The successes and limitations of this thermodynamic model in predicting the solubility of organic solutes in pure solvents

Transcription

1 PC-SAFT The successes and limitations of this thermodynamic model in predicting the solubility of organic solutes in pure solvents Gross and Sadowski s (21) modified SAFT, perturbed chain SAFT, was studied to determine its applicability to evaluating the solubility of organic solutes in pure and mixed solvents. The hard chain and dispersion contribution were included in the calculations but no additional terms such as association or multipolar were added. The model was programmed into MATLAB and experimental data for two solutes, benzoic acid and aspirin, in multiple solvents were input into the program. Both solutes and all solvents were parameterized using the model. These parameters were then used on their own for each system, averaged over solvent groups, and averaged as a whole to observe the quality of the solubility predictions. It was found that more polar solvents were more difficult for the model to handle and the predictions deviated significantly from the experimental data. Also higher molecular weight molecules where dispersion forces were more significant behaved better. Several non-polar solvents were examined and their paramterized values compared with those from literature and found to match well which indicates the model functions suitably well for nonassociating systems. For better predictive capabilities, additional terms for the residual Helmholtz free energy should be added that would be capable of handling polar forces and hydrogen bonding.

2 Table of Contents Introduction... 2 Materials and Methods... 3 Overview of the MATLAB code... 3 Parameterization... 4 Predicting Solubility... 5 Solutes and Solvents... 5 Results and Analysis... 7 Discussion Conclusions Recommendations Appendices Appendix A: Bibliography Appendix B: Nomenclature Appendix C: List of Equations... 3 Appendix D: Tables Appendix E: Figures

3 Introduction Solubility is of particular importance in the pharmaceutical industry where knowledge of this physical property determines processes such as extraction and crystallization. In this industry during research and development, there may only be a small amount of the product available for experimentation so excessive trials for characterizing solubility are not possible. Furthermore, small scale tests often provide misleading data on the saturation point and nucleation rates (Bouillot, et al., 211). Therefore, a thermodynamic model that could accurately predict solubility trends in pure and mixed solvents using only a minimal number of experimental data points. The application of thermodynamic models to solid liquid equilibrium was found to be limited. Many equations of state are used for vapour-liquid equilibrium and liquid-liquid equilibrium but few are applied to solids. It was desired to investigate the particular thermodynamic model PC-SAFT developed by Gross and Sadowski (21) with regards to its ability to model solubility. The implementation of the model used only the hard chain and dispersion terms, neglecting other contributions such as those from association forces. Two different solutes in multiple solvents were examined. Several of the solvents were significantly polar and all were able to engage in hydrogen bonding to some degree. This would indicate that the model would have trouble in predicting the behaviour as these forms of interactions are not taken into account in the model. Dispersion forces are included and the use of a hard chain reference fluid should mean the model is better able to model these interactions. The effects of the solvent type were considered with regard to the accuracy of the predictions. The model was used to parameterize the required variables for both the solutes and solvents which were then used to predict the solubility. Different initial guesses were examined to determine the best conditions to start the model with. The different characteristics of the solutes and solvents allowed the limitations of the model to be examined. The absence of contributions due to association forces and multipolar terms was evaluated along with the degree to which the model could correlate experimental data and predict solubility. 2

4 Materials and Methods The model used was perturbed chain SAFT which modifies the version of SAFT produced by Huang and Radosz (199); instead of using a hard sphere reference fluid for the dispersion term, a hard chain reference fluid was used. A full list of equations used for the model can be found in Appendix C: List of Equations. Overview of the MATLAB code The model was implemented into matlab which used several functions to predict solubility based on initial guesses for parameters or solubility. Figure 1: Model format for a non-associating system First a packing factor must be guessed; this value must be above zero which would describe an ideal gas, and less than or equal to.74 which is the closest packing of segments possible. Once an initial value was been selected, in this case.5 was deemed a good starting value, the number density of the mixture, compressibility factor, and eventually the pressure can be calculated. This calculated pressure was then referenced against the pressure at which the experimental solubility data was collected and the packing factor was adjusted until the calculated solubility was acceptably close to the provided experimental data. These parameters can then be used with a guessed value for the solubility to predict the solubility over a given temperature range. 3

5 This model only takes into account the hard chain and dispersion contribution and neglected any other terms such as association or multipolar. The model investigated by Gross and Sadowski (21) operated similarly but the systems studied were non-associating. In this report, organic solutes were investigated in some polar solvents which should require consideration of association forces. Parameterization An initial guess for each of the parameters is also required. For the non-associating parameters, m, σ, and ε, equations for the initial guesses were taken from the work of Gross and Sadowski (21). Segment diameter, σ: σ i = q 1 + M i M CH4 M i q 11 + M i M CH4 M i M i 2M CH4 M i q 21 Segment number, m: m i = q M 2 + M i M CH4 q i M 12 + M i M CH4 M i 2M CH4 q i M i M 22 i Pair depth potential, ε/k: ε i k = q 3 + M i M CH4 q M 13 + M i M CH4 M i 2M CH4 q i M i M 23 i (Equation 1) (Equation 2) (Equation 3) Table 1: Constants for Equations 1-3 i 1 2 qi1 Å qi2 mol/g qi3 K The pair depth potential is divided by the Boltzmann constant and this value is regressed. This is partly because ε always appears in equations as ε/k but also because without this division, the value for this last parameter is many orders of magnitude less than the other two non-associating parameters and this creates problems with more regression routines. Before parameterization can occur the binary interaction parameter, k ij, must be defined. This value adjusts the pair depth potential due to the intermolecular interactions occurring between unlike molecules. This parameter can be set to zero, given a single value, or set to be linearly dependent on temperature. A temperature independent k ij requires only one additional data point while the temperature dependent k ij requires two. The initial guess for any variation on the binary interaction parameter was set to zero and the effect of this initial guess was investigated (Figure 6) and it was found that a starting value of zero was appropriate as a wrong initial non-zero guess could have detrimental effects on the parameters. 4

6 Predicting Solubility Once the parameters were determined for each solute and solvent system, an initial guess for the solubility was given which then allowed the model to predict the solubility of the system over a defined temperature range. A constant solubility value was inputted into the model and different initial values were tested to find an optimal starting guess (Figure 8-Figure 1). A starting value of 1 gave consistently satisfactory results across the temperature range, solutes, and solvents. By simply using set parameters and adjusting a guess once, acceptable prediction were obtained; however this may not be the optimal solution. Another option may be to start with this process and then regress parameters from the predicted solubility. These new parameters would be compared with the initial parameters and the solubility guess would be suitably adjusted so that the new regressed parameters would match up more closely with the initial parameters. This method was found to be very difficult to implement and without very strong initial parameters, the solubility was found to often diverge from the experimental data. As the parameters regressed using the model for the chosen systems showed significant variation between solvents, it was decided the first, simpler method would be the best option. Solutes and Solvents The two solutes investigated were benzoic acid and aspirin as data for these systems in multiple solvents was made available. These were suitable as the project aimed to determine if PC-SAFT was a suitable thermodynamic model for predicting the solubility of organic solutes with a particular view to the pharmaceutical industry. Benzoic acid consists of a benzene ring and carboxylic acid functional group. The high electronegativity of the oxygen atoms compared with that of the carbon atom they are attached to in the carboxyl group creates a slightly positive charge on the carbon atom. Additionally, for the aspirin, the ketone also creates a slightly positive carbon atom. Both groups are capable of forming hydrogen bonds. It would initially seem aspirin would be considerably more polar than benzoic acid. The relative polarity of the two molecules was estimated by comparing each solute s solubility in water; aspirin has a solubility of 3.3 g/l in water (Fisher Scientific, 29) while benzoic acid has a solubility of about 3 g/l (Yalkowsky, et al., 21). Both solubilities were found at 2 C. This indicates there may not be much difference in polarity, with aspirin being slightly more polar due to the ketone functional group. In both cases, the large phenyl ring significantly reduces the overall polarity of the molecules but association forces are still an important aspect of their intermolecular interactions. Benzoic Acid Aspirin Figure 2: Molecular Structures of Benzoic Acid and Aspirin 5

7 The solvents used were group by their functional groups Table 2: Solvents and solvent groups for each solute investigated Benzoic Acid Aspirin Alcohols Methanol Alcohols Isobutanol 1-Propanol 1-Propanol 1-Butanol 1-Butanol Esters Ethyl Acetate Esters Ethyl Acetate Isopropyl Acetate Isopropyl Acetate Ketones Methyl Ethyl Ketone Ketones Methyl Isobutyl Ketone Methyl Isobutyl Ketone These solvents provided a variety of functional groups that would have different degrees and forms of association forces. By using several solvent groups, it was hoped that the error from not including an association term when regressing the parameters would be mitigated when average values for the solutes were then used for predictions. Alcohol Ester Ketone Figure 3: Molecular structure of solvent functional groups Figure 3 shows the molecular structure of the functional groups of the solvents. The alcohols are able to easily form hydrogen bonds with both like molecules and either of the solutes. Ketones are unable to act as hydrogen donors but may still engage in hydrogen bonding due to the difference in the electronegativity between the oxygen and carbon it is bonded to. Similarly the esters are able to engage in hydrogen bonding by acting as an acceptor but neither molecule is able to self-associate. Both the solvents along with the alcohol solvents are able to engage in self-association as they can act as both hydrogen acceptors and donors. The different molecular weights of the solvents within a group are also important. Typically, the relative importance of dispersion forces will increase with an increase in molecular weight. As the model is capable of predicting dispersion forces but not association forces, it is necessary to investigate the possible differences between lower weight molecules and higher ones. Lastly a few non polar solvents were used to check the parameterization of the model against values taken from literature (Gross & Sadowski, 21). Pentane, heptane, and cyclohexane were opted for as data for these systems was available (Thati, et al., 21). Furthermore, the difference in molecular weight could be observed between pentane and heptane along with the effect of a chain verses a cyclical structure. 6

8 Mol Fraction Benzoic Acid in 1-Propanol Mol Fraction Benzoic Acid in 1-Propanol Mol Fraction Benzoic Acid in Methanol Mol Fraction Benzoic Acid in Methanol Results and Analysis The first requirement for the PC-SAFT model was to take solubility data and regress the required parameters as needed. The model was provided with experimental data, needing at least as many data points as parameters that are to be regressed. In typical use, the parameters for the solvents would be well defined from literature or from other and only the solute parameters must be regressed. As it could not be assumed that these model parameters would match those from literature, the parameters for both the solute and solvent were regressed. All initial guesses for m, σ, and ε were calculated from Equations 1-3. Depending on the k ij parameter being used, the number of parameters required to define it were decided and initial set equal to zero. The parameters were adjusted so that the solubility predicted from those values is equal to the experimental solubility data within a given tolerance. It was found that in some cases where association forces dominate, the parameters were unable to be solved such that the solubility data matched up. The results from the parameterization are shown below in Figure 4. 1(a) 5 Temperature Dependent Kij 2(a) 5 Temperature Dependent Kij (b) 5 Temperature Dependent Kij 2(b) Temperature Dependent Kij

9 Mol Fraction Benzoic Acid in 1-Butanol Mol Fraction Benzoic Acid in 1-Butanol 1(c) 5 Temperature Dependent Kij 2(c) 5 Temperature Dependent Kij Figure 4: Correlation between experimental data (red circles) and PC-SAFT model (blue line) for benzoic acid in different solvents using (1) a temperature dependent binary interaction parameter, k ij, and (2) no binary interaction parameter for solvent (A) Methanol, (B) 1-propanol, and (C) 1-butanol. Experimental data points for the solubility of benzoic acid in three solvents, methanol, 1- propanol, and 1-butanol were used to regress the required parameters for a non-associating system; m, σ, and ε. In Figure 4, plots 1(a), 1(b), and 1(c) used a temperature dependent binary interaction parameter, k ij, which required the parameterization of two additional values whereas plots 2(a), 2(b), and 2(c) set the binary interaction parameter equal to zero, neglecting the effect of intermolecular interactions between unlike compounds on the pair depth potential. A minimum number of data points, eight or six depending on if a temperature dependent k ij value or no k ij value was used, was used to parameterize both the solute and the solvent. As observed in Figure 4, as the molecular weight of the solvent increases, the agreement between PC-SAFT and the experimental data improves. This is due to the absence of an association term in the calculation of the residual Helmholtz free energy which would account for the polarity of the molecules and their ability to form hydrogen bonds. As alcohols are quite capable of forming hydrogen bonds with like molecules, neglecting this contribution leads to poor correlation between experimental points and the model. Dispersion forces, however, are taken into account and the PC-SAFT model uses a hard chain reference fluid in order to model these forces better which provides more realistic predictions of behaviour. As the molecular weight of the solvent increases, the dispersion forces increases so the relative importance of dispersion forces for 1-butanol is greater than that for methanol which is reflected in the greater degree of correlation between the model and the experimental data (Table 3). Table 3: Correlation between molecular weight, polarity, and agreement between experimental data and model for the alcohol solvent group for benzoic acid Molecular Weight, g/mol Normalized Solvent Polarity (a) RMS (b) for Temperature Dependent k ij RMS for Temperature Independent k ij Methanol Propanol Butanol (a) (Reichardt, 23)values normalized to predicted solvent polarity of water (b) Root Mean Squared value given by RMS = n i=1 residuals2 n

10 The root mean squared equation was used as a measure of the correlation between the experimental data and the PC-SAFT solubility values for the regressed parameters. The RMS values for the temperature dependent, vs. no k ij systems seems to indicate the latter provides a better correlation; however, consideration of the parameters for benzoic acid regressed in each solvent system for each k ij, shows that the system with no k ij has more variation within the parameters (Table 4). Table 4: Non-Associating parameters for Benzoic acid in 3 solvents for a temperature dependent and no binary interaction parameter, k ij. Temperature Dependent, k ij No k ij Segment number, m Segment Diameter, σ Depth of Pair Potential, ε Methanol propanol butanol Average Std. Dev Std. Dev. As a % of the Average Methanol propanol butanol Average Std. Dev Std. Dev. As a % of the Average Should the model be able to sufficiently predict the solubility behaviour of an associating system, the parameters for the solute, benzoic acid, should remain relatively constant with some variation due to the error associated with the experimental solubility data. It is clear though, that there is much more significant variation when no binary interaction parameter is used than the temperature dependent k ij system. This is possibly due to the lack of an association term rendering the model unable to predict behaviour accurately. Parameterization then deviates from the true values to make up for this lack when attempting to match the calculated solubility to the experimental solubility. The error from excluding association interactions is then distributed amongst the non-associating parameters. By reducing the number of variable parameters from 5 to 3 by setting the k ij equal to zero, the error is then distributed amongst fewer parameters which may result in the greater variation observed. Additionally, the segment number and diameter for 1-butanol were significantly different from the other solvents in system without a binary interaction parameter and also from the 1- butanol values from the temperature dependent system. As stated before, the dispersion forces in 1-butanol in comparison to the polar contribution and hydrogen bonding are greater than those of the other two alcohol solvents covered in Table 4. As benzoic acid is relatively non-polar and dispersion forces play a decently significant role in its intermolecular interactions, it follows that 1-butanol will have more interaction with this solvent than the 9

11 Mol Fraction Aspirin in 1-Butanol Mol Fraction Aspirin in 1-Butanol Mol Fraction Aspirin in 1-Propanol Mol Fraction Aspirin in 1-Propanol lower molecular weight solvents in the same solvent group. This is backed up by the solubility data for these systems which show a greater degree of solubility in 1-butanol than 1-propanol which is in turn greater than that in methanol. As such, the binary interaction parameter is more important in this system so neglecting it in this case affects the model more significantly. This error is mitigated somewhat by allowing two data points for the fitting of a temperature dependent binary interaction parameter. The same simulation was then carried out for aspirin in 1-propanol and 1-butanol which would allow for good comparison between both the solute and solvent parameters. 1(a) 2(a) 4 Temperature Dependent Kij 4 kij= (b) Temperature Dependent Kij (b) No Kij Figure 5: Correlation between experimental data (red circles) and PC-SAFT (blue line) for aspirin in different solvents using (1) a temperature dependent binary interaction parameter, k ij, and (2) no binary interaction parameter for solvent (A) 1-propanol and (B) 1-butanol. As with the benzoic acid solvent systems shown in Figure 4, the correlation between experimental data and the model values improves as the molecular weight increases. Again, there is better agreement when no k ij value is used and for this solute this corresponds with little variation within the parameters. It is possible in this case that due to the low solubility of aspirin in alcohol solvents, the k ij parameter has little influence on the system as most intermolecular interactions would be solvent-solvent. In total, thirteen systems were analysed, benzoic acid in seven solvents and aspirin in six solvents. The solute parameters were collected for each system and the standard deviation within groups and overall was calculated and then listed as a percentage of the average. This 1

12 allows for comparison of the variation within one parameter with that in another solvent group, overall, or with another parameter. Table 5: Non-association parameters regressed for Benzoic Acid in several solvents with a temperature dependent k ij Solute: Benzoic Acid Solvent m sigma epsilon k ij slope k ij intercept Methanol Propanol Butanol Methyl Ethyl Ketone Methyl Isobutyl Ketone Ethyl Acetate Isopropyl Acetate Table 6: Non-association parameters regressed for Benzoic Acid in several solvents with no kij Solute: Benzoic Acid Solvent m sigma epsilon Methanol Propanol Butanol Methyl Ethyl Ketone Methyl Isobutyl Ketone Ethyl Acetate Isopropyl Acetate Table 7: Non-association parameters regressed for Aspirin in several solvents with a temperature dependent kij Solute: Aspirin Solvent m sigma epsilon k ij slope k ij intercept 1-Propanol Iso Butanol Butanol Ethyl Acetate E-7 Isobutyl Acetate E-6 Methyl Isobutyl Ketone E-7 It is clear from some systems investigated, such as benzoic acid in methyl isobutyl ketone that significant deviations from the average non-association parameter values can occur. Again, this is most likely due to the inability of the model to account for association forces and in an attempt to compensate for this the model adjusts the parameters excessively to provide good correlation with the experimental data. The variations between the non-association parameters were calculated and collected in Table 8 for both solutes. This table excludes the binary interaction parameters which were considered separately. 11

13 Table 8: Variation within the regressed parameters for each solute, temperature dependent k ij Benzoic Acid Aspirin m σ ε Average Std. Dev Std. Dev. as a % of the average Average Std. Dev Std. Dev. as a % of the average Usually, k ij values are established for solvent groups that share similar characteristics; an example of grouping might be strongly polar, weakly polar, and non-polar solvents where each group will use the same binary interaction parameters. For the benzoic acid, the three solvent groups used were alcohols, esters, and ketones; similarly for aspiring the groups included alcohols and esters (Table 2). These groupings are more specific than would usually be used but large variation in the k ij values was observed. Ketones and esters have very similar functional groups and so their binary interaction parameters would be assumed to be similar. Table 5 lists the k ij values for the solvents investigated in the benzoic acid systems. The difference between the alcohols and the other two groups is to be expected due to the ability of alcohols to self-associate; however, the k ij values for esters and acetates may be expected to be more similar. This variation is again likely due to a lack of compensation for the association forces and the error resulting from this significantly affects the binary interaction parameters (Table 9). Table 9: Variation in the binary interaction parameter for the different solvent groups with benzoic acid k ij slope k ij intercept Alcohols Ketone Esters Ketones and Esters Average Std. Dev 9.76E Std. Dev as a % of the average Average Std. Dev Std. Dev as a % of the average Ideally, the parameters regressed for each solute would have little variation from solvent to solvent. This would allow parameters regressed from one system to be easily applied to another without incurring significant error and can be accomplished by providing a model that is capable of accurately predicting the mixture behaviour. Similarly it would be best if the k ij values for solvents with similar functional groups and behaviours remained constant allowing them to be used over an entire solvent group. For this analysis, the temperature dependent binary interaction parameter had an initial guess of zero which neglects the intermolecular effects of unlike species. It was possible that for the system where association forces dominated, an initial guess of zero for the k ij was insufficient and would negatively affect the final parameters. For the benzoic acid methanol system, a temperature independent k ij was used and the effect of the initial guess was evaluated. It was found the initial guess for the k ij values had a significant 12

14 Mol Fraction Benzoic Acid in Methanol Mol Fraction Benzoic Acid in Methanol Mol Fraction Benzoic Acid in Methanol Mol Fraction Benzoic Acid in Methanol effect on the final parameters. The same values for the non-associating parameters were used each time and only the estimate for k ij was modified. No matter what initial k ij value was used, the final value remained approximately - but the correlation between the model solubility and the experimental data varied significantly as the other parameters were regressed to slightly different values. Initial k ij = Initial k ij = Temperature Independent Kij 5 5 Temperature Independent Kij Initial k ij =- Initial k ij =- 5 Temperature Independent Kij Temperature Independent Kij Figure 6: Effect of initial k ij guess on parameterization for a benzoic acid methanol system with a temperature independent k ij For values above zero, the difference between the experimental solubility and calculated solubility increases while for values below zero the difference decreases but the solubility trend becomes less accurate and eventually reverses. For an initial guess of - which is very close to the k ij value parameterized in each case, the solubility trend is significantly different from the actual trend. Overall, observation of associating systems without the use of a model designed to cope with those types of interactions provides little insight with regards to the accuracy of the parameters. It is difficult to differentiate between trends that are due to physical properties such as the relative importance of dispersion forces in intermolecular interactions and those caused by the lack of accuracy in the model. 13

15 Mol Fraction Benzoic Acid in CycloHexane Mol Fraction Benzoic Acid in Pentane Mol Fraction Benzoic Acid in Heptane In order to gain better insight into this, some systems for benzoic acid in non-polar solvents were investigated. The use of a non-polar solvent will minimize any non-dispersion forces acting between molecules; however; benzoic acid is capable of forming hydrogen bonds with like molecules so these systems were not entirely non-associating. Still, they should give a better indicator of the degree of error from the lack of an association term. The solvents considered were pentane, heptane, and cyclohexane with data taken from literature (Thati, et al., 21). No binary interaction parameters were used for this regression. Table 1: Regressed non-associating parameters for non-polar solvents compared with values from literature, (Gross & Sadowski, 21) m m (ref) σ σ (ref) ε ε (ref) Pentane Heptane Cyclohexane (a) 11 x 1-3 (b) (c) Figure 7: Correlation between experimental data (red circles) and PC-SAFT (blue line) for the solubility of benzoic acid in non-polar solvents, (a) Pentane, (b) Heptane, (c) CycloHexane. Non-Associating model. The alcohol solvents showed that an increase in molecular weight had an appraisable effect on the ability of the model to correlate with the experimental data. For the non-polar solvents in Figure 7, the increase in molecular weight from pentane (a) to heptane (b) has no noticeable effect. This is because these solvents engage in only dispersion forces so a decrease in accuracy of the model due to association forces is not an issue. Furthermore, no difference between a chain molecules and cyclical molecule, cyclohexane is observed. The parameters regressed from these systems for benzoic acid were collected in Table 11 and the standard deviation expressed as a percentage of the average shows there is little variation between the parameters and considerably less than was observed in the polar solvents. 14

16 Mol Fraction of Benzoic Acid in 1-Butanol Mol Fraction of Benzoic Acid in 1-Butanol Table 11: Non-association parameters for Benzoic Acid regressed from systems with non-polar solvents, k ij = Solvent m sigma epsilon Heptane Pentane CycloHexane Average Std. Dev Std. Dev. as a % of the average As a fundamental check for the performance of the model, the parameters regressed for the solvents themselves were compared with those taken from literature (Table 1). These parameters agree well with those from the work done by Gross and Sadowski (21) which considered non-associating systems. This along with the close agreement of the solute parameters in Table 11 would indicate that the model was sufficiently able to predict behaviour in non-associating systems. Therefore it can be assume that a significant portion of the error being observed in the benzoic acid and aspirin systems is a result of the lack of an association term. Once the parameters had been defined, they could be used to predict the solubility over a given temperature range. The temperature range considered varied from about K as most pharmaceutical applications would occur in this range. A. Initial guess x= B. Initial guess x=

17 Mol Fraction of Benzoic Acid in 1-Butanol Mol Fraction of Benzoic Acid in 1-Butanol Mol Fraction of Benzoic Acid in 1-Butanol Mol Fraction of Benzoic Acid in 1-Butanol C. Initial guess x= D. Initial guess x= E. Initial guess x=.8 F. Initial guess x= Figure 8: Predictions for solubility of benzoic acid in 1-butanol for different initial guess of a constant solubility value, x. (Initial guess-green dots; prediction-blue line; experimental data-red circles) The initial guess for the solubility at each temperature was one single value; in plot A of Figure 8 an initial value of was used which is about the solubility of benzoic acid at the middle of the temperature range. In plot B a value of was chosen as the max solubility for the temperature range and good agreement was still observed for the solubility trend and magnitude. Plot C used which was the minimum solubility for the temperature range which resulted in a prediction with a more exaggerated trend and much higher solubility values. In plot D value of was used and the model appeared to break down and returned numbers that are physically impossible for the system. The last two plots, E and F, show initial values of.8 and 1 respectively which gave much better agreement between the experimental and predicted values while also correctly predicting the trend. This was repeated for benzoic acid in ethyl acetate and methyl ethyl ketone to see if this trend persists in other solvent types. 16

18 Mol Fraction of Benzoic Acid in Ethyl Acetate Mol Fraction of Benzoic Acid in Ethyl Acetate Mol Fraction of Benzoic Acid in Ethyl Acetate Mol Fraction of Benzoic Acid in Ethyl Acetate A. Initial guess x= B. Initial guess x= C. Initial guess x=.8 D. Initial guess x= Figure 9: Predictions for solubility of benzoic acid in Ethyl Acetate for different initial guesses (Initial guess-green dots; prediction-blue line; experimental data-red circles) From Figure 9 it is still clear that an initial value of zero (plot B) causes serious issues in the model. In Figure 8 this behaviour resulted in solubility parameters far higher than can be physically possible while in Figure 9 it caused a reversal in the solubility trend. For an initial value of, plot A, the predicted solubility varied little from the initial guess. This effect was observed for any initial value between and.6. Above.6 the agreement between the values and trend improves. Plot C used an initial value of.8 while plot D used a value of 1. The initial value of 1 gave much better overall agreement while also giving a reasonable idea of the solubility trend over the temperature range being considered. 17

19 Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone A. Initial guess x= B. Initial guess x= C. Initial guess x= D. Initial guess x= E. Initial guess x=.7 F. Initial guess x= Figure 1: Predictions for solubility of Benzoic Acid in Methyl Ethyl Ketone for different initial guesses (Initial guess-green dots; prediction-blue line; experimental data-red circles) The same trends were found for the predictions of solubility in Benzoic acid in methyl ethyl ketone. For all systems, an initial guess of 1 gave consistently satisfactory results and so this guess was then used for subsequent predictions. Parameters were regressed for systems with a temperature dependent binary interaction parameter and no interaction parameter; the difference in their predictive capabilities then considered. 18

20 Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Methyl Ethyl Ketone Mol Fraction of Benzoic Acid in Ethyl Acetate Mol Fraction of Benzoic Acid in Ethyl Acetate Mol Fraction of Benzoic Acid in 1-Butanol Mol Fraction of Benzoic Acid in 1-Butanol Temperature dependent k ij ; 1-butanol 1 No k ij ; 1-butanol 1 No kij Temperature dependent k ij ; Ethyl Acetate Temperature dependent k ij ; Methyl Ethyl Ketone No k ij ; Ethyl Acetate No k ij ; Methyl Ethyl Ketone 1 No kij Figure 11: Comparison of predictions (blue line) in systems with a temperature dependent k ij value (left) and no k ij value (right) for Benzoic acid in 3 solvents with experimental data (red circles). Green dots represent the initial guess The magnitude of the predicted solubilities in Figure 11 agrees reasonably well with the experimental data and the solubility trend with respect to temperature is satisfactorily described for each system. 19

21 Experimental Solubility of Benzoic Acid butanol (kij) 1-Butanol (no Kij) Ethyl Acetate (kij) Ethyl Acetate (no kij) Predicted Solubility of Benzoic Acid Methyl Ethyl Ketone (kij) Methyl Ethyl Ketone (no kij) Figure 12: Comparison of experimental and predicted solubility for benzoic acid in multiple solvents for both temperature dependent and no binary interaction parameter In Figure 12 it became clear that the model works better for some solvents than others. Both methyl ethyl ketone and 1-butanol had reasonably good results for both temperature dependent and no k ij parameters. Ethyl acetate, however, deviated significantly more from the experimental values. This was perhaps simply due to the higher degree of solubility of the solute in ethyl acetate meaning the model has difficulty predicting behaviour at higher solubilities. This is backed up by the slope of the lines for the other two solvents in Figure 12 which indicates that if the temperature range extended and the solubility continued to increase the performance of the model would decrease. There was no appreciable difference between using a temperature depended k ij parameter and no k ij parameter noted for these systems. Overall more variation in the acetate parameters was observed. This is possibly due to the two oxygen atoms bonded to one carbon which would create a more positive charge on that atom than in the ketone where only one oxygen atom bonds to a carbon. The rest of the solvents were then analysed similarly and compared. 2

22 Experimental Solubility Data Methanol 1-Propanol 1-Butanol Ethyl Acetate Isopropyl Acetate Methyl Ethyl Ketone Methyl Isobutyl Ketone Predicted Solubility Data Figure 13: Comparison of predicted and experimental solubility data for benzoic acid in various solvents with a temperature dependent kij Figure 13 shows that the model continues to be unable to accurately predict the behaviour of the esters while the alcohol solvents are all reasonable well modelled. It is also clear that for each solvent, as the solubility increases, the deviation from experimental data increases with the model typically predicting higher solubilities than are actually experienced Methanol 1-Propanol 1-Butanol Ethyl Acetate Methyl Ethyl Ketone Methyl Isobutyl Ketone Figure 14: Comparison of predicted and experimental solubility data for benzoic acid in various solvents without a binary interaction parameter 21

23 Experimental Solubility The data for isopropyl acetate was not included in Figure 14 as the predicted solubility was found to be outside of physical limitations. It was again seen that the model was unable to predict accurate values for the ester solvents; however, the other systems were in good agreement. It was still unclear if the use of a binary interaction parameter was beneficial overall, some systems performed better with the temperature dependent k ij while others did with no k ij. The predictions for solubility in 1-butanol matched experimental data better when a binary interaction parameter was used which was discussed above with regards to the deviation in parameters of this solvent in Table 4. The esters also performed better with the use of a k ij value which may be due to the increase polarity of the solvent in comparison to the ketones Isobutanol 1-Butanol Ethyl Acetate Isobutyl Acetate Methyl Isobutyl Ketone Predicted Solubility Figure 15: Comparison of predicted and experimental solubility data for aspirin in various solvents with a temperature dependent k ij. The model has significant difficulty in predicting the solubility of only slightly soluble solutes. The predictions for benzoic acid agreed better with experimental data than that of those for aspirin show above in Figure 15. All the results experience a similar degree of under prediction which suggests this is a fault in the model itself and not an issue with any particular solvent or solvent group. This may again be a result of the lack of an association term and the error from is more evident when the degree of solubility is small. 22

24 Experimental Solubility Isobutanol (kij) 1-Butanol (kij) Isobutanol (no kij) 1-Butanol (no kij) Predicted Solubility Figure 16: Comparison of predicted and experimental solubility data for aspirin in various solvents with and without a binary interaction parameter, k ij. The trends observed in Figure 16 suggest that for aspirin, not using a binary interaction parameter may produce better results. This is possibly due to the low level of interactions between unlike molecules due to the limited solubility and in this case the binary interaction parameter simply propagates the error due to the lack of an association term. These predictions for each particular system were determined from parameters regressed for that specific system. Ideally, parameters regressed from one system can be used for the same solute or solvent in any other system. The following predictions used averaged solute and solvent parameters for benzoic acid and, as this model was unable to predict behaviour due to the association forces in the system, parameters from one system will initially only be used in another system with the same solvent group. Table 12: Averaged solute parameters for each solvent group Benzoic Acid in: Alcohols Ketones Acetates m σ ε k ij slope k ij intercept

25 Experimental Solubility Experimental Solubility Table 13: Averaged solute parameters for each solvent group, no kij Benzoic Acid in: Alcohols Ketones Acetates m σ ε Methanol 1-Propanol 1-Butanol Methyl Ethyl Ketone Methyl Isobutyl Ketone Ethyl Acetate Isopropyl Acetate..5 Predicted Solubility Figure 17: Pure prediction for solubility of benzoic acid in various solvents compared with experimental data, temperature dependent k ij.5 Methanol 1-Propanol 1-Butanol Methyl Ethyl Ketone Methyl Isobutyl Ketone Ethyl Acetate Isopropyl Acetate.5 Predicted Solubility Figure 18: Pure prediction for solubility of benzoic acid in various solvents compared with experimental data, no k ij 24

26 Figure 17 showed good prediction of the solubility in the alcohol solvents but poor prediction for the ketones and esters when a binary interaction parameter is used. Figure 18 shows the model predictions verses the experimental solubility data where the parameters were regressed without a binary interaction parameters. Overall, the predictions behave better when no binary interaction parameter is used. Both Figure 17 and Figure 18 used averaged solute parameters for each solvent group but an ideal model would be able to use parameters from any one solvent system in another. As a test of this the average solute parameters over all solvents was used for solubility predictions. In the cases where a binary interaction parameter was used, the k ij value was averaged over the solvent group..5 Methanol 1-Propanol 1-Butanol Methyl Ethyl Ketone Methyl Isobutyl Ketone Ethyl Acetate Isopropyl Acetate..5 Figure 19: Pure prediction for solubility of benzoic acid in various solvents compared with experimental data using overall averaged solute parameters, temperature dependent k ij 25

27 .5 Methanol 1-Propanol 1-Butanol Methyl Ethyl Ketone Methyl Isobutyl Ketone Ethyl Acetate Isopropyl Acetate..5 Figure 2: Pure prediction for solubility of benzoic acid in various solvents compared with experimental data using overall averaged solute parameters, no k ij It appears as though using averaged parameters for the solute improves the accuracy of the predictions. This is likely due to mitigation of error in one system by averaging it with others. This suggests that if parametrization is done using multiple solvents more accurate parameters will be obtained. Interestingly, predictions made from parameters where no binary interaction parameter was considered had better agreement with the experimental results. Discussion It was found that the initial guesses used for the parameters during regression had a significant effect on the final values and the solubility trends predicted. Equations 1-3 provided suitable starting guesses for the non-associating parameters and these were used for all systems. The initial guess for the binary interaction parameter was investigated and it was found that if the guess was set further from the final value, the correlation between the experimental data and model decreased; if it was set near the final value, the solubility trend shown by the model was negatively affected, in some cases showing an opposite trend. Due to this an initial guess of zero was deemed the most suitable. Molecules with a higher molecular weight with a corresponding increase in the importance of dispersion forces in their intermolecular interactions had better agreement between the model and experimental data during the parameterization step. 26

28 The parameters collected for both systems with a temperature dependent binary interaction parameter and no binary interaction parameter were analysed and the standard deviation was calculated. The variation, taken as the standard deviation as a percentage of the average, showed that the parameters for the system with no k ij agreed better than those with a temperature dependent k ij. This was then confirmed when the parameters were used for predicting the solubility for each solvent system where it was found that no k ij gave better agreement with the experimental data than a temperature dependent k ij. Overall it appears that the absence of the association term creates error in the model parameters as it is unable to account for association forces in its solubility calculations. This error is picked up by the parameters to varying degrees depending on the polarity and potential for hydrogen bonding of the solvents. The predictions of one system using the parameters regressed specifically for that system had greater error than those where the parameters were averaged over several systems. This indicates that by averaging the parameters the associated error is mitigated somewhat allowing for more accurate predictions. It would also suggest that for better determination of the parameters, it would be useful to regress the parameters from systems with different solvents. Conclusions While this model did not account for association contributions, the predictions from the regressed parameters satisfactorily reflected both the magnitude and trend of the solubility over the designated temperature range. The error associated with neglecting these terms was found to be mitigated by averaging the parameters from several different solvent systems. A temperature dependent binary interaction parameter actually decreased the accuracy of the predictions while also requiring additional experimental data points for regression. It would therefore be best to use no binary parameter and run additional experiments in a different solvent if possible. Recommendations The model is readily adjustable to allow for the inclusion of the association terms and preliminary code for this was attempted based on the work done by Angel Martin (21). By including this term the variation between the parameters can be reduced and the binary interaction parameter may improve the predictive capabilities of the model instead of hindering it. It would also mean that fewer systems would need to be analysed to ensure accurate parameters are regressed. Furthermore, the method of making predictions discussed in the materials and methods section may not be the optimal approach and others should be investigated as discussed in that section. It is also possible to adjust the method of initial guesses; for example, instead of selecting a single value for an initial solubility guess a temperature dependent guess could be made. 27

29 Appendices Appendix A: Bibliography Bouillot, B., Teychene, S. & Biscans, B., 211. An evaluation of thermodynamic models for the prediciton of drug and drug-like moecule solubility in organic solvents. Fluid Phase Equilibria, Volume 39, pp Chapman, W., Gubbins, K., Jackson, G. & Radosz, M., Equation of state solution model for associating fluids. Fluid Phase Equilibria, pp Chen, C. & Crafts, P., 26. Correlation and Prediction of Drug Molecule Solubility in Mixed Solvent Sytems with the Nonrandom Two-Liquid Segment Activity Coefficient (NRTL-SAC) Model. Industrial & Engineering Chemistry Research, 45(13), pp Fisher Scientific, 29. Acetylsalicylic Acid MSDS, New Jersey: Fisher Scientific. Gross, J. & Sadowski, G., 21. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res., Issue 4, pp Laube, F. & Sadowski, G., 214. Predicting the Extraction Behaviour of Pharmaceuticals. Industrial & Engineering Chemistry Research, 53(2), pp Martín, Á., 211. PC-SAFT open source code. Spain: University of Valladolid. Reichardt, C., 23. Solvents and Solvent Effects in Organic Chemistry. Third Edition ed. Weinheim: Wiley-VCH. Ruether, F. & Sadowski, G., 29. Modelling the Solubility of Pharmaceuticals in Pure Solvents and Solvent Mixtures for Drug Process Design. Journal of Pharmaceutical Sciences, 98(11), pp Spyriouni, T., Krokidis, X. & Economou, I., 211. Thermodynamics of Pharmaceuticals: Prediction of Solubility in Pure and Mixed Solvents with PC-SAFT. Fluid Phase Equilibria, 32(12), pp Thati, J., Nordstrom, F. & Rasmuson, A., 21. Solubility of Benzoic Acid in Pure Solvents and Binary Mixtures. J.Chem.Eng.Data, Issue 55, pp Wei, J. S. & Sadus, R., 2. Equations of State for the Calculation of Fluid-Phase Equilibria. AIChE Journal, 46(1), pp Yalkowsky, S., He, Y. & Jain, P., 21. Handbook of Aqueous Solubility Data. Second Edition ed. Boco Raton: CRC Press. 28