Bjerrum Plots

The plot of pH against titrant volume added is called a potentiometric titration curve. The latter curve is usually transformed into a Bjerrum plot [8, 24, 27], for better visual indication of overlapping pKiS or for pffjS below 3 or above 10. The actual values of pKa are determined by weighted nonlinear regression analysis [25-27]. 

Avdeef, A., pH-metric solubility. 1. Solubility-pH profiles from Bjerrum plots. Gibbs buffer and pfCa in the solid state. Pharm. Pharmacol. Commun. 1998, 4,165-178. 

Figure 3.1 Four-step construction of the Bjerrum difference plot for a three-pi 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.]...

Figure 6.8 shows the Bjerrum plots for an weak acid (benzoic acid, pKa 3.98, log So — 1.55, log mol/L [474]), a weak base (benzydamine, pKa 9.26, log So —3.83, log mol/L [472]), and an ampholyte (acyclovir, pKa 2.34 and 9.23, log So — 2.16, log mol/L I/40N ). These plots reveal the pKa and pA pp values as the pcH values at half-integral % positions. By simple inspection of the dashed curves in Fig. 6.8, the pKa values of the benzoic acid, benzydamine, and acyclovir are 4.0, 9.3, and (2.3, 9.2), respectively. The pA pp values depend on the concentrations used, as is evident in Fig. 6.8. It would not have been possible to deduce the constants by simple inspection of the titration curves (pH vs. volume of titrant, as in Fig. 6.7). The difference between pKa and pA pp can be used to determine log So, the intrinsic solubility, or log Ksp, the solubility product of the salt, as will be shown below.

Figure 6.8 Bjerrum plots for (a) benzoic acid (black circle = 87 mM, unfilled circle = 130 mM, gray cicle = 502mM), (b) benzydamine (black circle = 0.27 mM, unfilled circle = 0.41 mM, gray circle = 0.70 mM), and (c) acyclovir (black circle = 29 mM, unfilled circle = 46 mM). The dashed curves correspond to conditions under which no precipitation takes place.

Since soy lecithin ( 20% extract from Avanti) was selected as a basis for absorption modeling, and since 37 % of its content is unspecified, it is important to at least establish that there are no titratable substituents near physiological pH. Asymmetric triglycerides, the suspected unspecified components, are not expected to ionize. Suspensions of multilamellar vesicles of soy lecithin were prepared and titrated across the physiological pH range, in both directions. The versatile Bjerrum plots (Chapter 3) were used to display the titration data in Fig. 7.33. (Please note the extremely expanded scale for %.) It is clear that there are no ionizable groups... 

Figure 7.33 Bjerrum plot for titration of a suspension of 1 mM soy lecithin.

The charge-state section highlighted the value of Bjerrum plots, with applications to 6- and a 30-pKa molecules. Water-miscible cosolvents were used to identify acids and bases by the slope in the apparent pKa/wt% cosolvent plots. It was suggested that extrapolation of the apparent constants to 100% methanol could indicate the pKa values of amphiphilic molecules embedded in phospholipid bilayers, a way to estimate pAi m using the dielectric effect. 

Avdeef, A. Kearney, D. L. Brown, J. A. Chemotti, A. R. Jr., Bjerrum plots for the determination of systematic concentration errors in ttitration data, Anal. Chem. 54, 2322-2326 (1982). 

A difference plot, also called a Bjerrum plot, is an excellent means to extract metal-ligand formation constants or acid dissociation constants from titration data obtained with electrodes. We will apply the difference plot to an acid-base titration curve. 

In acid-base titrations, a difference plot, or Bjerrum plot, is a graph of the mean fraction of protons bound to an acid versus pH. The mean fraction is nH calculated with Equation 13-59. For complex formation, the difference plot gives the mean number of ligands bound to a metal versus pL(= —log[ligand]). 

To extract acid dissociation constants from an acid-base titration curve, we can construct a difference plot, or Bjerrum plot, which is a graph of the mean fraction of bound protons, H, versus pH. This mean fraction can be measured from the quantities of reagents that were mixed and the measured pH. The theoretical shape of the difference plot is an expression in terms of fractional compositions. Use Excel SOLVER to vary equilibrium constants to obtain the best fit of the theoretical curve to the measured points. This process minimizes the sum of squares [nH(measured) -nH( theoretical) 2. 

Table II. Slope of Brpnsted—Bjerrum Plot for Some Horse-Heart Cytochrome c Reactions (25 °C)...

With the potentiometric approach, determination of intrinsic solubility is based upon the measurement of the pH shift caused by compound precipitation during acid-base titration of ionizable compounds. Two commercial potentiometric methods currently available are pSol [30, 39] and Cheqsol [40-42], In the pSol method developed by Avdeef, a minimum of three titrations in the direction of dissolution are performed. Normal pH versus volume titration plots are reexpressed as Bjerrum plots, that is, average number of bound protons versus pH. The Bjermm plots enable the shift in compound pKa to be more readily observed and are used to determine intrinsic solubility (S0) via Equation 2.5 ... 

Titration in a mixture with resolution of difference UV-visible spectra Data analysis for up to nine components with TITAN program Bjerrum plots for determinationn of systematic cone, errors Evaluation of digital potentiometric titns. by the Tubbs method... 

The key element in the data analysis is the construction of the difference or Bjerrum plot. This plot shows the average number of bound protons versus pcH. The difference plot is obtained from the difference between two titration curves one is the titration of an ionizable substance and the other is a blank titration. Graphically, the pKa corresponds to the pH where the average number of bound proton equals 0.5 (or a multiple of 0.5 if multiple ionization takes place). Nowadays nonlinear regression calculations are used to derive pKa values form Bjerrum plots. 

Avdeef A, Kearney DL, Brown JA, Chemotti Jr AR (1982) Bjerrum plots for the determination of systematic concentration error in titration data. Anal Chem 54 2322-2326... 

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