Correlation of Solid Solubility for Biological Compounds in Supercritical Carbon Dioxide: Comparative Study Using Solution Model and Other Approaches Jaw-Shin Cheng, Muoi Tang and Yan-Ping Chen * Department of Chemical Engineering National Taiwan University Taipei, Taiwan, ROC

Abstract Solid solubility of biological compounds in supercritical carbon dioxide was calculated using the solution model approach. These solutes include steroids, antioxidants, xanthines and drugs. The regular solution model coupled with the Flory-Huggins equation was employed. The molar volume of the solute in the supercritical phase was optimally fitted. With two parameters in the solution model approach, satisfactory results were obtained which are comparable to those calculated from the equation of state method or semi- empirical equations with more adjustable parameters.

Calculation of Solid Solubility  The solubility of a solid solute (component 2) in a supercritical fluid (component 1) is:  The modified regular solution model coupled with the Flory-Huggins equation is applied to represent the limiting activity coefficient of the solid solute.

 The solid solubility is expressed as:  The value of  1 is calculated from the Peng-Robinson equation of state (PR EOS).  The value of  2 is calculated from  U = internal energy change of vaporization  Molar volume v 1 is determined from EOS. Molar volume v 2 is regressed in this study.  For comparative studies, solid solubility is also calculated by EOS or semi-empirical equations.

Results and Discussion  The relationship between the molar volume of the solid solute in the supercritical fluid phase and the pure solvent density is shown in Fig. 1. A simple relation is given as:  Two methods are applied in the solution model approach: Two-Parameter Method ( ,  are optimally fitted) One-Parameter Method (  is optimally fitted,  is fixed)

 An overall AAD of solid solubility calculations is 16.48% using two parameters in the solution model approach. With the single parameter, the AAD for all systems increases to 24.64%, which is still in acceptable tolerance for complex systems.  The correlated solubilities from the solution model are compared with the experimental data for DDT and piroxicam. Good agreement is obtained as shown in Figs. 2 and 3.  When the value of  is fixed, it is shown in Fig. 4 that the optimally-fitted value  is linearly correlated with the internal energy change of vaporization of the solute. The generalized correlation is:

 With this generalized equation, about half of the calculated systems have an AAD less than 50%. The overall calculated deviation using the generalized correlation for all solid systems in this study is 56%.  The calculated results of stigmasterol using this generalized correlation are illustrated in Fig. 5 with acceptable accuracy. Fig. 6 shows the predicted results of antibiotic penicillin G at different temperatures with the generalized correlation.  Comparison of the typical calculated results using the solution model is shown in Table 1. Comparison of the calculated results from various approaches is shown in Table 2. The solution model generally gives comparably good performance with less parameters.

Fig. 1. Relationship between the solid molar volume of caffeine in the supercritical phase and the pure solvent density.

Fig. 2. Comparison of the experimental and calculated solid solubility of DDT from the solution model.

Fig. 3. Comparison of the experimental and calculated solid solubility of piroxicam from the solution model.

Fig. 4. Relationship between the single parameter  of the solution model and the internal energy change of vaporization of the solute.

Fig. 5. Comparison of the experimental and calculated solid solubility of stigmasterol from the generalized correlation of solution model.

Fig. 6. Comparison of the experimental and predicted solid solubility of penicillin G from the generalized correlation of solution model.

, NP is the number of data points Table 1. Typical correlation results of the solid solubility using the solution model method

Table 2. Comparison of the overall calculated deviations for biological compounds from various methods I: Solution model with two adjustable parameters  and  II: Solution model with a single adjustable parameter  III: 3-parameter semi-empirical model of Méndez-Santiago and Teja IV: 4-parameter semi-empirical model of Jiang et al. V: PR EOS with 4 parameters VI: PR EOS with 2 parameters

Conclusion  The solubility of complex biological compounds in supercritical carbon dioxide is correlated with acceptable accuracy by the simple solution model which requires less adjustable parameters than either EOS approaches or semi- empirical equations.  The parameters in the solution model are generalized with the pure solid property. Feasible prediction is illustrated with only estimated properties of pure solid.