Figure 3.1 Four-step construction of the Bjerrum difference plot for a three-pKa molecule, whose constants are obscured in the simple titration curve (see text): (a) titration curves; (b) isohydric volume differences; (c) rotated difference plot; (d) Bjerrum plot. [Avdeef, A., Curr. Topics Med. Chem., 1, 277-351 (2001). Reproduced with permission from Bentham Science Publishers, Ltd.]

concentration divided by that of the sample gives the average number of bound hydrogen atoms per molecule of substance The Bjerrum curve is a plot of % versus pcH.

Operationally, such a plot can be obtained by subtracting a titration curve containing no sample (''blank'' titration; left curve in Fig. 3.1a) from a titration curve with sample (right curve in Fig. 3.1a) at fixed values of pH. The resultant difference plot is shown in Fig. 3.1b. The plot is then rotated (Fig. 3.1d), to emphasize that nH is the dependent variable and pH is the independent variable [163]. The volume differences can be converted to proton counts as described in the preceding paragraph, to obtain the final form, shown in Fig. 3.1d.

The Bjerrum plot in Fig. 3.1d reveals all the pKa terms as pcH values at halfintegral % positions. The three pKa values of M6G are evident: 2.8, 8.2, and 9.4. In contrast to this, deducing the constants by simple inspection of the titration curves is not possible (Fig. 3.1a): (1) the low pKa is obscured in Fig. 3.1a by the buffering action of water and (2) the apparent pKa at pH 8.8 is misleading. M6G has two overlapping pKa terms, whose average value is 8.8. M6G nicely illustrates the value of Bjerrum analysis. With Bjerrum analysis, overlapping pKas pose no difficulty. Figure 3.2a shows an example of a 6-pKa molecule, vancomycin [162,166]. Figure 3.2b shows an example of a 30-pKa molecule, metallothionein, a small heavy-metal-binding protein, rich in sulfhydryl groups [167]. (The reader is challenged to identify the six ionization sites of vancomycin.)

3.3.2 pH Definitions and Electrode Standardization

To establish the operational pH scale [168-170], the pH electrode can be calibrated with a single aqueous pH 7 phosphate buffer, with the ideal Nernst slope assumed. Because the nH calculation requires the ''free'' hydrogen ion concentration (as described in the preceding section) and because the concentration scale is employed for the ionization constants, an additional electrode standardization step is necessary. That is where the operational scale is converted to the concentration scale pcH (=— log [H+]) using the four-parameter equation [116,119,171,172]

where Kw is the ionization constant of water [173]. The four parameters are empirically estimated by a weighted nonlinear least-squares procedure using data from alkalimetric titrations of known concentrations of HCl (from pH 1.7 to 12.3) or standard buffers [116,174-180]. Typical aqueous values of the adjustable parameters at 25°C and 0.15 M ionic strength are a = 0.08 ± 0.01, ks = 1.001 ± 0.001, jH = 1.0 ± 0.2, and jOH = — 0.6 ± 0.2. Such a scheme extends the range of accurate pH measurements and allows pKa values to be assessed as low as 0.6 (caffeine [161]) and as high as 13.0 (debrisoquine [162]).

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