When a weak acid is very dilute, it is necessary to take into account the autoionization of water. We are going to write the two equilibria that take place in solution:

$H_2O + H_2O \rightleftharpoons H_3O^+ + OH^-$

                               x             x

$HA +H_2O\rightleftharpoons H_3O^+ + A^-$

M-y                        y              y

Both M, x and y are expressed in molarity. It must be taken into account that the concentration of protons in the medium is given by x+y.

We write the equilibrium constants:



Solving x from the equation of $K_a$:



Replacing these values in the $K_w$:


The value of y is obtained by solving the equation by the method of successive approximations.

Once y is obtained, its value is replaced in the equation of x. The sum x+y gives us the $[H_3O^+]$ and therefore the pH of the dilute weak acid solution.