<>  <>Abstract Sublimation pressure from solubility data of solids in different supercritical solvents is determined and solvent effects are discussed. Tree binary gas-solid systems containing anthracene, napthalene and phenanthrene as solutes and carbon dioxide, ethane, fluoroform as solvents are considered in the study. The Peng-Robinson equation of state with the mixing rules proposed by Wong and Sandler are used to evaluate the fugacity coefficient f^s that appears in the classical solubility equation which relates the mole fraction of a dissolved solid in a compressed gas phase: y=(P^sub/Pf^s)[exp{(V^s(P-P^sub)/RT}]. The sublimation pressure P^sub of the solid is considered as a parameter to be determined by regression analysis of experimental solubility data (TPy), and theoretically it should be independent of the solvent. Thus, an optimization problem, in which the difference between correlated and experimental solubility is to be minimize, is solved using a method based on genetic algorithms. The results show that the determination of P^sub using high pressure solubility data is reliable and produce unique values of sublimation pressure independent of the solvent present in the mixtures.

  <>Introduction

 

Phase equilibrium calculations of a solid dissolved in a compressed gas, at a pressure P and a temperature T can be performed using the fundamental equation of phase equilibria which leads to a simple equation that relates the solubility y the pressure P, and the temperature T. The equation also contains other properties such as the sublimation pressure P^sub , the molar volume of the solid V^s and the fugacity coefficient of the solid component in the high pressure gas f^s [1]. Of all these properties involved in the calculation of the solubility of the solid in the high pressure gas, the sublimation pressure has received low attention in the literature, although it is directly related to the solubility. The molar volume does not have a strong influence on the calculations and the fugacity coefficient f^s can be estimated from an appropriate equation of state and mixing rules. The sublimation pressure is usually small for common industrially important solids and in many cases available experimental techniques cannot be used to obtain accurate values. In theory, the calculated sublimation pressures should be independent of the solvent used, but this statement must be proved. In this article the influence of the type of solvent in which the solid is dissolved on the calculation of P^sub, is analyzed. For this, the sublimation pressure of anthracene, naphthalene and phenanthrene in solvents of different nature (carbon dioxide, ethane and fluoroform), are determined to prove what in theory seems to be clear. Solubility Calculations The theory of solid solubility in a compressed gas is found in standard books [1, 2]. If subscript 2 stand for the solid component in the gas-solid mixture, the so-called solubility equation is: ₍₁₎ Here, is the sublimation pressure of the pure solid, is the solid molar volume, all at the temperature T and j₂ is the fugacity coefficient of solid (2) at the pressure P. The fugacity coefficient is calculated from standard thermodynamic relations [2]. The Peng-Robinson equation (PR) [3], with the mixing rules proposed by Wong and Sandler [4], are used as the thermodynamic model to evaluate the fugacity coefficient j₂. The excess Gibbs free energy that appears in the Wong-Sandler mixing rule (WS) is represented by the van Laar model (VL). Therefore, the model contains three parameters (the van Laar constants A₁₂ and A₂₁ and the k₁₂ parameter included in the mixing rule). This model is designed here as PR/WS/VL and the expressions for the fugacity coefficient can be found elsewhere [1]. The problem is then reduced here to determine the three parameters of the thermodynamic model and the sublimation pressure that appears in the solubility equation (1), using available high pressure TPy data for gas-solid systems.

 

Binary system containing anthracene, napthalene and phenanthrene as solute in various solvents (carbon dioxide, ethane, fluoroform) were considered in the study. The data consisted of 30 isotherms and a total of 381 PTy data points.

  <>Results and Discussion The accuracy of the model and the method used to calculate the sublimation pressure is evaluated by considering the deviations between experimental [5-8] and calculated values of the solubility of the solid in the high pressure gas. Literature information indicate that deviations are less than 10% using the PR/WS/VL model [9], deviations that were found of the same order in this study. The Table 1 shows the results obtained using PR/WS/VL model for gas-solid systems considered in this study. The calculated sublimation pressure and the deviations with respect to experimental data are given. For systems which contain naphthalene as solute, the relative deviations vary from -0.22% to 7.5%. For the systems which contain phenanthrene as solute, the relative deviations vary from 0.09% to -6.8% and for systems which contain anthrace as solute, the relative deviations vary from -0.05 % to 7.7%.

 

Fig. 1 shows sublimation pressure calculated from napthalene for all isotherms with different solvents. The symbols in the Fig. are (*) CO₂ for T= 308, 328, 333 and 338 K; and (D) ethane for T= 318 K. The values for fluoroform are not shown in the figure since the isotherms and the calculated values coincide with those of the system naphthalene + CO₂ and naphthalene + ethane. In the figure, sublimation pressure is in bar and T in Kelvin. The results shown in Fig. 1 indicate that the proposed method is reliable enough to estimate the sublimation pressure of solids using solubility data of the solid in high pressure gases. The solvent present in the mixture shows to be irrelevant in determining P^sub as expected. Table 1: Sublimation pressures of the solid solutes at all temperatures studied and relative deviations between calculated and experimental values. N is the number of experimental data.

 

 

Fig. 1: Calculated sublimation pressure of Naphthalene.: (*) CO₂ for T= 308, 328, 333 and 338 K; and (D) ethane for T= 318 K. (·) experimental sublimation pressure.

  <>Conclusions Sublimation pressure of solid from high pressure solubility data are calculated using genetic algorithms. Based on the results and discussion presented in this study, the following main conclusions are obtained: i) the low deviations between experimental and calculated values of the sublimation pressure show that the thermodynamic model PR/WS/VL is appropriate to model these systems and to estimate the sublimation pressure of solids. ii) the genetic algorithms method has shown to be a good tool to solve the optimization problem studied here, providing accurate global optima. Acknowledgements

 

The authors thanks the support of the National Commission for Scientific and Technological Research (CONICYT-Chile), through the research grants FONDECYT 1050410 and 1040285. JOV also thanks the Direction of Research of the University of La Serena-Chile for permanent support through several research grants and the Center for Technological Information (CIT, La Serena-Chile), for computer and library support. CAF thank the Dept. of Physics of the University of Concepción-Chile for special support that made possible the preparation of this work.

 

References

 

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