Acid/Base dissociation constants relationship

Question

From what I understand, the concentration of hydronium and hydroxide molecules is constant in pure water (and equals $10^{-7}\ \text{M}$, which is measured experimentally).

What I don't understand is why this remains true for an aqueous solution with an acid or base in it. Shouldn't it affect these concentrations?

When you write the equilibrium reactions for a weak/weak acid/base pairs (one with its conjugate), and get that $K_\mathrm a\cdot K_\mathrm b = [\ce{H+}][\ce{OH- }]$, what's the explanation for it? Why does their multiplication (either as pure water or in a acidic/basic solution) remain constant?

Answer 1

The concentrations do change in the presence of acid or base and are no longer $10^{-7}$M.

Instead, [H3O+][OH-] $= 10^{-14}M^2$

This comes from:

$\ce{2H2O <=> H3O+ + OH-}$

$K = \frac{a(\ce{H3O+})a(\ce{OH-})}{(a(\ce{H2O}))^2}$

Where "a(x)" is activity of species "x".

Then, approximating activity of $\ce{H2O}$ as 1, and the activity of the ions as the concentration of the ions there is a constant $K_w$, such that:

[H3O+][OH-] $= 10^{-14}M^2 = K_w$