where K_{1} = 6.5 x 10^{-3} ; K_{2} = 6.2 x 10^{-8} ; and K_{3} = 3.6 x 10^{-13} . Each of these three equilibrium equations can be expressed mathematically in several different ways. Taking the equation for K_{2} as an example, each of the following is equivalent:
where pH = -log[H^{+} ], and pK_{2} = -logK_{2}. The latter logarithmic expression (the "Henderson Hasselbalch Equation") is a convenient form to use in buffer calculations, particularly when the pH is within one unit above or one unit below the pK. (Where the ratio [HPO_{4}^{2-}]/[H_{2}PO_{4}^{-}] is between 0.10 and 10). Although all four species of phosphate are always present in solution, the two forms in the equation are the predominant ones near the pK and the others can usually be ignored in calculations. As one gets more than two units above or below the pK, however, other species become involved and the calculations get more complicated.
The activity coefficients of ionic species depend upon both its charge and the "ionic strength" of the medium, a relationship quantitatively described by the Debye-Hückle equation. The effect is greater for multivalent ions, hence polyprotic acids such as phosphate are affected more strongly than monoprotic acids. (Even the "apparent pK_{2}" will vary slightly with pH because ionic strength increases as a proton dissociates). The pH meter is actually measuring -loga_{H+}, rather than -log[H^{+}]. The quantitative details of this relationship are beyond the scope of this introductory course.
The main lesson for you with respect to this laboratory experiment is that one should use the appropriate pK_{2} value for calculations, depending on the total concentration of phosphate buffer involved, and recognize that even then there may be a slight deviation from expected pH values. Below is a table and a graph showing the effect of the phosphate concentration on the apparent pK_{2} value.
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These values were calculated from data in A. A. Green, "The Preparation of Acetate and Phosphate Buffer Solutions of Known PH and Ionic Strength", J. Am. Chem. Soc. 55, 2331, (1933). as the average pK_{2} from the Henderson Hasselbalch equation over the pH range from 5.9 to 7.7. |