 4.916 σn3
(7)
σn ≡ σ Vm /b0
(8)
Here, b0 = 1 dm3 mol−1 is the volume unit. The entire perturbation term corresponding to Eq. (10) in [23] is then ppert = ppert,rep − 8RbT∗ Ψ (T )σ 2
(9)
The remaining unknown function such as the density and temperature dependence of σ or the temperature dependence of the attraction term Ψ (T) can be drawn from the aforementioned paper [23]. At this stage, it is possible to calculate the pVT-data of the supercritical solvent, here carbon dioxide, with these expressions. Referring to the object of our investigations, the extension to mixtures of a solid substance dissolved in the supercritical solvent has been accomplished by a fugacity approach [26]. What we finally want to obtain is the equilibrium solubility of the low-volatile solid in the supercritical solvent as a concentration or mole fraction, respectively. This approach requires two quantities, namely the values for the molar volume of the pure solute, which has been taken from the literature where available or fitted to experimental data instead. The other one, the isothermal compressibility κ T of the solute can be taken from the literature. Alternatively, it has also been used as adjustable. Here, we set it equal to zero due to the lack of experimental data, what is very often the case. For the van der Waals parameters of the equation of state T∗ and b quadratic one-fluid mixing rules have been employed. Throughout this work the Lorentz–Berthelot combining rules are used. Furthermore, the saturation pressure of the pure solute has to be known as the third quantity. However, saturation pressure data have been measured only for a few solids, especially for low-volatile solids. Here, we have chosen an indirect way of estimating the saturation pressures. Based on the correlation of the solubility isotherms for different temperatures, we have adjusted the parameters of a Clausius–Clapeyron type function modeling the saturation pressure of the solute: Asat psat = p0 exp − (10) + Bsat T
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with p0 = 1 MPa. In order to obtain the values for Asat and Bsat from the solubility data we have correlated all isotherms corresponding to different temperatures together. Since Eq. (10) is implemented in the model, the parameters Asat and Bsat can be obtained directly in this way. 3. Experimental section We have applied our model to correlate solubility data of three anthraquinone-based disperse dyestuffs. The particular substances (see Fig. 1) are 1-amino-4-hydroxy-2-phenoxy-9,10-anthraquinone (C20 H13 NO4 , C. I. Disperse Red 60; DR60), 1,4-bis-(propylamino)-9,10-anthraquinone (C20 H22 N2 O2 ; AQ03), and 1,4-bis-(octylamino)-9,10-anthraquinone (C30 H42 N2 O2 ; AQ08); the solvent is CO2 . The particular experimental p,T-conditions for these systems have been the following: 7.2 ≤ p (MPa) ≤ 97.6 and 303.1 ≤ T (K) ≤ 322.5 for DR60, 7.86 ≤ p (MPa) ≤ 18.99 and 305 ≤ T (K) ≤ 340 for AQ03, as well 9.04 ≤ p (MPa) ≤ 19.98 and 310.1 ≤ T (K) ≤ 340.1 for AQ08. For all three substances solubilities have been determined spectrophotometrically, in case of DR60 by a similar static technique, of which a schematic diagram is already published in a paper by Tuma and Schneider [27]. The only difference to the equipment reported there is that a cell with two pairs of sapphire windows (a sectional drawing is given in [28]), thus enlarging the accessible concentration range, has been employed. The technique to investigate the two other substances AQ03 and AQ08 in CO2 has been an adapted supercritical fluid chromatography (SFC) (see Fig. 2). For both methods, equilibrium concentrations, i.e. solubilities s, are calculated from the integral absorbance, being the area below the scanned spectrum.
Fig. 1. Structures of the three disperse dyestuffs investigated: 1-amino-4-hydroxy-2-phenoxy-9,10-anthraquinone (Disperse Red 60, DR60), A; 1,4-bis-(propylamino)-9,10-anthraquinone (AQ03), B; 1,4-bis-(octylamino)-9,10-anthraquinone (AQ08), C.
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Fig. 2. Outline of the modified SFC device for dynamic solubility investigations: A, amplifier; C, column filled with charcoal; CR, cryostat; DAD, diode array detector; DMM, multimeter; E, extraction column; F, flow-through cell; FM, flowmeter; H1, H2, heat exchangers; I, current supply; IN1, IN2, injectors; M, multiplexer; P, pressure gauge; PC1, PC2, computers; PR, printer; PT, platinum resistance thermometer; SP, syringe pump; T1–T3, thermocouples; TH, air thermostat; V1–V7, valves.
Calibration is done both in suitable organic solvents, meaning the integrated molar absorption coefficient Bν˜ being similar to the solvent gas, and in CO2 itself [28]. Prior to all solubility investigations, the dyestuffs have been purified by elution of by-products and impurities with CO2 in the SFC-apparatus [28]. Concerning the static method, an excess amount of solid has to be introduced into the spectroscopic autoclave to ensure saturation at all p,T-conditions investigated; whereas for the dynamic method there the low-volatile solid is brought into the supercritical solvent by continuous elution from a reservoir, being the extraction column E in Fig. 2. Both methods have in common, as they base on spectroscopy, cf. the flow-through cell F in Fig. 2, that equilibrium is achieved when the recorded spectrum remains constant. All data presented have been drawn from the doctoral theses of Tuma for DR60 [29], of Wagner for AQ03 [30], and of Kautz for AQ08 [31], respectively. Concerning a more profound description of experiment, analysis, and pretreatment, the authors refer to these references [4,28,32].
4. Results and discussion 4.1. 1-Amino-4-hydroxy-2-phenoxy-9,10-anthraquinone (DR60) + CO2 In Fig. 3 solubility data obtained experimentally and two correlations are plotted for the system DR60 + CO2 . With increasing temperatures the isotherms at T = 303.1, 314.8, and 322.5 K become less steep, meaning a stronger solubility increase within a given pressure interval. From approximately 40 MPa upwards the increase of solubility diminishes as characterized by the curvature. The mutual intersection of the isotherms at p ≈ 20 MPa does not appear when plotting against density ρ instead. The correlations reproduce the complete course of the measured data satisfactorily. It has become apparent that the correlation is improved by a temperature-dependence of the characteristic temperature
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Fig. 3. Experimental (symbols) and calculated solubilities of DR60 in CO2 . The solid curves correspond to a temperature-dependent parameter set (T∗ , b), whereas the dashed curves result from these two parameters set constant, thus temperature-independent. The parameters are listed in Table 1. The data at T = 393.14 and at 423.15 K are taken from the literature [36].
T∗ and the covolume parameter b for modeling the saturation pressure of DR60. This modification is very useful for the calculation of the solute vapor pressure, however, such empirical extensions should be carried out carefully, especially for extrapolations. All equation of state parameters are listed in Table 1. Several investigations on this particular system have been published by other groups so far [33–37]. Muthukumaran et al. [37] have studied the solubility enhancement upon addition of the polar cosolvents ethanol and acetone at T = 40 and 60 ◦ C and up to p ≈ 34 MPa. Common to all five is that their results have been obtained by a dynamic flow-method. The p,T-range investigated amounts to four isotherms between 15 ≤ p (MPa) ≤ 29 and 333.2 ≤ T (K) ≤ 423.2 for Bae and Her [33], four isotherms covering 10 ≤ p (MPa) ≤ 30 and 323.7 ≤ T (K) ≤ 413.7 for Lee et al. [35], and ultimately five isotherms between 10 ≤ p (MPa) ≤ 34 and 313.15 ≤ T (K) ≤ 423.15 measured by Sung and Shim [36]. Özcan et al. [34], however, have only published one single datapoint for DR60 in CO2 at p = 20 MPa and Table 1 Parameters for the solutes dissolved in supercritical carbon dioxide
DR60 + CO2 ; T = 303.1 K DR60 + CO2 ; T = 314.8 K DR60 + CO2 ; T = 322.5 K DR60 + CO2 ; T = 393.15 K [36] DR60 + CO2 ; T = 423.15 K [36] DR60 + CO2 ; 303.1 ≤ T (K) ≤ 322.5 AQ03 + CO2 ; 305 ≤ T (K) ≤ 340 AQ08 + CO2 ; 310.1 ≤ T (K) ≤ 340.1
T22∗ (K)
b22 (cm3 mol−1 )
Asat (K)
Bsat
Vm,s (cm3 mol−1 )
799.872 790.230 784.632 749.937 794.369 705.714 754.423 758.937
81.7009 84.7281 85.3309 80.5537 80.3058 94.1156 80.5019 127.6910
9200.0 9200.0 9200.0 9200.0 9200.0 7709.73 10423.5 15730.4
5.80788 5.58534 5.83209 5.37428 5.70381 1.25074 11.8918 21.4659
312.0 312.0 312.0 312.0 312.0 364.37 322.4 462.7
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T = 353.15 K, respectively. If we compare these data sets both mutually and to our own results, large discrepancies are noticed, in a manner that our data coincide best with those of Sung and Shim [36], their data being approximately 25% higher than ours (exemplarily our value taken at p = 32.83 MPa and T = 314.8 K). Sung and Shim have applied two solubility-density relations originating from Chrastil’s theory, the s(ρ)-plot in semilog and log–log presentation, to correlate their results, showing good consistency. Additionally, they have optimized their correlation by the introduction of a temperature-dependent parameter. However, correlations of this type prove to be unapplicable for an extended density range, as we have investigated for this system. Due to the significant curvature of the isotherms shown in Fig. 3, both the semilog and the log–log relation between solubility and density do not give straight parallel lines for the solubility isotherms any more [4]. Our equation of state overcomes this shortcoming. It is universally applicable and not restricted to a limited density region where Chrastil’s conceptions are applicable [2]. Nevertheless, Chrastil’s model provides a fast and easy tool for correlating solubilities of low-volatile substances in supercritical fluids in a region roughly between 10 and 30 MPa, where, e.g. supercritical fluid chromatography (SFC), supercritical fluid extraction (SFE), and other applications are normally performed. But any extrapolations into regions especially of high (and also very low) densities must be handled with caution. It follows from experience that just the shape of the s(ρ)-isotherm is the crucial point for the success of an extrapolation. In those density regions where the s(ρ)-isotherm shows an extremum, calculations with Chrastil’s model will fail. The correlation of the solubility by our equation of state [23,25] requires the saturation pressure of the solute. These data can either be taken from literature sources or they have to be estimated indirectly from the solubility isotherms. As shown in Fig. 4, several studies on sublimation and vaporization of DR60 have been published because of its particular use in, e.g. thermal-sublimation printing or textile-dyeing [43–46].
Fig. 4. Comparison of the sublimation and vapor pressure of DR60 obtained from solubility correlations (diamonds, derived from the three isotherms shown in Fig. 3) and from direct measurements. In order to fill the gap between our data at low temperature and the directly measured vapor pressures at high temperature [38–41], we have also calculated the vapor pressure from the solubility data of Sung and Shim [36]. The temperature of fusion T fus = 459 K has been taken from [42].
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Fig. 5. Experimental (symbols) [30] and calculated solubility isotherms for AQ03 in CO2 for the temperatures 305, 310, 315, 320, 330, and 340 K. The corresponding parameters are listed in Table 1, being temperature-independent.
4.2. 1,4-Bis-(propylamino)-9,10-anthraquinone (AQ03) + CO2 Fig. 5 shows the experimental data [30], as well as the calculated curves for the system AQ03 + CO2 . There is a good agreement between both within the investigated p,T-range. In contrast to DR60, also Chrastil’s equation gives remarkably good results [30], mainly since the particular p,T-range (pressure region not exceeding 20 MPa) for this system can comply with the specific requirements concerning validity and applicability. The effect of different saturation pressures on the calculation of solubility isotherms is illustrated in Fig. 6. If a Clausius–Clapeyron type function modeling this quantity is based on fitting the solubility data only (dashed curves in Fig. 6, equivalent to the solid curves in Fig. 5 as the best fit in the low-pressure region and based on a temperature-independent parameter set), measured and calculated isotherms coincide well, whereas our model produces very high solubilities at higher pressure. With the employment of vapor pressure data measured by Hickman et al. [47] extrapolated to much lower temperatures below the melting point (see Fig. 7) the solubility maxima are at realistic solubility values (solid curves in Fig. 6). The difference between both correlations is relatively small in the low pressure region in which the solubility was fitted to experimental data. Moreover, a more accurate correlation for the experimental psat -data is obtained for higher temperatures. The (thermodynamically correct) solubility maxima, which do exist in both cases, are at drastically different solubility and pressure. The p,s-coordinates for the maxima in the first correlation (dashed curves) are located at 150 MPa and 2.0 × 10−3 mol dm−3 for T = 305 K and 220 MPa and 2.2 × 10−2 mol dm−3 for T = 340 K, thus, being unrealistic high. In the second (solid lines) correlation a distinct solubility maximum occurs at p = 20 MPa and s = 2.5 × 10−4 mol dm−3 for T = 305 K and p = 35 MPa and s = 9.7×10−4 mol dm−3 for T = 340 K, respectively. This comparison shows that it is not reliable to use an extrapolation of the vapor pressure to low temperatures instead of using the sublimation pressure representing a solid phase in equilibrium with the solution. The change in the saturation pressure has a significant effect on the solubility at the solubility maximum, however
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Fig. 6. The influence of psat on the calculated solubility isotherms of AQ03 in CO2 for the temperatures 305, 310, 315, 320, 330, and 340 K. Dashed curves: best fit in the low-pressure region with adjusted Clausius–Clapeyron parameters; solid curves: correlation for given experimental vapor pressure data published by Hickman et al. [47] extrapolated to low temperatures below the melting point.
the effect on the low pressure data near the crossing of the isotherms is relatively small. Furthermore, for an overall correlation the data for at least one solubility maximum should be known. If that is not possible, mostly because of experimentally disadvantageous pressure conditions, one should at least be sure of the isotherms’ course, such as eventual changes in their trend, any indications for an extremum, and alternatively an extremum at another temperature. Fig. 7 shows the vapor pressure of AQ03 as obtained by direct measurement [47] and the saturation pressure as obtained here from solubility data at temperatures
Fig. 7. Comparison of the vapor/sublimation pressure of AQ03 obtained from the solubility correlations (dashed line) and direct measurement of Hickman et al. [47] (solid line). Also shown is the sublimation pressure of AQ08 as obtained from the solubility correlation (dot-dashed line). The melting temperatures Tfus of AQ03 and AQ08 are 406 and 349 K, respectively [42].
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Fig. 8. Correlation of solubility isotherms for AQ08 in CO2 for the temperatures 310.1, 320, 330, and 340.1 K [31]. The corresponding parameters are listed in Table 1, likewise temperature-independent.
below Tfus . The general behavior of the coexistence pressure over the complete temperature range is comparable to that of DR60 and several other dyestuffs as reported in the literature [43,44] from direct measurements. Another interesting feature from these calculations is that the maximum is moderately shifted to higher pressures with increasing temperatures. This effect has already been found experimentally for the octadecylamino-substituted member of this series AQ18 + CO2 [48]. 4.3. 1,4-Bis-(octylamino)-9,10-anthraquinone (AQ08) + CO2 The system AQ08+CO2 has been subject to the same procedure as the previous two. The results shown in Fig. 8 are similar to those of AQ03 + CO2 , with the correlations even better. In contrast to AQ03, no vapor pressure data have been published so far for AQ08. That is the reason why only the correlation working with fitted saturation pressures (see Fig. 8) has been executed. For AQ08, however, a considerable amount of solubility data both in different solvents (CO2 , N2 O, CClF3 , SF6 ) and (because of the static technique used) covering a by far broader p,T-region has been published by Tuma and Schneider [27,49]. From using this data fund, we expect a better prediction of the coordinates of the solubility maxima. This work is currently in progress. 5. Conclusions A recently developed equation of state has been successfully applied on the modeling of solubility behavior of low-volatile organic compounds, in this case of three anthraquinone derivatives. Solubilities in a remarkably broad density range of the solvent have been modeled showing congruity between our model and the experiments. It is moreover noteworthy to mention that our model predicts the occurrence
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of solubility extremes regardless whether independently determined or along solubility isotherms fitted saturation pressures are considered. As an additional result, the saturation pressure of the particular solute, i.e. the dyestuff, can be deduced from the solubility isotherms, the results (concretely for DR60) being in good agreement with literature data. List of symbols A, B, C, c1 coefficients in the simplified perturbation term Asat , Bsat equation parameter for the saturation pressure b covolume parameter b0 unit volume Bν˜ integrated molar absorption coefficient based on wavenumbers k association number in Chrastil’s model [2] p pressure p0 unit pressure R ideal gas constant s solubility (concentration of saturated solution) T absolute temperature T∗ characteristic temperature, attraction parameter of the equation of state Tfus melting point at normal pressure V volume y packing fraction Greek letters κT ρ σ Ψ (T)
isothermal compressibility density perturbation parameter of the equation of state temperature dependence of the attraction term
Subscripts m n pert ref rep sat T
molar special notation for the parameter σ perturbation state reference state repulsive state state of saturation temperature dependence
Acknowledgements The authors thank the Deutsche Forschungsgemeinschaft (DFG) and the Fonds der Chemischen Industrie e. V. for financial support. One of us (KOL) wishes to thank the Käthe-Hack-Stiftung for support. Additionally, we thank Professor Jae-Jin Shim from Yeungnam University, Kyongsan, Korea, for providing DR60.
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