Fig. 1. Schematic representing the three-state model (see text). Depicted is the case of a weak base, B, titrated with HCl in an alkaline solution at a pH near the pK_a. The vertical direction represents concentration of reactants; the horizontal direction represents distance from the surface of the solid. The solid is represented by rectangles. The lighter shaded rectangle represents the amount of solid which dissolves as a result of HCl titrant addition. Block (a) represents the initial state, Σ_i, before titrant addition. Block (b) represents the situation right after titrant addition, where the bulk solution is equilibrated by convective mixing, but none of the solid has dissolved (t=0). Block (c) represents the equilibration of the solution adjacent to the surface of the solid and the establishment of the initial steady state (t≤2 s). Block (d) represents the equilibrium state (t=∞), where a portion of the solid has been dissolved, and a lower-pH solution has been formed. The transition from states (a) to (b) is nearly instantaneous; from (b) to (c) takes less than 2 s; from (c) to (d) takes minutes or hours. The vertical dashed lines represent the boundary between the diffusion layer and bulk solution. The horizontal dashed lines assist in visualizing that pH has decreased in block (d) compared to block (a), but that the concentration of the uncharged (free) base has remained the same in (d) as in (a), as is characteristic of a saturated solution.

Fig. 2. Template simulation for 0.51 mg propoxyphene hydrochloride in 5.1 ml 0.15 M KCl, using the pK_a 9.06 and log P_ow 4.37.

Fig. 3. Bjerrum plots: (a) benzoic acid, indicating ideal behaviour (squares, 96 mM; triangles, 130 mM; hexagons, 500 mM), and (b) propoxyphene hydrochloride, showing some incomplete dissolution (squares, 0.11 mM; triangles, 0.25 mM; hexagons, 0.73 mM).

Table 1. Diffusion layer and bulk solution concentrationsa

Table 2. Simulated dissolution rates during titration of propoxyphene

Table 3. Potentiometric titrations

Table 4. Intrinsic solubilities and solubilities at selected pH valuesa