pH calculation - diluted strong acids and bases

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:

and then:

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  :

shows that

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.