Four-parameter Equations Research Articles

Mobile phase effects in reversed-phase chromatography : VII. Dependence of retention on mobile phase composition and column temperature

The relationship between logarithmic retention factor, K, in reversed-phase chromatography and operating conditions including mobile phase composition and column temperature is still controversial. Earlier, the following analytical relationship was proposed for use with alkyl benzenes: K = A 1 ø(1 - T c/ T) + A 2/ T + A 3 where ø is volume fraction organic co-solvent of the hydro-organic mobile phase, T is absolute temperature and A 1- A 3 are parameters appropriate to the eluite and stationary phase. Insofar as its use had not been tested with polar eluites that are retained by only one mechanism, its description of the retention of 54 polar and apolar eluites was examined and compared to equations with other, usually more complex, relationships between retention and operating conditions. The following four-parameter equation described best most data: K = A 1 ø(1 - T c/ T) + A 2/ T + A 3 + A 4ø. But neither this nor any other four-parameter equation under examination here emerged solely as the best expression for the dependence of retention on the eluent composition and temperature. The average relative errors were 7.8% and 6% with use of the three- and the four-parameter equations which were found to be best for a given eluite, respectively. In view of the small decrease in error but significant increase in complexity and in data points with the use of the four-parameter equation, the three-parameter equation is recommended for general use.

Expanded Solubility Parameter Approach II: P-Hydroxybenzoic Acid and Methyl P-Hydroxybenzoate in Individual Solvents

The recently introduced four-parameter extended Hansen approach was used to study the solubility of p-hydroxybenzoic acid and methyl p-hydroxybenzoate in 32 and 35 individual solvents, respectively. The results are compared with those for benzoic acid in 40 solvents. Seventeen of the thirty-two or 53% of the calculated solubilities of p-hydroxybenzoic acid were within the established solubility criterion (i.e., <30% error from the experimental value). Twenty-two of thirty-six or 61% of the calculated solubility values for methyl p-hydroxybenzoate met the solubility criterion. Experimental excess free energies of solution for p-hydroxybenzoic acid and methyl p-hydroxybenzoate were plotted against theoretical values using the expanded four-parameter solubility regression equations. From such results, adjustments may be made in the partial solubility parameters to bring the calculated solubilities in line with experimental results. Multiple regression analyses were used to estimate the total solubility parameter and the four partial solubility parameters of the two benzoic acid derivatives. Satisfactory results were obtained for methyl p-hydroxybenzoate, but poor agreement was found for p-hydroxybenzoic acid for the total parameter when compared with the Fedors group contribution method. Both the multiple regression and group contribution methods may yield inaccurate solubility parameters for relatively polar solid solutes. Factor analysis was used to test the adequacy of three- and four-parameter approaches in the evaluation of drug solubility. A principal factor method without iteration and orthogonal factor rotation were used to compare the two expanded solubility parameter approaches. Factor analysis showed that four solubility parameters were significantly more independent and presumably more satisfactory than three parameters.

Parametric correlation of formation constants in aqueous solution. 1. Ligands with small donor atoms

It is shown that four-parameter equations proposed by previous authors have poor predictive powers for data of formation constants in aqueous solution. This relates partly to the paucity of data on complexes of ligands such as ammonia which, with most metal ions, cannot exist in water because of hydrolysis. Equations previously proposed that relate the formation constants of polyamine and poly(aminbcarboxy1ate) complexes to those of the ammonia and acetate complexes are used to calculate formatio‘n constants for these hydrolysis-prone ammonia complexes. An equation of the type log K, = EAEB + CACB, where E and C are identified with the tendencies of the Lewis acid A and base B to undergo ionic and covalent bonding, was used to correlate the F-, OH-, and NH, formation constants of 27 Lewis acids to a standard deviation of 0.24 log unit. Hardness parameters l!fA and HB were defined as EA/CA and EB/CB for acids and bases, respectively, and gave reasonable orders of hardness It was found that for ligands with large donor atoms, such as C1-, and for sulfur or phosphorus donor atoms, deviations from the predictions of this equation were observed that appeared to be related to the size of the acid, so that no deviations were observed for large cations such as Ag’ and Pb2+, aith Occurrence of large deviations for smaller cations such as Cu” or Ni2+, with the very largest occurring for the proton. These deviations were attributed to steric hindrance between the large donor atom and adjacent coordinated water molecules. The important contribution of the Edwards equation’ lies in illustrating that formation constant data for complexes of unidentate ligands in aqueous solution need at least a dualbasicity scale for any kind of correlation to be obtained at all. There have been several other multiparameter equation^^-^ proposed for correlating formation constant data. The need for at least a dual-basicity equation has manifested itself in the classification of metal ions’into Schwart~enbach’s~ and Ahrland and Chatt’s6 A- and B-type metal ions and Pearson’s more general classification7 of Lewis acids and bases into hard and soft acids and bases (HSAB). Drago and co-workers* have had considerable success with a more generalized form of a four-parameter equation similar to eq 1 for the correlation of enthalpy changes on adduct formation in solvents of low dielectric constant. In eq 1, which resembles Drago’s expression, except that -AHo has been replaced with log K1, C and E are identified with the tendency of each Lewis acid A or base B to undergo covalent or ionic bonding. This interpretation of the significance of the C and E parameters seems most reasonable in the light of the work of Klopman9 on the origin of hardness and softness in acids and bases. Klopman9 found softness to be associated Rith “frontier-controlled’’ (covalent) and hardness with “charge-controlled’’ (ionic) bonding in calculations based upon polyelectronic pertubation theory. We have therefore adopted the E and C formalism of Drago and co-workers8 in this paper as probably being the best interpretation of the parameters in a four-parameter equation such as (1).