## Bjerrum Difference Plots

The Bjerrum difference plots [3, 21,27, 28] are probably the most important graphical tools in the initial stages of equilibrium analysis. The difference curve is a plot of hH, the average number of bound protons (that is, the hydrogen ion-binding capacity), versus pcH (-log [H+]). Such a plot can be obtained by subtracting a titration curve containing no sample ("blank" titration) from a titration curve with sample, hence the name "difference" curve. Another way of looking at it is the following. Since one knows how much strong acid and strong base have been added to the solution at any point and since one knows how many dissociable protons the sample substance brings to the solution, one knows the total hydrogen ion concentration in solution, regardless of what equilibrium reactions are taking place. By measuring the pH (and after converting it into pcH), one knows the free hydrogen ion concentration. The difference between the total and the free concentrations is equal to the concentration of the bound hydrogen ions. The latter concentration divided by that of the sample substance gives the average number of bound hydrogen ions per molecule of substance, hH.

Fig. 2 shows the difference plots for the simple monoprotic examples flumequine and diacetylmorphine. Fig. 3 shows plots for the diprotic examples salicylic acid, morphine, and nicotine. The difference plots in Fig. 2 and 3 reveal all the pK.,s and p0/fas as pcH values at half-integral nH positions. By mere inspection, the pKas of salicylic acid can be estimated to be 2.9 and 13.3, while the pQ^as are 5.1 and very slightly greater than 13.3 (as indicated in the inset in Fig. 3a). It would not have been possible to deduce the constants by simple inspection of the titration curves, pH vs base equivalents. For morphine, the approximate pKas are 8.1 and 9.3, while the corresponding p,K„s are 7.2 and 10.2; for nicotine, the p/Cas are 3.2 and 8.1, while the p^s are 3.2 and 6.8. The differences between the pATas and p0Kas can be used to determine log P constants, using equations slightly more complicated than Eqs. (2) and (4), as detailed elsewhere

Flumequine

Flumequine

Diacetylmorphine

Diacetylmorphine

Figure 2. Water (solid curves) and 1:1 octanol/ water (dotted curves) difference plots (0.15 m KCl) of (a) flumequine (pKa 6.27), (b) diacetylmorphine (p*. 7.96). Po h
 —^ZS'C 0.15M NoCl - P>Ko2 F>oKqS
4 ' 6 ' 8 ' Ï^O

Nicotine

---

25°C 0.15M KCl

Pa H

Figure 3. Water (solid curves) and 1:1 octanol/water (dotted curves) difference plots (0.15 m KC1) of (a) salicylic acid (pAT, 13.31 and 2.88), (b) morphine (pK, 9.26 and 8.17), and (c) nicotine (pK¡¡ 8.11 and 3.17).

, Difference curve analysis often gives one the needed "seed" values for refinement of equilibrium constants by mass-balance-based nonlinear least squares [4, 25].

### 7.2.4 pH Definitions and Electrode Standardization

To establish the operational pH scale, the pH electrode is first calibrated with a single aqueous pH 7 phosphate buffer and the Nernst slope is assumed. (Most users of pH electrodes are familiar with this step.) Because the hH calculation requires the "free" hydrogen ion concentration (as described in the preceding section) and because we employ the concentration scale for the ionization constants (next section), and 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 [5,20], pH = a + S pcH + jH [H+] + /'oh *w/[H+] (5)

where Kv is the ionization constant of water. The four parameters are empirically determined by a weighted nonlinear least squares procedure using data from alkalimetric titrations of known concentrations of HC1 (from pH 1.8 to 12.2) or standard buffers . Typical aqueous values of the adjustable parameters at 25 °C and 0.15 M ionic strength are a = 0.08 ± 0.01, S = 1.001 ± 0.001, ;H = 1.0 ± 0.2 and ;OH = -0.6 ± 0.2.

### 7.2.5 Definitions of Constants

All equilibrium constants in the present discussion are based on the concentration scale. This is a perfectly legitimate thermodynamic scale, provided the ionic strength of the solvent medium is kept constant. Most of the results reported here were determined in 0.15 m KC1 or NaCl.

An example will illustrate the definitions of the equilibrium constants used here. Consider the diprotic amino acid phenylalanine (Phe) ; let us say X" represents the fully deprotonated Phe species. At the physiological level of background salt (0.16 m NaCl) X partitions both as the zwitterion and the cation: log PiXH4) = -1.38 and log P(XH2+) = -1.41; the dissociation constants are pital = 2.20 and pA"a2 = 9.08. Ion-pair partitioning can be characterized either by a log P constant (which is only valid at a particular level and type of background salt, hence a "conditional" constant) or by an extraction constant, log K,,. For the above example, log Kc = -0.62. The equilibrium reactions corresponding to the above constants (in the order of mention) are

 XH+ î=» (XH+)oct- P = [XH±]ocr/[XH±] (6) XH2+ (XH2+)oct: P = [XH2+]OCT/[HX2+] (7) XH2+±5XH±+ H+: K.dl = [XH*] [H+]/[XH2+] (8) XH±Ê5X' + H+: K* = [X ] [H+]/[XH±] (9) XH2+ + Cl~i=> (XH2+, CI ) oct : K, = [XH2+, CF]ocT/[XH2+][Cr] (10)

Eqs. (7) and (10) are two ways of expressing the same ion-pair partitioning. One can convert log P(XH2+) to log Kc according to the relation.

Since all components of the extraction equilibrium expression are explicitly identified in Eq. (10), the ionic-strength dependence of the extraction constant can be predicted by the Debye-Huckel theory. That is, in principle, a result determined at 0.15 m KC1 background, could be "corrected" to another background salt concentration, provided the ionic strength is within the limitations of the theory (<0.5m for the Davies  variant of the Debye-Hiickel expression). It is often convenient to convert constants to "zero ionic strength" in order to compare values to those reported in older literature.

The "log p" form of constants are also used in the discussion. For the above example,

These are called stability constants, and are quite useful in many calculations involving multiprotic substances. The double-digit fi subscript indices refer to the stoichiometric coefficients of the species on the right side of the equilibrium equation, in the order X, then H. For phenylalanine, log ft, = 9.08 and log pn = 11.28 (9.08 +2.20). ## Cigarette Crusher

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