Figure 5. (a) The family of lipophilicity profiles for a series of hypothetical molecules which all have pK^ 10, log P 5 and ionpair log P from 4 to 1. (b) The corresponding first derivative plots of the curves in Fig. 5 a.
curves on the right side at pH 10, the pKa. (This is not precisely so for the curve corresponding to log PXH 4, where the intersection is at about 9.6).
Most shakeflask analyses, if they get this far, stop at this point. However, there is also an intersection of the curves at slopes 0.5 one the left sides in the curves in Fig. 5 b, at pHs 9.2, 8, 7, 6, 5, and 4. This is also a pKa but very unusual kind. It was first noted by Scherrer [23] in onephase octanol titrations, and it is appropriate to call it the Scherrerp^a, or simply p0ATaSC H (the subscript o is a reminder that the pKa is observed in an octanolcontaining medium). Scherrer made the valuable observation that 41ogP = p^ap0iCaSCH, thus suggesting a way to measure log Pmn.
The above discussion describes the ideal of shakeflask analyses: log D is measured directly, and log P, log Pion, pK.d are derived if measurements are done at several judiciously selected pHs. By contrast, in the pHmetric analysis, one gets log P, log Plan, pKa from difference curve analysis (followed by massbalance least squares refinement) as a starting position and calculates the log D as the final step, using using Eq. (19).
How does one get log Pi0„ by the pHmetric technique? Eqs. (2) and (4) deal with only a single species. The answer is that it is necessary to do two octanolcontaining titrations (rather than one, as in the cases associated with Figs. 13), selecting a very small octanol/water volume ratio for one (< 0.02) and a very large ratio for the other (> 1). Just as Eq. (20) is a general form of case la and lb equations in Table 1, Eqs. (2a) and (2b) in [5] are the generalized forms of Eqs. (2) and (4), where ion pairing is
Figure 6. (a) Difference plots for ibuprofen based on octanol/water titrations: (squares) 0.1 mL octanol: 20 ml 0.15 m KC1, (triangles) 0.25 :20, (hexagons) 1 :19, (diamonds) 10:10, (pentagons) 15 :5. (b) Calculation illustrating the effect of ionpair partitioning on the apparent pKr of ibuprofen in octanol  0.15 m KC1 solutions as a function of the octanol/ water volume ratios.
taken into account. Fig. 6a shows not two, but five octanol/water Bjerrum plots for ibuprofen, each at a different octanol/water volume ratio. Eq. (2) predicts that as one adds more octanol to an aqueous solution of a weak acid, the apparent pKa, p0^a> increases. So, for 0.1 mL octanol/20 mL 0.15 M KC1, paKa is 6.0, compared with pK„ 4.35; for 15 mL octanol, 5 mL 0.15M KC1, p0A"a would be expected to be 8.7. Such would be the prediction of Eq. (2). However, the observed p0ATa is about 8.1. Ion pairing causes an attenuation effect. Fig. 6b shows that Eq. (2) would predict the upper curve, which would correspond to pn/Ca 9.35 at Vocr! Kvatcr of 10. The lower curve in Fig. 6b is the one actually observed. At volume ratio of 10, the true paK3 is 8.35. In fact, if one measured it in 1000 :1 volume ratio, it would only be 8.37. This latter value is the Scherrer pK.A\ that is, in the limit of highest octanol: water volume ratios, p0Ka = p0£aSCH. One does not need to do the onephase octanol titration that Scherrer describes [23,31], Our experience suggests that pH electrodes do not function well in such a solution.
In the pHmetric technique, one can calculate the lipophilicity profile from the derived ionization and partition constants, using Eq. (19). The uppermost curve in Fig. 4a shows the Scherrer pKa at pH 8.37, where the slope of the lipophilicity plot is 0.5.
Scherrer did not extend his analysis to multiprotic substances. It was possible to do so with our approach. Fig. 7 is an illustration of the pHmetric relationships for quinine, which is a diprotic base. One distinct limiting pDKa in Fig. 7b is 6; the other is below 3, compared with pKas 8.55 and 4.24 in 0.15 m KC1. The topic of the limiting p„Kas will be better illustrated when we consider lipophilicity plots of diprotic substances in later sections.
Quinine Apparent pKas in Octanol—Water
Quinine Apparent pKas in Octanol—Water 
 0 10 20 30 40 50 vOCT(mL) / 20mL 0.15M KCl
Figure 7. (a) Difference plots for quinine based on octanol/water titrations: (squares) 0.25 mL octanol: 20 mL 0.15 m KC1, (triangles) 0.5 :20, (hexagons) 5 :15, (diamonds) 10:10, (pentagons) 15 : 5. Note that no curves intersect, (b) Calculation illustrating the effect of ionpair partitioning on the apparent pXas of quinine in octanol  0.15 m KC1 solutions as a function of the octanol/water volume ratios.
7.3.2 Further Insights into the Scherrer pKa
Fig. 8 is a scheme of reactions depicting the most general octanol/water partitioning of a weak acid. In the octanol phase, all ions are treated as ion pairs. It is not as likely to expect an octanolsolvated H+, free of other ions, as it is to expect to find (H+, CI ) solvated by watersaturated octanol, because the dielectric constant of the wet organic solvent is much lower than that of water. The scheme uses extraction reactions formalism. The octanol: water extraction constants log Ke are 3.12 for KC1 and 1.34 for HC1, whose values were carefully determined by conductivity measurements by Westall et al. [32], We have already defined the Kc, K;,, and P equilibrium expressions associated with the scheme except that of the ionpair exchange reaction associated with the constant K,. This reaction corresponds to proton exchange in watersaturated octanol between the weak acid HA and the weak base KC1, whose equilibrium quotient is

 Figure 8. Scheme illustrating the octanol/water partitioning of a weak acid, along with the extraction reactions of various salts.
From the relationships in the scheme of Fig. 8, we can derive the expression pK,  pK;t = logPHA  log KCKA + logKcKa  log Kc"a (22)
For example, ibuprofen (log KeKA 0.77) has pK, = 4.35 + 3.97  0.77  3.12  (1.34) = 5.79 in 0.15 m KC1. This number is not one that we can easily have a intuitive feel for. Eq. (22) is not as simple an expression as the one Scherrer derived, since it involves the partitioning of HC1 and KC1, in addition to the weak acid HA and the ionpair KA. We can relate Eq. (22) to that derived by Scherrer. Scherrer did not explicitly consider counter ions. We can safely assume that the concentration [K+A"]0ct and [A~]0ct are the same; since PA = [A ]oct/[A"], then /CeKA = PA/[K+]. Substituting this into Eq. (22) produces
{log [HA]0CT  log [K+A ]OCT  PcH}  pKa = log PHA  log PA . (23)
The term in braces is the Scherrer pKa. The advantage of our derivation is that it precisely states what equilibrium corresponds to p0#SCH, it is
Scherrer's intuitive assessment was correct: the constant refers to the aqueous pcH at which point the octanol concentrations of HA and A" are equal. It is a mixed constant. This is delightful from an experimental point of view, since pH electrodes work in aqueous media; it has been our experience that they are extremely drifty in wet octanol.
In Scherrer's onephase titration, he used 0.1 M NaOH as titrant, dissolved in methanol/isopropanol (1:4). The pH electrode was effectively working in an isopropanol/ methanol medium. We know that such a solvent can shift pH readings by about 0.20.5 pH units [5] (K. J. Box, in preparation). So, we believe that the Scherrer pK^ is best determined not in onephase wet octanol titrations but from the analysis of the lipophilicity profile at slope 0.5 (above discussions), determined from much better behaved dualphase titrations. When the pcH is at the Scherrer value, what is the pH in octanol?
7.3.3 pH Scale in Lipids?
We can answer the question from the last section if we define the pH scale in octanol to be based on the concentration of the ion pair (H+Cr)0cr If there were no species other than KC1 and HC1 partitioning into the octanol phase, the two extraction constants K,,Ha and KtKa would fix the relationship between pcH and log [H+Cr]ocx. A massbalance calculation of concentrations, using Westall et al. [32] extraction constants, produces the linear relationship p[H+Cr]ocr = PcH + 2.18 (25)
in the pH interval from 2 to 12 when 0.15 M KC1 is considered. At very low pH, the difference is very slightly smaller, due to the common ion (CI") effect. At very high pH, the extraction of (K+OH~) into octanol would have to be considered. We do not have reliable constants for that extraction process.
We did a similar calculation in the presence of 0.5 mm ibuprofen and 0.15 m KC1. The difference between the pcH and p[H ' Cr]0CT scales was 2.20, and was also nearly constant in the pH interval from 2 to 12.
We repeated the calculation at 0.03 m KC1, still with 0.5 mm ibuprofen, and found that the difference between the two pH scales was still constant, but the value increased to 2.86, reflecting the role that salt plays in the extraction of hydrogen ion into the lipid phase. The background salt appears to play a key role in maintaining a constant difference between pcH and p[H+Cr]0{T values across the useful pH range. It appears that the shift factor 2.18 in Eq. (25) is approximately equal to pi£eHCI + pCl. It seems that Eq. (25) is worth further study.
Is "what is the pH inside of cell membranes?" a fruitful question? Water can diffuse across the lipid layer quite rapidly in phospholipidbased membranes. Even the tight bloodbrain barrier is permeable to small molecules like water. The transition between the bulk aqueous environment and the lipid of the biological membrane is not abrupt. The phosphatidylcholine zwitterion and the ester groups of the phospholipid are polar and can accommodate a quasiaqueous environment for small ions. Hydrogen ions may be found in such a zone, but with reduced activity compared with that found in bulk aqueous solution. Watersaturated octanol has 27 mol % water, hydrogen bonded to octanol molecules in clusters. Perhaps such an environment can model some of the interfacial properties of the biological membranes.

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