(60b) Sublimation Pressure from Solubility Data of Solids in Supercritical Solvents

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^subas 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

[1] J.M.
Prausnitz, R.N. Lichtenthaler, E. Gomes de Azevedo, Molecular Thermodynamics of
Fluid-Phase Equilibria. Prentice Hall International Series, New Jersey, USA, (1999).

[2] S.M. Walas, Phase
Equilibria in Chemical Engineering, Butterworth Pub.-USA, (1985).

[3] D.Y. Peng,
D.B. Robinson, Ind. Eng. Chem. Fundam., 15, 59-64 (1976).

[4] D.S. Wong, S.I.
Sandler, AIChE Journal 38 671-680 (1992).

[5] K.P. Johnston, D.H. Ziger, C.A. Eckert, Ind. Eng. Chem. Fundam. 21, 191-197 (1982).

[6] G. Anitescu, L.L Tavlarides, J. Sup. Fluids 10, 175-189 (1997).

[7] W. J. Schmitt, R.C.
Reid, Fluid Phase Equilib. 32, 77-99 (1986).

[8] M. McHugh, M.E. Paulaitis, J.Chem.Eng. Data, 25 326-329 (1980).

[9] J.O.
Valderrama, J. Zavaleta, Ind. Eng. Chem. Res., 44, 4824-4833 (2005).