## Distribution Function D and Lipophilicity Profile log D vs pH

The distribution function, D [1,30], is a useful parameter related to the partition coefficient, P, and the two parameters are unfortunately sometimes interchanged in the literature. The partition coefficient, P is a constant and refers to a single molecular species partitioning between two phases. On the other hand, the distribution coefficient, D, can be thought of as an effective (or apparent) partition coefficient which varies with pH when ionizable substances are considered. In the pH region where a substance is unionized, D = P, and the substance is usually maximally partitioning in the organic phase. However, as pH is changed (increased for weak acids or decreased for weak bases), the substance ionizes. This in turn causes its redistribution between the two phases, with more substance shifting to the aqueous phase. The concentration ratio P is still the same, but D decreases, reflecting that there is more of the substance in ionic form, which favors the water phase.

Let us consider the protophilic substance X (neutral or charged). The general definition of the distribution coefficient is

= {([X] org +[XH]ORG + [XH2]org + ■ • •) / ([X] + [XH] + [XH2]+...)}/r (14)

where we use the notation (italic) X to represent the total concentration of the substance in all of its protolytic forms. The primed quantity is defined in concentration units of moles of species dissolved in the organic phase per liter of aqueous phase; the octanol/water volume ratio, r, takes into account the change in volume scale. Assumptions must not be made as to which species partitions predominant into the organic phase. If the fully-deprotonated base X partitions into the organic phase, we have

Here, lower case (italic) x refers to the concentration of the fully deprotonated substance X (the conjugate base). The index 10 refers to stoichiometric coefficients of the partitioning species: the 1 refers to one unit of substance X and the 0 refers to the number of protons associated with X (i. e., fully-deprotonated base). This type of indexing has been used already for the P cumulative protonation constants in Eqs. (12) and (13). We will find this general notation increasingly useful as we consider increasingly complicated protonation reactions. If a j-protonated from of X partitions into the organic phase, we have

We can rearrange Eqs. (15) and (16) into very useful forms.

Here h denotes [H+], Substituting Eqs. (17) and (18) into the last line of Eq can state the distribution function in terms independent of x.

_ Pw + hl3uPn+h2l312Pn + ... I + h pu + h2 ¡3U+...

The distribution coefficient is only a function of the variable pcH and the pK.¿ (expressed in ¡3 form) and log P constants. The latter constants depend on ionic strength. So, the calculation of D associated with a titration can be quite complicated, depending on the level of ionic strength adjuster, the sample concentration, and dilution effects as a result of weak titrant concentrations or extreme starting and /or ending pH values in an assay. Ion-pair partitioning is also an added complication when the background salt changes in concentration due to dilution effects. A further subtlety may arise: the lipophilicity profile constructed from measurements by the shake-flask technique is invariably expressed as a function of the operational (activity) pH scale. All calculations here are based on the pcH concentration scale. This distinction must be kept in mind when comparing literature values.

Eq. (19) is applicable to all lipophilicity calculations. It may be helpful to identify special cases where the lipophilicity equation is considerably simplified, and made more easily comparable to forms found in the literature. Table 1 shows the simplified

Table 7.1. Lipophilicity equàtions Monoprotic substance la. XHpartitions (weak acid) (e.g., flumequine)

log D

lb. Xpartitions (weak base) (e.g., diacetylmorphine)

log D

Diprotic substance

2 a. XH2 partitions (weak acid) (e.g., salicylic acid)

log D = log PXH2-log (1 + 10-p^i+p=h + îO-P^-P^i+^H)

2b. XH partitions (ampholyte or zwitterion) (e.g., morphine)

log D = log Pxh " log (1 + lO -P^- l PcH + 10+pK,l-pcH)

log D = log Px - log (1 + lO+P^-P^ + lO+PKtf+PK'i-ZPcH)

Table 7.1. Continued

Triprotic Substance

3a. XH3 partitions (weak acid) (e.g., citric acid)

log D = log Pxh, - log (1 + KTPKo'+P'--" + 10"P^-piial+2pcH + 10-p^-p^2-p/i:,i+3ptH^

3b. XH2 partitions (e.g., terbutaline)

log D = log PxH, - log (1 + 10+P*"-P«H + 10-P^+PcH + 10-p^p^+2pcH)

3c. XH partitions log D = log Pxh-log (1 + KrPfe3+P'H + 10+PK'2"P'H + io+P^+pK.i-2pcH)

3d. Xpartitions (weak base) (e.g., diethylenetriamine)

log D = log Px - log (1 + io+PAr«-,-P':H + lo+P^+P^^P'H + io+pA'a:,+PA:i!+P/i"]~3pcH)

equations for the cases of mono-, di- and triprotic substances. The equations in Table 1 apply only to cases where a single species partition into the lipid phase. This is seldom realized in real systems. Invariably, a monoprotic substance partitions in two forms: neutral and ion pair, often with a log P difference of 3-4 units. For example, the weak acid ibuprofen has a neutral log Pha 3.97 and an ion-pair log PA -0.05, in 0.15 M KC1 (Fig. 4a). For ibuprofen, case la equation in Table 1 correctly represents the lipophilicity profile for pH <7; above that pH, ion-pairing becomes substantial, but the case la equation has no provision to describe it. The weak base propranolol has a neutral log PB 3.47 and an ion-pair log PBh 0.88, also in 0.15 M KC1 (Fig. 4b). Likewise, case 1 b equation in Table 1 is only accurate for pH > 8. To describe a monoprotic substance completely, we need an expression that incorporates partitioning of the ion pair. We can proceed in the same way that led us to derive the equations in Table 1 from Eq. (19), to get.

log D = log (Px + PXH 10 "PcH + p/Cil) - log (1 + 10 "P'H + p*») (20)

This equation is valid for both ibuprofen and propranolol, for the entire pH range. For a weak acid, PXH > Px and the log D curve decreases with pH; for a weak base, Px > PXH) and the log D curve increases with pH, as the above equation describes.

Fig. 4a shows several examples of lipophilicity profiles for weak acids, with log Ps ranging from 4.0 to -0.4. Fig. 4b shows several examples of weak bases, with log Ps from 5.4 to -1.3. Eq. (20) can be used to represent any one of the curves in Fig. 4.

Lipophilicity Profiles: Weak Acids

Ibuprofen prostaglandin I

Lipophilicity Profiles: Weak Acids

Ibuprofen prostaglandin I

25°C 0.15M KCl |
chlorpromaztne |

propranolol | |

,,—' ' Udocoine | |

ephedrine | |

— |
N—methyl — D — |

a/ucamme | |

Figure 4. Examples of lipophilicity curves of (a) weak acids and (b) weak bases.

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