Three-Parameter Equations

Sodium bromide has a very high water solubiUty. At 25°C a saturated solution contains 48.6% sodium bromide by weight. Values for the solubiUty at several temperatures are known (1). Three parameter equations for calculating the solubiUty iu terms of mole fraction of both the anhydrous and dihydrate salts are available (2). Convenient equations for calculating the solubiUty iu weight percent of sodium bromide iu water at various temperatures, t iu °C, are as follows ... 

This three-parameter equation behaves linearly in the Henry s law region and reduces to the Langmuir isotherm for m = 1. Other well-known isotherms include the Radke-Prausnitz isotherm [Radke and Prausnitz, Ind. Eng. Chem. Fundam., 11, 445 (1972) AIChE J., 18, 761 (1972)]... 

Another three-parameter equation that often fits data well and is linear in the Henry s law region is the UNILAN equation [Honig and... 

There are also three-parameter equations. See, for example, de Ligny, C.L. van Houwelin-gen, H.C. J. Chem. Soc., Perkin Trans. 2, 1987, 559. 

Mackay, D., Hughes, A.I. (1984) Three parameter equation describing the uptake of organic compounds by fish. Environ. Sci. Technol. 18, 439 144. 

The three-parameter equation was chosen for the analytical description of the dependence 0 = f(mi) ... 

The inductive parameter, aL, is the same in both the meta and para positions the resonance parameter, aR, is, of course, appreciably different in the two positions the inductive reaction constant is pv This three-parameter equation was employed to calculate reaction types of meta- and para-substituted benzene derivatives. It was shown that free radical processes yielded different values, and a common set of resonance parameters was not possible. The conclusion is, of course, identical to that of van Bekkum and his co-workers (1959). The utility of a unique set of resonance parameters for electrophilic reactions is obscured by the inclusion of both electrophilic side-chain and electrophilic substitution reactions in a single series. 

Eqn.(3.70) is a three-parameter equation which describes the relationship between retention and pH for weak acids. If the dissociation ratio of a buffer (rB = [B ]/[HB]) is used as a variable instead of pH, then a combination of eqns.(3.68) and (3.69) leads to... 

However, a simple linear relationship does not usually exist. A clear example is the optimization of the pH in RPLC. The window diagram approach was applied to this problem by Deming et al. [550,551,552]. They measured the retention of each solute at a series of pH values (9 in ref. [550], 4 in refs. [551,552]) and fitted the experiments to eqn.(3.70). This is a three-parameter equation and hence a minimum of three experiments is required for it to be applied as a description of the retention surface. If more data points are available, the equation can be fitted to the data by regression analysis. 

El Rassi and Horvath185 could fit the logarithmic retention factor of proteins on metal-chelate sorbents by a three-parameter equation ... 

In formal analogy to the KAT equation (7-53), Catalan has combined his sol-vatochromically derived solvent parameters SPP, SA, and SB in the three-parameter equation (7-60) [296, 335-337] ... 

The present paper deals with the application of the fluctuation theory of solutions to the solubility of poorly soluble drugs in aqueous mixed solvents. The fluctuation theory of ternary solutions is first used to derive an expression for the activity coefficient of a solute at infinite dilution in an ideal mixed solvent and, further, to obtain an equation for the solubility of a poorly soluble solid in an ideal mixed solvent. Finally, this equation is adapted to the solubility of poorly soluble drugs in aqueous mixed solvents by treating the molar volume of the mixed solvent as nonideal and including one adjustable parameter in its expression. The obtained expression was applied to 32 experimental data sets and the results were compared with the three parameter equations available in the literature. 

There is only one adjustable parameter (e) in our equation. However, the solubilities of the solute in the individual constituents of the mixed solvent are also needed. Therefore, one can consider our equation as a three-parameter one. For this reason, our results in Table 2 are compared to the best three-parameter equations. One can see from Table 2 that Eq. (23) with the molar volume given by Eq. (25) provides slightly better results than the three-parameter equations available in literature. 

In a previous paper (Ruckenstein and Shulgin, 2003), the Kirkwood-Buff theory of solutions (Kirkwood and Buff, 1951) was employed to obtain an expression for the solubility of a solid (particularly a drug) in binary mixed (mainly aqueous) solvents. A rigorous expression for the composition derivative of the activity coefficient of a solute in a ternary solution (Ruckenstein and Shulgin, 2001) was used to derive an equation for the activity coefficient of the solute at infinite dilution in an ideal binary mixed solvent and further for the solubility of a poorly soluble solid. By considering that the excess volume of the mixed solvent depends on composition, the above equation was modified empirically by including one adjustable parameter. The modified equation was compared with the other three-parameter equations available in the literature to conclude that it provided a better agreement. 

As one can see, M1 contains no adjustable parameter, M2 contains one adjustable parameter, M3-one and M4-two adjustable parameters. However, in addition, all equations (M1-M4) require the solubilities in individual solvents, pure water and organic cosolvent. By considering them as parameters. Ml becomes a two-parameter equation (Xj and xf) M2 - a three-parameter equation (x ,x and e) M3 -a three-parameter equation x, x and F) and M4 - a four-parameter equation (x ,x, Ii3 and I31). Of course, expressions for the activity coefficients with any number of parameters can be used. 

A generalized three-parameter equation for surface tension, y, valid from the triple point to the critical temperature, (where y = 0), has been proposed [1922a] ... 

Iglesias etal. (1975) proposed the following three-parameter equation to fit sorption isotherm data for a range of foods ... 

This three-parameter equation enables solubility to be simulated and correlated quite accurately over a wide range of temperatures. 

The rms fit values of Table 7 are small enough compared to our estimated +100 cm 1 accuracy of measuring tJmax to call fmr linear with y. Although we do not believe it can be stated that solvent donicity causes the changes in vmaK that are observed, use of the three-parameter Equation (17) produces A values that we argue are close to 7.v because of their dependence upon structure of the IV compound. As shown in Scheme 7, the z.v values obtained in Table 7 for similar structural units are... 

Iglesias-Silva et al. [87-iglhol] have proposed an accurate, three parameter equation that can fit data from the triple point to the critical point ... 

Fig. 3.15). In GTA, the isoviscous temperature dependence almost disappears at the TST-invalid viscosity region. In MPD, however, a large proportion of the isoviscous temperature dependence still remained even at 1 x 10 Pa s. The results in MPD suggest that the two coordinates are partially correlated and the fluctuation-limited rate constant kf would show temperature dependence even at a constant viscosity. The fcobs values at TST-valid pressures were fitted to either one of the following empirical three-parameter equations [28, 44] to estimate the TST-expected rate constants. 

---