ABRAHAM SOLVATION PARAMETER MODEL: PREDICTION OF ENTHALPIES OF VAPORIZATION AND SUBLIMATION OF MONO-METHYL BRANCHED ALKANES USING MEASURED GAS CHROMATOGRAPHIC DATA

Abraham model L solute descriptors have been determined for 174 additional mono-methyl branched alkanes based on published linear-programmed gas chromatographic retention indices. Standard molar enthalpies of vaporization and sublimation at 298 K are calculated for the 174 mono-methylated alkanes using the reported solute descriptors and our recently published Abraham model correlations. Calculated vaporization and sublimation enthalpies derived from the Abraham model compare very favourably with values based on a popular atom-group additivity model. Unlike the additivity model the Abraham model gives different predicted values for each mono-methyl alkane having a given CnH2n+2 molecular formula.


INTRODUCTION
Gas-liquid chromatographic measurements [1][2][3][4][5][6][7][8][9][10] have been used in the indirect determination of both standard molar enthalpies of vaporization, ∆Hvap,298K, and standard molar enthalpies of sublimation, ∆Hsub,298K, of organic compounds at 298 K. For example, Hamilton 1 determined the ∆Hvap,298K of eleven herbicide esters based on experimental gas chromatographic retention volumes, Vg, measured on a nonpolar SE-30 stationary phase. The method assumed that the ratio of the enthalpy of vaporization of each herbicide ester to that of the reference compound (which in this case was dibutyl phthalate) was independent of temperature. The ∆Hvap,298K of each individual ester herbicide was calculated from the slope of the graph of ln(Vg,ester/Vg,reference) versus the natural logarithm of the vapor pressure of the reference compound at the column temperature T, ln Preference,T, in accordance to Eqn. (1). Peacock and Fuchs 2-4 developed a method for determining ∆Hvap,298K based on solution calorimetric measurements of liquid organic compounds being dissolved in the stationary phase solvent. The enthalpy of vaporization was calculated as the difference in the measured enthalpy of solution of the organic liquid, ∆Hsoln,298K, minus the chromatographicallymeasured enthalpy of solution of the gaseous compound in the stationary phase liquid. The later value was determined from the variation in the compound's retention volumes with temperature, and then corrected back to 298 K using liquid-phase and gas-phase heat capacities.
Chickos and coworkers 5 proposed a method for determination of ∆Hvap,298K based on linear plots of the chromatographically-measured ∆Hsoln values of gaseous reference compounds in the liquid stationary phase versus the compounds' known ∆Hvap,298K values. Enthalpies of vaporization of additional compounds can then be calculated from the linear mathematical relationship established by the reference compounds. The authors demonstrated the applicability of their method using 102 hydrocarbon and mono-functional hydrocarbon derivatives. Enthalpies of vaporization based on the authors' method differed from published literature values by a standard deviation of 1.27 kJ mol -1 . The method was later extended to the determination of ∆Hsub,298K by combining ∆Hvap,298K values measured by correlation gas chromatography with calorimetric enthalpy of fusion, ∆Hfus,298K, adjusted to 298 K. 6 Numerical values of ∆Hvap,298K and ∆Hsub,298K determined in this fashion depend on the reference compounds used in establishing the ∆Hsoln versus ∆Hvap,298K mathematical correlation.
Our method of obtaining ∆Hvap,298K and ∆Hsub,298K values is more of a computational method that uses gas chromatographic retention data to calculate Abraham model solute descriptors. Once calculated, the numerical values of the solute descriptors are then used in conjugation with our published Abraham model correlations 11,12 to calculate the desired ∆Hvap,298K and ∆Hsub,298K values of organic, organometallic and inorganic compounds. The Abraham solvation parameter model is among the most widely used linear free energy relationship in the prediction of solute properties having chemical and biological significance. To date predictive mathematical correlations have been reported for describing solute transfer into more than 130 different organic nonelectrolyte mono-solvents [13][14][15][16][17][18][19] and into more than 100 different ionic liquid solvents. [20][21][22][23][24][25][26][27][28][29] Mathematical correlations have also been developed for predicting enthalpies of solvation of organic vapors and inorganic gases into water and 35 common organic solvents [30][31][32][33][34][35][36][37][38][39][40] blood-to-body tissues/fluids partition coefficients, [41][42][43][44][45] lethal median concentrations of organic compounds towards fish and other aquatic organisms, [46][47][48][49] nasal pungency, 50-53 eye irritation thresholds and Draize eye scores, [53][54][55] and many other solute properties. [56][57][58][59][60][61] More recently the Abraham model has been extended to predicting enthalpies of vaporization 11 and sublimation 12 and the vapor pressure of organic and organometallic compounds. 62 In the present communication we illustrate the application of the Abraham solvation parameter model in predicting ∆Hvap,298K and ∆Hsub,298K values. First, we calculate the Abraham model solute descriptors of mono-methyl branched alkanes from published gas chromatographic retention indices of Krkosova and co-workers. 63 Once calculated, the solute descriptors will be substituted into our previously published Abraham model correlations. 11 Thus enabling the estimation of ∆Hvap,298K and ∆Hsub,298K values for those compounds for which solute descriptors are known. Solute descriptors are identified in Eqns. 2 and 3 by the capitalized alphabetical characters, and are defined as follows: the solute excess molar refractivity expressed in units of (cm 3 mol -1 ) / 10(E); the solute dipolarity/polarizability (S); the overall or summation hydrogenbond acidity and basicity (A and B, respectively); and the logarithm of the gas-to-hexadecane partition coefficient at 298 K (L). Both Abraham model correlations use indicator variables (Iamine, INH, Inon-αω-diol, Iαω-diol, IOH,adj, IOH,non) to improve the predictions or organic compounds having amino-and more than one hydroxy-functional group. Mono-methylalkanes do not contain either of these functional groups, so no further discussion of indicator variables is needed. The two mathematical correlations were developed based on ∆Hvap,298K and ∆Hsub,298K values for N = 703 and N = 864 compounds, respectively. As indicated by the standard deviation (SD), squared correlations coefficient (R 2 ), and Fisher F-statistic (F), both Abraham model correlations provide reasonably accurate mathematical correlations of the ∆Hvap,298K and ∆Hsub,298K data for wide range of organic compounds.
Several earlier publications have illustrated the calculation of Abraham model solute descriptors from either liquidliquid partition coefficients, 64 or high-performance liquid chromatographic retention data, 65 or in the case of crystalline nonelectrolyte compounds from saturation solubilities. [66][67][68][69][70] The latter papers primarily focused on using the calculated solute descriptors to select organic solvents for recrystallization and/or biphasic partitioning systems for liquid extraction. The intended audience of the solubility studies were chemical engineers and industrial working in the chemical manufacturing sector. Recrystallizations and liquid extractions are commonly used purification methods in chemical syntheses. A more recent publication 71 reported Abraham solute descriptors of terpene esters determined from gas-liquid chromatographic retention data of solutes eluted on several stationary phase liquids. Here the application was to predict the human odor thresholds of the terpene esters. Solute descriptors of terpene hydrocarbons 72 had been reported previously. There was very little information in the afore-mentioned studies that would attract the attention of chemical thermodynamic experts or computation chemists, which is the intended audience of the current communication. The calculated solute descriptors of mono-methyl branched alkanes will be used to predict thermodynamic properties, namely ∆Hvap,298K and ∆Hsub,298K values. These thermodynamic quantities are required in the calculation of gas-phase standard molar enthalpies of formation from measured enthalpies of combustion, and in describing how the vapour pressure of a compound varies with temperature. Such information is also needed by individuals working in the chemical manufacturing sector.

CALCULATION OF ABRHAM MODEL SOLUTE DESCRIPTORS
Determination of solute descriptors generally involves constructing a series of Abraham model correlations that involve solute transfer between two condensed phases (Eqn. 4) or solute transfer from the gas phase into a condensed phase (Eqn. 5).
Solute property = ck + ek · E + sk · S + ak · A + bk · B Solute properties used in these computations have included the logarithms of partition coefficients, logarithms of molar solubility ratios, logarithms of chromatographic retention factors, and chromatographic retention indices. Two of the solute descriptors, E and V (McGowan volume), can be reasonably estimated from the solute's molecular structure. For solutes that lack an acidic hydrogen capable of hydrogen-bond formation, the A solute descriptor can be set equal to zero. This leaves either four solute descriptors  Table 1. The derived mathematical relationship then allows us to calculate the L-solute descriptors of the remaining 174 mono-methyl branched alkanes. These calculations are summarized in the last column of Table 1. Examination of the numerical entries reveals that eqn. (6) provides reasonably accurate backcalculation of the known L descriptor values as one might expect from the correlation's small standard deviation, SD = 0.022, and near unity value for the squared correlation coefficient, R 2 = 1.000.

PREDICTION OF STANDARD MOLAR ENTHALPIES OF VAPORIZATION AND SUBLIMATION
The chromatographic retention measurements performed by Krkosova and coworkers 63 allowed us to have a complete set of solute descriptors for an additional 180 saturated hydrocarbons. Previously we had only the five solute descriptors (E, S, A, B, and V) needed for Eqn. (4). Published studies have shown, however, that Eqn. (5) of the Abraham model provides the better set of predicted values for several thermodynamic properties such as enthalpies of vaporization 11 and enthalpies of solvation of organic vapours and inorganic gases dissolved both in water and in organic solvents. [30][31][32][33][34][35][36][37][38][39][40] Having a complete set of solute descriptors will provide better applicability for these important thermodynamic quantities.
We illustrate the application of the Abraham model by calculating the enthalpies of vaporization (Eqn. 7) and enthalpies of solvation (Eqn. 8) of the 174 mono-methyl branched alkanes for which we have just determined the L descriptor. For the convenience of the reader we have simplified the predictive expressions to contain only the non-zero terms.
∆Hvap,298K (kJ mol -1 ) = 6.100 + 9.537 L ΔHsub,298K (kJ mol -1 ) = 13.93 + 13.57 L -0.05 L L (8) Enthalpy of sublimation predictions given in Table 2, start with the C20-compounds as most of the smaller compounds is liquid at 298 K. Predicted values of ∆Hvap,298K are given in Table 3 for all compounds as vaporization enthalpies of compounds that are crystalline at 298 K can be easily determined using the method of correlation gas chromatography. 5 We are unable to find experimental ∆Hvap,298K and ∆Hsub,298K data in the published chemical literature to compare our calculated values against. What we offer in the way of a comparison is to compare our calculated values against the calculated values of a popular group-additivity method 73 that has been shown to predict ∆Hvap,298K and ∆Hsub,298K values for a wide range of organic and organometallic compounds to within standard deviations of SD = 4.30 kJ mol -1 (N = 3460 compounds) and SD = 10.33 kJ mol -1 (N = 1866 compounds), respectively. The basic method (Eqn. 9) sums the contributions that each atomic group makes to the given thermodynamic or physical property, where Ai is the number of occurrences of the ith atom group, Bj is the number of times each special group occurs, ai and bj are the numerical values of each atom group and special group, and C is a constant. For both the ∆Hvap,298K and ∆Hsub,298K computations a CnH2n+2 mono-methyl branched alkane would be fragmented into 3 sp 3 carbons (with an environment of 3 hydrogen atoms and 1 carbon), 1 sp 3 carbon atom (with an environment of 1 hydrogen atom and 3 carbon atoms), n-4 sp 3 carbon atoms (with an environment of 2 hydrogen atoms and 2 carbon atoms), and one special alkane group that is multiplied by the number of carbon atoms in the molecule. Numerical values of the groups values and constant are different for each property. In Eqns. (10) and (11) below we have filled in the numerical group values and constants for predicting ∆Hvap,298K (kJ mol -1 ) and ∆Hsub,298K(kJ mol -1 ) of CnH2n+2 mono-methyl branched alkanes: ∆Hvap,298K = 3 x 3.07 + (n-4) x 4.67 + 3.57 + n x 0.09 + 8.61 (10) ∆Hsub,298K = 3 x 5.99 + (n-4) x 6.88 + 2.28 -n x 0.53 + 21.03 (11) Examination of the numerical entries in Tables 2 and 3 reveals that the predictions based on the Abraham model are similar to predictions based on the group-additivity model of Naef and Acree. 73 The group-additivity method though is not able to distinguish between the placement of the methyl group within the molecule, and gives the same predicted values for a given molecular formula. In other words, the predicted values of all methylheneicosane molecules are the same. This limitation is a common feature of most groupaddivity and group contribution methods. The Abraham model, on the other hand, does provide different predicted values for a given molecular formula, and does not require fragmentation of the molecule into atom groups or functional groups. Fragmentation of molecules into functional groups can be difficult at times, particularly in the case of more complex molecules having many different functional groups. Moreover, the solute descriptors for a given molecule can be used to predict many other properties of chemical and biological importance, such as vapour pressure, water-to-organic solvent partition coefficients, gasto-water partition coefficients, solubility ratios and the infinite dilution activity coefficients of the compound in water. 74,75 CONCLUSION Numerical values of the Abraham model L solute descriptor have been reported for the first time for 174 different C12-C30 mono-methyl branched alkanes. The numerical values were determined by regression analysis of published linear-programmed gas chromatographic retention indices versus known L solute descriptors of linear alkanes and smaller mono-methylated alkane molecules. Calculated L solute descriptors were used to predict the standard molar enthalpies of vaporization and standard molar enthalpies of sublimation of 174 mono-methyl alkanes at 298 K based on recently published Abraham model correlations. 11,12 The predicted values compare very favorably with calculated values based on an atom-group additivity model. 73 Unlike the additivity model the Abraham model gives different predicted values of ∆Hvap,298K and ∆Hsub,298K for each monomethyl alkane having a given CnH2n+2 molecular formula.