What is the useful equation for calculating the pH of a weak base?

To calculate the pH of a weak base, it is essential to utilize the relationship between the base's dissociation constant ((K_b)) and the acid dissociation constant ((K_a)). In this case, using the equation for hydroxide ion concentration and deriving it via the ion product of water ((K_w)) is the key to understanding the equilibrium involved.

The equation indicated involves using the equilibrium constant for the conjugate acid along with the ion product of water to find the hydronium ion concentration. This results from the relationship defined by the equation:

[ K_a = \frac{[H_3O^+][A^-]}{[HA]} ]

By rearranging and substituting (K_w) (where (K_w = [H_3O^+][OH^-]) at 25°C is (1.0 \times 10^{-14})), one can solve for ([H_3O^+]):

[ [H_3O^+] = \sqrt{\frac{K_a \times K_w}{[A^-]}} ]

This shows how the concentration of hydronium ions at equilibrium in a weak base solution can be calculated