In the following discussion, only the basic equations that allow the calculation of the solubility of a solid material (solute, component 2) in a dense gas will be reported. The equilibrium condition for component 2 is:
f s = j~sr (2.3-26)
where the superscript S indicates the solid phase and the superscript SF the supercritical phase. Since normally the solid phase is considered to be the pure solid solute, according to eq. (2.2-17) the left hand side of this equation is given by:
f s = f s = P.~"~'qk~ exp b RT ) (2.3-27)
where p sue is the sublimation pressure of the solid', ~ is the fugacity coefficient of the pure component 2 at the pressure p su~ and temperature T in the gas phase, and v2 s is the molar volume of the pure solid component, 2, at the temperature T. The fugacity of component 2 in the supercritical phase is given by the well-know relationship:
fsr = y2qksy p (2.3-28)
and, finally, the solubility is given by the following equation [2]"
Vex (i vSd ll
,p~_~~ ~2 4 RT ) | (2.3-29)
Y2 = ~SF ]
The quantity contained in square bracketts represents the ratio between the real solubility and the ideal solubility (the supercritical phase is described by the ideal gas law); it is always greater than 1 and it is also called the enhancement factor (E). E can have values of 10 +3 or higher.
The enhancement factor contains three terms" ~b st, the fugacity in the supercritical phase,
~b s , which takes into account the non-ideal behaviour of the pure component 2 in the vapour phase at the sublimation pressure, and the Poynting factor that describes the influence of the pressure on the fugacity of pure solid 2.
Since the sublimation pressure is normally very low, the fugacity coefficient ~b s is close to 1 and also the Poynting correction is not very different from 1 (usually less than 2). As a consequence, the most important term that contributes to a value of E is the fugacity coefficient in the supercritical phase.
The effect of pressure on ~b sp is given by the equation:
_ v 2 1
a P R T P
T,y
(2.3-30)
where 7SF is the partial molar volume of component 2 in the supercritical phase. When the solubility y2 is very low, and the temperature T, and pressure P are close to the critical conditions of the component 1, ~SF assumes large and negative values.
2 . 0 E - 4
1 . 0 E - 4
5 . 0 E - 5
2 . 0 E - 5
1 . 0 E - 5 -
5 . 0 E - 6 , i , i , i
50 100 150 2 0 0 2 5 0
P / b a r
Figure 2.3-2 Solubility S (mole fraction) of a solid compound as a function of pressure at 312.5 K (.) and 331.5 K (o).
As a consequence a small variation of the pressure causes a large variation of the fugacity coefficient and of the solubility.
In Figure 2.3-2, as an example, typical data of the solubility as a function of pressure, at two different temperatures, are reported.
It is necessary to observe, in particular, the effect of temperature on the solubility. At pressures below a given value, which is typical for each binary solute-solvent system, the solubility increases with decreasing temperature. At higher pressures the opposite effect is observed. This characteristic pressure is normally referred to as crossover pressure and it is very important when a process involving solids must be optimized.
According to eq. (2.3-30), the temperature mainly influences the sublimation pressure and the fugacity coefficient in the supercritical phase. The sublimation pressure always increases with increasing temperature, and if this is the main effect the solubility must always increases with temperature. To the contrary, at relatively low pressures (close to the critical pressure of the supercritical fluid) the fugacity coefficient in the supercritical phase plays the most important and preponderant role.
For the evaluation of the solubility it is necessary to know the pure component properties and to use an equation-of-state model for the evaluation of the fugacity coefficients. In general, two problems arise:
9 pure-component parameters required by the EOS are lacking for the heavy component;
9 the sublimation pressure of the solute is not known.
The critical properties, that are essential basic data if a cubic equation of state is used, can be evaluated using group contribution methods but the numerical values obtained depend on the method used. In particular, this fact represents a problem for multifunctional components that are generally involved in processing natural products and/or pharmaceuticals. As an example, depending on the prediction method used, a critical temperature ranging from 817.8 to 1254.0 K can be obtained for cholesterol [60].
The evaluation of the sublimation pressure is a problem since most of the compounds to be extracted with the supercritical fluids exhibit sublimation pressures of the order of 10 -14 bar, and as a consequence these data cannot be determined experimentally. The sublimation pressure is thus usually estimated by empirical correlations, which are often developed only for hydrocarbon compounds. In the correlation of solubility data this problem can be solved empirically by considering the pure component parameters as fitting-parameters. Better results are obviously obtained [61], but the physical significance of the numerical values of the parameters obtained is doubtful. For example, different pure component properties can be obtained for the same solute using solubility data for different binary mixtures.
Some of these ambiguities can be partially solved using a simple approach recently proposed by Gamier et al. [62]. The sublimation pressure of a solid can be estimated using experimental fusion properties and the vaporization enthalpy derived from the equation of state. Using the Clapeyron equation P2 sub can be approximated by:
'n( "2 I -AH:"b 1
r e ) R (2.3-31)
where Tt and P t are the reference conditions chosen at the triple-point of the pure component 2. The quantityAH~ ue is the sublimation enthalpy in the reference state, and is calculated from the fusion and vaporisation enthalpies AM ''b = A n f~ + A M v'p
~ A 2 ~ A 2 "
Using the approximation that for a solid the triple-point temperature Tt is equal to the normal fusion temperature, T f us , it is possible to calculate, from the same equation of state that is used for the calculation of ~b sF"
9 an estimate the reference pressure P t at the temperature Tt z Tz :us ; 9 an estimate of the vaporisation enthalpy AH~ ~ at the temperature Tt.
Hence, the estimation of the sublimation pressure requires only the fusion properties, T2 '~' and AH{ ~ which are more easily available in the literature. Using this approach more reliable results are obtained.
The so-called subcooled liquid approach was also suggested in the literature in order to overcome the difficulties connected with the pure component properties of the solid compound. This approach is commonly used for the calculation of the liquid-solid equilibria, and for solubility calculation of solids in supercritical fluids was already suggested [2] and subsequently extensively applied to different supercritical fluid processes [63]. The solubility y2 is then expressed as"
- --ff ~b-~ exp R T exp - R T ~ (2.3-32)
where P0 is the normal pressure and ~b~ is the fugacity coefficient for the supercooled component 2, considered as a liquid at temperature T and pressure P0.
Also with this approach the problem of the pure-component properties necessary for the determination of the parameters of the equation of state remains: the advantage lies in the use of experimentally accessible properties of heat-of-fusion and melting point, instead of the sublimation pressure.