In principle, when a strong acid or a strong base produces in solution a concentration of ions H3O+ or OH- greater than 10-6 M you can just neglect the contribution given to the concentration of these ions (and then to the pH) by the water autoionization (or self-ionization) . But what happens when the concentration of strong base or acid drops below this altitude?
Let's take an example. You want to calculate the pH of a 10-8 M solution of HNO3. Nitric acid is a strong acid (Arrhenius or Bronsted-Lowry theory) therefore completely dissociated in water.
If we calculate the pH as we have seen for strong acids and bases not particularly diluted we would have:
Paradoxically, adding a strong acid to a neutral solution you would get an alkaline pH. But this is (obviously) not possible because we have a very dilute solution of strong acid!
In this case, we must take in account the coming from water self-ionization:
Calling again with "x" the concentration of ions that comes from the self-ionization:
Let's say that:
Substituting in the expression of Kw :
We can now find the total hydrogen ion concentration:
and therefore the pH:
As you can see, the very low concentration of our acid has very little impact on the pH, which is then only slightly below neutrality.
To calculate the pH of a very diluted strong base we can run the same type of calculation, this time finding , which corresponds to exactly according to the same logic.