A Simplified Cubic Equation of State Approach to Model the Solubility of Solids in Supercritical Carbon Dioxide

: The modelling of the solubility of a solute in a supercritical solvent system using the cubic equation of state ( cEoS) requires the critical properties (Critical temperature (Tc) & Critical pressure (Pc)), vapour pressure, acentric factor

, where y 2 is the mole fraction of solubility of the solute in the scCO 2 , P is pressure, and T is temperature; molar volume (v 2 ) is expressed as a function of solvent density as g(ρ 1 ) = exp(β 1 × ln(ρ 1 ) + β 2 ), where β 1 and β 2 are the model constants and 2 φ ∞ is the solute's fugacity at infinite dilution which is a function of the solute's equation of state parameters a 2 and b 2 .The proposed approach was evaluated with the solubility of parabens (Methylparaben, Ethylparaben, and Propylparaben), Aspirin, Griseofulvin, Ibuprofen, and Salicylic acid in supercritical carbon dioxide (scCO 2 ).Further, the accuracy of the proposed cEoS approach was compared with existing EoS model correlations.Finally, the proposed approach was observed to give satisfactory results in terms of the relative deviation and Akaike's information criteria.

Introduction
Supercritical fluid technologies (SFT) are finding use in petrochemicals, chromatography, biotechnology, pharmaceuticals, dyeing, and several industries [1]- [4].Usually, in all these applications, targeted solute compounds are extracted or dissolved using a supercritical fluid (SCF).Substances include colourants and pigments, nutraceuticals, active pharmaceutical ingredients, and food additives.These solutes are frequently solid at room temperature as a result of their high molar masses.Furthermore, they only dissolve in the SCF.In principle, any substance can be used in its supercritical form, but carbon dioxide and water are the most frequently utilised supercritical fluids [1]- [4] due to their chemical and physical properties.
In recent years, the use of supercritical carbon dioxide (scCO 2 ) in the dyeing, food processing, and pharmaceutical industries has grown in importance.Supercritical fluid technology, especially via a green solvent such as supercritical carbon dioxide (scCO 2 ) has many applications in various topics [5]- [8].The solubility data determines the utility of scCO 2 , and it can also act as a substitute for organic solvents that are usually used in regular unit operations [9]- [12].As scCO 2 has distinctive qualities, including non-toxicity, non-flammability, and adjustable density, it can be a substitute for many solvents.It has a moderate critical pressure of 7.39 MPa and a critical temperature of 304.12 K.A precise understanding of the compound's solubility is required for the effective implementation of scCO 2 -based technology in industry.However, acquiring solubility data under the necessary pressure and temperature circumstances is a difficult process.Thus, modelling is important for solubility data interpolation [13], [14].
In recent years, a variety of methods have been used to model solubility data.These methods can be grouped broadly into five categories, of which the three most user-friendly are equation of state (EoS) models, density models, and mathematical models.The first model needs the compounds' physical characteristics, such as vapour pressure, molar volume, critical properties, and acentric factor.The equation of state modelling would be quite beneficial if it had all the required physical characteristics [13]- [15].Unfortunately, many compounds lack the necessary property data.On the other hand, the remaining methods that do not require these data have drawn more interest and succeeded.Density and mathematical models are quite effective at modelling solubility because of their ease of use [14]- [16].The main feature of these models is that solubility is treated as a function of temperature, solvent density, and pressure.Since there won't be a single model that accounts for all compounds, research is always being done, and many new models are being developed [14].The cEoS method has a more sound theoretical basis than the density and mathematical models.Thus, there is a need to address EoS models appropriately.Solubility is highly nonlinear, which poses challenges in developing solubility models.
The work is presented in two stages.In the first stage, the strategy for the model development is done, and in the second stage, the proposed cEoS strategy is validated with the help of literature-reported solubility data of parabens (Methylparaben, Ethylparaben, and Propylparaben), Aspirin, Griseofulvin, Ibuprofen, and Salicylic acid in scCO 2 .

Modelling
Although there are many cEoS models in the literature, the Peng-Robinson equation of state (PR EoS) is frequently used to correlate the solubilities of solids in scCO 2 due to its better correlating ability [17], [18].Thus, PR EoS [19] is used for model development.

Thermodynamic modelling
The solubility of a solute in a solvent is denoted as [17]- [19]: where all parameters have their usual meaning.The pure solid solute saturation fugacity coefficient is assumed to be unity.The fugacity of the solute 2 ˆscf φ in the supercritical phase is calculated using the cubic equation and the mixing rules. 2 ˆscf φ is obtained using pressure explicit form of EoS by eq. ( 18).

PR EoS [19]
The pressure explicit form is , 1 (0.37464 1.5422 0.26992 1 Mixing rules Type 1 by van der Waals, (7) ( ) ( ) Mixing rules Type 2 by van der Waals, (11) ( ) ( ) ( ) The expression for the fugacity coefficient is Combining equations ( 1) to ( 13), we get the solubility expression as a function of ( ) , , , , , ,  , , If all the physical properties of the solute and the solvent are available, the left-out adjustable parameters would be k 12 , l 12 , depending on the mixing rules.
To implement the above method, the critical properties (Tc & Pc), vapour pressure, acentric factor, and molar volume of the solute and solvent are required.Quite often, it is difficult to get the solute properties.Thus, a new approach is proposed in the following section that does not require the critical properties of the solute (i.e., Tc, Pc, vapour pressure, acentric factor, and molar volume), but the critical properties of the solvent are necessary.Mode details are presented in the following section.

Simplified cubic equation of state approach
The basic equations relevant to the solubility model mentioned in the previous section are applicable.In the proposed approach, the following assumptions are applied to the modelling.
Assumption 1: The solute's solubility in the solvent is not high; therefore the solute is infinitely diluted.This results in the following expression for the Fugacity coefficient, in which solute parameters are treated as adjustable parameters [20], [21].Thus, the following expression is obtained.
Assumption 2: Solute molar volume (v 2 ) is a function of scCO 2 (solvent) density (ρ 1 ) [22].The following expression is used: Applying assumptions 1 and 2 results in the solubility expression as a function of (17) It is important to note that the present study is exemplified only with PR EoS and vdW mixing rules, but the same idea can be extended to any EoS along with mixing rules.This study is important when the molar volume, critical properties, and acentric factor of the solute are unavailable.In case of the non-availability of the solute's sublimation pressure, an appropriate temperature function can be used [15] in place of sublimation pressure but it has not been

Results and discussion
The proposed strategy for the correlating solubility of solids in supercritical fluids is tested with a solubility of parabens (Methylparaben, Ethylparaben, and Propylparaben), aspirin, griseofulvin, ibuprofen, and salicylic acid in supercritical carbon dioxide (scCO 2 ).[12], [23]- [26].Parabens are the derivatives of para-hydroxybenzioc acid (PBHA), and they have been widely used in the personal care, cosmetics, drug, and food industries as preservatives for almost a century.Thus, validating the proposed model with them is useful.In literature, the solubilities of Methylparaben (at 308, 318, 328, and 338 K), Ethylparaben (at 308, 318, and 328 K), Propylparaben (at 308, 318, and 328 K), Aspirin (at 308.15, 318.15, and 328.15 K), Griseofulvin (at 313.15 and 333.15 K), Ibuprofen (at 308.15, 313.15, and 318.15 K), and Salicylic acid (at 313.15 and 333.15 K) are available.The data ranges of compounds considered in this work and other properties [12], [23]- [28] obtained from literature are reported in Tables 1, 2, and 3.For the case of aspirin, griseofulvin, ibuprofen, and salicylic acid compounds, the required saturation pressures are obtained from their critical properties data using the lee-Kessler expression [29].Table 4 shows the simplified cubic equation of state approach results and Tables 5  and 6 show PR EoS model (Single and two binary interaction parameters) results.For data correlation, the following objective function, eq. ( 18), is used [30]. ( where N is the number of solubility data points.With the help of MATLAB's (fminsearch) built-in functions, the correlation exercise with and without the solute's critical properties has been carried out, and the results are reported in Tables 4, 5, and 6.Figures (1) to (7) indicate the correlating ability of the proposed cEoS approach for the compounds considered in the study.
The accuracy of the simplified cubic equation of state approach model correlations is compared with the existing PR EoS model correlations, which are based on the critical properties of the solute.Tables 4, 5, and 6 show present study correlations and PR EoS model correlations respectively along with the coefficient of determination (R 2 ) values and absolute average relative deviation percentage (AARD%).Figures (4), ( 5), (6), and (7) clearly indicate the relative performance of the present study and the existing model correlation.Further, the relative performance of the two approaches with and without critical properties of the solute is quantified with Akaike's information criteria (AIC) and corrected AIC (AICc) [31]- [34].
AIC and AICc [8], [9] are related as follows ( ) In eqs.( 19) and ( 20), σ, N, and Q represent the variance of deviations, number of data points, and model parameters, respectively.The calculated AIC and AICc values are shown in Table 7.The best model will have the least AIC and AICc values.From Table 7, the present study is observed to provide satisfactory correlations for the Methylparaben, Propylparaben, and Aspirin compounds compared to other compounds.AICc values are the least for the proposed study for those compounds; hence, the new approach is acceptable.

Conclusions
Correlating the solubilities of solid solutes in supercritical solvents is essential for the effective deployment of supercritical fluid technology.This study successfully deals with the solubility modelling of solids in scCO 2 without the solute's properties.The model results clearly show that, in the case of Methylparaben, Propylparaben, and Aspirin, the present study is performing on par with the existing PR EoS models (with two binary interaction parameters).AARD, AIC, and AICc analyses also indicate that the suggested new approach is yielding acceptable correlation results.Finally, the suggested simplified approach can be extended to any solute-supercritical fluid solvent system with appropriate EoS and mixing rules.

Table 4 .
New PR EoS correlation results (Present study)

Table 5 .
PR EoS correlation results with single interaction parameter

Table 6 .
PR EoS correlation results with two interaction parameters

Table 7 .
Summary of SSE, AIC, and AICc of the systems