Solubility of vitamin A in supercritical CO2: experimental study and thermodynamic modeling

One of the best methods of extracting Vitamin A, as a helper of the immune body system and vision, was measured in supercritical carbon dioxide (SC-CO2); Mole fractions were gained at practical conditions in which the temperature was in the range of 303–323 K and the pressure range was 90–235 bar, respectively. Moreover, four Equation of States [Soave–Redlich–Kwong, Peng–Robinson, Stryjek–Vera and Dashtizadeh–Pazuki–Taghikhani–Ghotbi (DPTG)] were compared with the experimental data. Also, the mixing rules of Van der Waals (vdW1 and vdW2) selected to correlate the solubility data of vitamin A. The outcomes indicate that each of EOSs coupled with vdW2, as a method of estimating the physicochemical and critical properties, were correlated with the solubility data of vitamin A in SC-CO2 with more accuracy, in comparison with vdW1. Among the cubic EOSs, the DPTG model with vdW2 generated the most suitable correlation with the percentage average absolute relative deviation (Average Absolute Relative Deviation%) of 6.


List of symbols a(T)
Energy parameter of in the cubic EOS (Nm 4 mol −2 ) b Parameter depend on Volume in cubic Equations-Of-State (m 3  www.nature.com/scientificreports/ • predicting gas-phase properties of solubility compounds, in vapor-liquid equilibria.
• In this EOSs, the effect of intermolecular forces and size of molecules are considered.
In order to evaluate its ability and estimate the deviation from the ideal state, the two-parameter mixing rules of van der Waals are practiced. The experimental data under the anticipated operating conditions for the binary vitamin A-SC-CO 2 system was used.
The deviation of the calculated outcomes was calculated by the experimental data in order to evaluate the abilities of the selected models represented in terms of the AARD%. Moreover, the sublimation pressure is essential for the accuracy of the calculations which is anticipated to be used at the end of the calculations as a lever to validate the state equations.
In this work, a new study has been done for investigation the solubility points of vitamin A in supercritical solvent in two separate parts. In the first part (experimentally), the amount of molar fraction extracted by the setup designed (Setup description section) is recorded. In the second part (theoretically), these solubility points are compared with the calculated points by the EOS. Finally, total deviations between two numbers (experimental and calculations points) are investigated as a measure for select a more accurate EOS to predict the solubility points of vitamin A in the SC-CO 2 solution.

Methods
Materials. Zagrosgas Company (Tehran, Iran) supplied the carbon dioxide (CO 2 ) (CAS Number 124-38-9) at a least clarity of 99.99%. Retinol (Vitamin A) (CAS Number 68-  with purity (GC) > 95% was obtained from Safirazma (Tehran, Iran). Also, analytical grade methanol (CAS Number 67-56-1) with purity of 99.8% was purchased from Safirazma (Tehran, Iran). Figure 1 indicates the system used for measuring the equilibrium solubility. Before it was introduced into the fridge, the CO 2 gas was passed through the pump by means of revealing the valve on a stainless-steel flask. Consequently, the gas was liquefied through dropping the temperature to − 20 °C so that it was prepared for pumping function by means of pressure controlling instruments (TEBCO, Iran) with an uncertainty of ± 0.1 MPa. www.nature.com/scientificreports/ After CO 2 pumped into the inlet solvent stream, the methanol tank (as a co-solvent in this section) and modulator valve controlled the solvent concentration. The inlet solvent concentration should be regulated. Thus, it is an adjustable factor as function of solute content based on simultaneous data from experimental setup. The pressure of the circuit was taken into the collection container through altering amount of modulator on the injection tap. This includes a certain volume of solvent (CO 2 and methanol). Providing the thermal equilibrium condition with the inner part of the oven in shorter time, this equilibrium spiral pipe can be utilized for efficient mixing of CO 2 and modulators. The cell was put inside carefully (FG Model, BF400E, Iran) to keep the operating temperature (with ± 0.5 K).

Setup description.
Roughly before the start of the extraction, 500 mg of powder Retinol were put into the static cell with the volume of 10 mL and due to the pressure drop, downward flowrate (in line with weight force of the materials) was used. Eluding physical entrainment of the unextracted Vitamin A (Rich Retinol), the tops of the pressurized equilibrium cell were connected by means of the stainless-steel disks before SC-CO 2 was distributed into the cell, its pressure was enhanced up to the anticipated pressure before it is moved through the equilibrium cell at operational conditions. In the present research, 60 min was estimated to achieve the state of equilibrium (in the cell). It was shown to be sufficient through basic experiments.
In fact, the final sample (in the standard state) in two phases of gas (carbon dioxide) and liquid phase (solution containing methanol and vitamin A) It was stored in accumulation glass and entered the spectrophotometer. Finally, the solvent was used to clean the circuit.
As solubility values of Vitamin A were obtained in various conditions, they were noted by absorbency mensuration's at max using a model 2100 UNICO UV-VIS spectrophotometer with 1000 nm Wavelength. The equilibrium mole fraction y A and equilibrium solubility, S(g/L) in SC-CO 2 over the ranges of temperature and pressure obtained from below equation 21 : where where n A and n C are sample moles of vitamin A and solvent, C A is the concentration of vitamin A g/L in the accumulation glass calculated by the curve of calibration. The volumes of the accumulation vial are denoted by V A (L) and V l (L) is the sampling circuit. Furthermore, M A and M C are the molecular weights of Vitamin A and CO 2 , respectively. By combining the aforementioned, the subsequent relation is achieved (Eq. (4)).
In order to obtain the g/L of Vitamin A in SC-CO 2 Eq. (4) was used 21 . Sublimation pressure estimation. Using supercritical solvents is considered as a new method in the purification process of pharmaceutical solids. When the solubility reliance on pressure and temperature is explicitly defined, it is possible to optimize the supercritical extraction. Notably, one of the significant design parameters is detected to be the sublimation pressure in the industrial solid-liquid separation processes.
In the present research, by means of the Clapeyron relation joined from the triple point pressure P t and temperature T t , the sublimation pressure P s at a temperature T is assessed with assuming an insignificant functionality of the sublimation enthalpy to temperature 22 : In which, H s presents the sublimation enthalpy at the triple point which can be analyze by the equation below due the singularity of the triple point 23 .
In which H f and H v are the fusion enthalpies and vaporization at the triple point, correspondingly. In many conditions, the triple point conditions are indefinite experimentally and the use of (Eqs. (1) and (2)) only needs some specific thermodynamic conditions. The method planned with this is established in two stages. Initially, it is supposed that the triple point temperature T t can be assessed by DinT of the normal fusion temperature T f . Certainly, the experimental values of transition temperatures in various literatures are scattered less than 0.1 K which is the difference between these two temperatures for the majority of heavy compounds 24 .
Based on this assumption, H f in (Eq. (6)) can be projected by the synthesis enthalpy measured at the normal boiling point, thus application of Eq. (5) just needs specifying of Pt and H v at this temperature. In the next step, a correlation of vapor pressures is used to compute Pt and H v In this situation, experimental values of P sat are first interrelated through a relation due to temperature. In the current research, experimental statement of the vapor pressure applied 25 .
(1) www.nature.com/scientificreports/ In which bounds A, B, C, D, E were adjusted with the practical data. The equation of Yaws was used to obtain this statement in the case that very limited data points are available 26 .
Let subscript C stand for the light (SC-CO 2 ) component and let subscript A stands for the heavy (Vitamin A) component. First, the general equation of equilibrium for Vitamin A at distinct operating conditions is written to calculate the solubility in the gas phase.
In which, superscript S characterizes the solid state. In this study, the supercritical fluid is presumed to not to be soluble in the solute stage. Additionally, the molar volume of the solute is pressure-independent and at sublimation point, the solute fugacity coefficient equals to 1. Solid phase is incompressible and pure. By considering the aforementioned assumptions, the derivation of equations is performed as following: where P sub A is the sublimation pressure, φ sub A is the fugacity coefficient at sublimation pressure point, and V s A is the solid molar volume, all at temperature T . For the fugacity of vapor-phase, we present fugacity coefficient φ sub A through eliciting its explanation.
We achieve the anticipated solubility of the heavy component in the gas phase by replacing and solving for y A : where The enhancement factor E comprises three correction terms. The first term is the Poynting factor showing the effect of pressure on the pure solid fugacity which is considerable for the case of enhancement factor less than 2 or 3. The next correction, φ sub A , considers non-ideality of the pure saturated vapor; the sublimation pressure of the solid is very small leading φ sub A to be almost equal to unity in most applied cases. The final term, φ SCF A , is the last but the most significant. Nevertheless, φ SCF A is always far from unity and can yield great enhancement factors. With the given numerical value of the y A parameter from previous experimental data, we can calculate the sublimation pressures at those conditions (Eq. (11)). Moreover, with this new data points about the sublimation pressure and the EOSs, we can predict new data points about solubility mole fractions ( y A ).
In fact, this fulfills the fugacity equation (Eq. (8)) and it is essential to measure the solution to the fugacity equation to confirm the minimum Gibbs energy or since fugacity is needed for an adequate condition for a steady equilibrium 10 . As the temperatures, at which the solubility of Vitamin A was examined (303, 313 and 323 K), are well underneath its melting point, it is not necessary do a thermodynamic constancy analysis. Evidently, none of the considered temperatures are very close to the critical point of CO 2 and they are amid the lower and upper critical end point.
To calculate φ SCF A we used different EOS-based models as described in later section. Consequently, we conclude an exclusive thermodynamically solution to the fugacity condition at each T and P allocating to the stable solid-fluid equilibrium 27 . The values for critical parameters, acentric factor, and sublimation pressure of Vitamin A should be put on Eq. (11) in order to calculate the solubility. Afterwards, we practice diverse EOS-based models for estimation of φ SCF A .

Equation of state-based models.
According to the thermodynamic models mentioned in the previous sections, now they are examined in more detail. As already stated, the SRK 17 , PR 18 , SV 19 and DPTG 20 EOSs have been employed to calculate the fugacity constants in order to conclude the solubility of Vitamin A in SC-CO 2 accompanied with relating vdW1 and vdW2 mixing rules (supplementary for more detail). Holistically, cubic equations of state like (RK (and SRK), PR, SV and DPTG) [17][18][19][20] are identified as the best choice modeling for phase equilibrium computations for multicomponent mixtures.
Moreover, it is required to conduct research so that the analytical cubic equations of state can be entirely developed. In this regard, improving the temperature dependency of attraction terms and the P(v) functional form so as to control the vapor pressure (estimation and enhance the prediction of volumetric properties are the main development of this study.
Based on perturbation theory, a new two-parameter cubic equation of state is suggested.
In the meantime, DPTG equation of state 20 are used for the sake of precise calculation of the solubility factor since the outstanding role of solubility parameter has a significant consequence on the results of the model. Furthermore, the exactness of this EOS has been deliberated through estimation of the solubility in some (8) ln P sat = A + B/(T + C) + DinT + ET Compressibility fugacity. The fugacity coefficient of the system and component i that shown in equations below, ( ∅ and ∅ i respectively) are identified as the two types of fugacity coefficients in thermodynamics 28,29 . In addition, ln ∅ i and ln ∅ can be gained over the subsequent relation due to the functionality of fugacity coefficient and pressure: Although the aforementioned approaches are really suitable in case the EOS for fluids is volume-explicit 30 , this is not the case for pressure-explicit EOS (where P is a function of T, v , and x i ), such as the numerous cubic EOS and several non-cubic EOS. In such circumstances the independent P in Eqs. (14) and (15) should be distorted into v and this can be understood by means of the relations specified in the supplementary for more details.
Meanwhile furthermost EOS for fluids is pressure-explicit, Eq. (16) is more general than Eq. (14). Considering the component fugacity coefficients, most of the problems can be solved by the afore-said methods. Nonetheless, they bring about rough inadequacies. For instance, due to the intricated EOS in the first integral in Eq. (16) will be is problematic, or it cannot be recognized because the partial differentiation (∂P/∂n i ) T,v c ,n j(j =i) which remarkably enhances the intricacy of integrated function. Necessarily, alternative approaches were taken in recent works 31 . Thus, they were easier than Eq. (16).
According to the previous studies ln ∅ i can be derived from ln ∅ and also, A straight method for the derivation of ln ∅ i from ln ∅ comes from the following relation 31 : Solution procedure. The optimal evaluation of the global value of the model binary interaction parameter is performed in terms of using the mixing rules of vdW1 and vdW2. The mixing rules comprise of adjustable parameters to consider exact chemical bonding forces such as hydrogen bonding and interaction due to the various size of mixture constituents. Depending on the mole fractions of mixture components, these mixing rules are employed in order to reach the slightest error when the predictions of the EOSs are associated with practical data. The aforementioned factors are recognized as binary interaction parameter which are used as a scale for assessing the deviation from the behavior anticipated in non-ideal systems. At a certain temperature the binary interaction parameter is achieved through regressing the model against experimental data of Vitamin A solubility. Figure 2 displays the stages needed so as to calculate the solubility. Furthermore, these stages indicate the several calculations done to calculate the solubility of Vitamin A in SC-CO 2 at different operating conditions. The compatible function is the AARD% between the calculated and experimental solubility. In which N is the number of experimental points at each temperature. The whole relations concluding the equations of state as well as mixing rules practiced in this research are presented in the supplementary for more detail.

Result and desiccation
Experimental data. In this study when the pressure is in the range of 90-240 bar and the temperature range set at 303-323 K, equilibrium solubility data S (g/L) and the mole fraction (y) data of Vitamin A in SC-CO 2 were specified. Table 2 features the outcomes. It is worth noting that all experiments were repeated three times so that accurate and precise measurements was ensured. Remarkably, the comparative normal uncertainties were less than 6%. Furthermore, Fig. 5 divulges the uncertainty associated with the each of EOS. (The Span-Wagner equation 32 was applied to contribute the density of SC-CO 2 . This equation is specifically formulated for CO 2 ). In this regard, (14) ln    Fig. 3, it can be deduced that solubility improves through increasing pressure at a constant temperature. This was brought about by the improved density along with the strong solvation property of SC-CO 2 at higher pressures. A double effect on solubility is detected by the temperature in SC-CO 2 . This relies on in what way the solvent density and solute vapor pressure are well-adjusted34 , 35 . Indeed, increasing the solute vapor pressure might arise from the enhancement of the solution temperature.  Fig. 3, in pressure 110 bar, there is a cross-over pressure (at 303,313 and 323 K). This is despite the fact that in all cases, the solubility of vitamin A in SC-CO 2 was over than 0.15 g/lit.
Henceforth, it affects stronger solvating power of SCF. Simultaneously, it is recognized that the SC-CO 2 density might be diminished by a rise of temperature. This is detected to decline the total solvation property of the fluid. Figure 3 indicates that the pressure range of 90-240 bar is in the beyond crossover pressure region for Vitamin A. Nonetheless, the density and solute vapor pressure are the leading factors at pressures under and beyond the crossover pressure region, correspondingly. For instance, the solubility may grow with temperature at pressures above the crossover pressure region. On the contrary to the 303 K which is recognized as the lowest one, the 323 K isotherm is highest isotherm. The solubility enhances through growing temperature for all pressures in the range examined. The behavior is influenced by two effects.
The primary is related with the increment of the solvating power of CO 2 for the sake of density and the other one is the enhancement of solubility according to lessening the vapor pressure of Vitamin A. In this regards, similar results have been reported by researchers in clarification of the binary effect of temperature on the solubility in SC-CO 2 33 .
EOS-based models. As aforementioned, the EOS-based model was practiced in order to relate the solubility data of Vitamin A in SC-CO 2 . Furthermore, the consequences were associated in terms of AARD%. To calculate the Vitamin A solubility in SC-CO 2 , diverse combinations of EOSs such as the SRK 17 , the PR 18 , the SV 19 and DPTG 20 with two mixing rules (VdW1 and VdW2) are put into practice. At three operation temperatures (303, 313 and 323 K) the PR 18 and SRK 19 EOSs with mixing rules of vdW1 and vdW2 contributed to the correlation results ( Table 3). The optimum binary interaction factors ( k ij and l ij ) as well as AARD% are held by them. The experimental data (27 data points) in extensive pressure at three levels of temperature were practiced to protect the generality of the current study. Figure 4 indicates that less average absolute relative errors (5.77% at 303 K, 4.51% at 313 K and 4.09% at 323 K) can be obtained from the combination of DPTG EOS 20 along with applying vdW2 mixing rule. Definitely, it would be suitable for industrial uses. The deviations of the EOS-based models perhaps mostly owing to the point that cubic equations of state usually end to weak solubility predictions in supercritical fluid areas. In comparison with the results from other EOSs for this region, more reliable outcomes have been obtained from the DPTG EOS 20 .  Fig. 4, it is implied that the growth of solubility of such compound is made through the enhancement of pressure at constant temperature. The same influence is contributed by enhancing the temperature at fixed pressure.
In such conditions, the solubility of Vitamin A is successfully represented by the chosen set of equations. Figure 5 illustrates the outcomes of AARD% according to the application of the (SRK, PR, SV and DPTG) EOSs [17][18][19][20] .
It is revealed that achievement of adjustable parameters is possible to estimate for wide pressure ranges. This figure likewise demonstrates that the assumption of the vdW2 mixing rules to gain consistent outcomes for the solubility of Vitamin A as a compromise has been obtained at a temperature range of 303-323 K for the parameters of the objective function.
Evidently, the two correlation parameters of k 12 and l 12 are properly extracted that the temperature reliance on the aforementioned parameters considered by means of the fitting procedure.  The binary interaction parameters ( k ij and l ij ) for the vdW1 and vdW2 mixing rules are linear functions of temperature as the following: In which the coefficients A 1 , A 2 , A 3 ,A 4 were assessed through the analysis of linear regression (Table 4). These linear equations are appropriate to relate the interaction parameters for assessing the solubility of Vitamin A in SC-CO 2 within the operating temperature range. Such linear equations are suitable to transmit the interface bounds for measuring the solubility of Vitamin A in SC-CO 2 in the functioning temperature range. The consequences specified that the interaction parameter of kij and lij are linearly reliant on temperature in reverse.

Conclusion
Experimental measurement of fat-soluble Vitamin A in SC-CO2 from retinol (with 95% purification as a raw material) carried out from 303 to 323 K and at pressures from 90 to 235 bar. In this regard, solubility of Vitamin A (S (g/L)) and mole fractions (y) dissolved in SC-CO 2 were in the variety of 0.096 to 1.036 and 2.18 × 10 -5 to 1.964 × 10 -4 , respectively. The results were shown in Fig. 3, indicated the solubility dependency towards the variation of temperature and density. Also, with the density increases, the solubility of all substances increases at constant temperature. As long as the density remained constant, the solubility is increased resulting from the temperature increment although the improvement in the solid vapor pressure is caused by it. According to the results, the range of 0.7 g/kg was the one at which the solubility of fat-soluble vitamin A was measured in SC-CO 2 under certain conditions. For fitting curves onto the solubility data of Vitamin A compound in SC-CO 2 , comparative studies were assumed so as to examine the employment of numerous cubic equations of state (SRK, PR, SV and DTPG) 17-20 by various mixing rules (vdW1 and vdW2). Regressing the model against experimental solubility data was applied to find the optimized values of the model parameters. In the range of operating condition, 27 experimental data points were utilized for the thermodynamic equilibrium calculations.
As observed, applying DTPG EOS 20 and VdW2 mixing rule directs to not as much complete average relative deviation (AARD 6%) of the outcomes from the consistent experimental values in comparison with others being capable of the higher accuracy in the Vitamin A solubility data in SC-CO 2 .