Autoprotolysis constant

If you consider the equilibrium constant of water protonating itself, and once again assume the solution is dilute enough to ignore [H₂O], you get the autoprotolysis constant K_w.

The experiment value of K_w at room temperature is 1 x 10^-14. We know that pure water has a ph of 7; a pH of 7 corresponds to a H₃O⁺ concentration of 1 x 10^-7. Since with pure water [H₃O⁺] = [OH^-], then we can get K_w by multiplying [H⁺] and [OH^-], which is 1 x 10-7 x 1x-10-7 = 1 x10^-14.

What makes the autoprotolysis constant really useful is that it remains constant even when adding acid or base in dilute quantities, and nearly all acid or base used in a lab can be called dilute. So because K_w = [H⁺][OH^-] , doubling the concentration of hydrogen ions will half the concentration of OH^-. Unfortunately I have never been able to grasp why intuitively. But the equation works.

We can also express [H⁺] and [OH^-] as an acidity and basicity constant respectively, and substitute them into the autoprotolysis equation:

Or in log format:

The K_a is the strength of the acid (ability to give a proton to water), the K_b is the strength of the conjugate base (the ability to receive a proton from water). So the stronger the acid, the weaker its conjugate base. The stronger a base, the weaker its conjugate acid. We already knew that of course, but the equations both confirm this fact and gives a more precise relationship.

pK_w = 14. Similar expressions apply to other solvents, with pK_w replaced by the autoprotolysis constant of the solvent, pK_sol.