p-Functions

Sometimes it is inconvenient to use the concentration units  during a reaction a species’ concentration may change by many orders of magnitude. If we want to display the reaction’s progress graphically we might plot the reactant’s concentration as a function of time or as a function of the volume of a reagent being added to the reaction. Such is the case in Figure 1.1 for the titration of HCl with NaOH.

Figure 1.1: Graph showing the progress for the titration of 50.0 mL of 0.10 M HCl with 0.10 M NaOH. The [H+] is shown on the left y-axis and the pH on the right y-axis.

The y-axis on the left-side of the figure displays the [H+] as a function of the volume of NaOH. The initial [H+] is 0.10 M and its concentration after adding 80 mL of NaOH is 4.3 × 10^-13 M. We can easily follow the change in [H+] for the first 14 additions of NaOH. For the remaining additions of NaOH, however, the change in [H+] is too small to see. 

When working with concentrations spanning many orders of magnitude, it is often more convenient to express concentration using a p-function. The p-function of X is written as pX and is defined as

The pH of a solution that is 0.10 M H⁺ is

and the pH of 4.3 × 10^-13 M H⁺ is