Book seems to say that we can have weak acid and weak conjugate base

[zenterix]

Homework Statement

In the book I am reading there is a snippet that seems to indicate

Relevant Equations

that it is possible to have a weak acid with a weak conjugate base.

Here is the relevant section from the book

Consider the equilibrium

$$\mathrm{HA(aq)+H_2O(l)\rightleftharpoons H_3O^+(aq)+A^-(aq)}\ \ \ \ \ K_a=\mathrm{\frac{[H_3O^+][A^-]}{[HA]}}$$

The values of $\mathrm{[HA]}$ and $\mathrm{[A^-]}$ that appear in $K_a$ are the equilibrium concentrations of acid and base in the solution, not the concentrations added initially.

However, a weak acid $\mathrm{HA}$ typically loses only a tiny fraction of its protons, and so $\mathrm{HA}$ is negligibly different from the concentration of the acid used to prepare the buffer, $\mathrm{[HA]_{initial}}$.

This is all fine, but then there is the following

Likewise, only a tiny fraction of the weakly basic anions $\mathrm{A^-}$ accept protons, so $\mathrm{[A^-]}$ is negligibly different from the concentration of the base used to prepare the buffer, $\mathrm{[A^-]_{initial}}$.

I am under the impression that if an acid is weak (ie, only a small fraction of the molecules donate protons) then the conjugate base is strong (a large fraction of the molecules accept protons).

But the snippet above seems to say that that both the acid and the conjugate base are weak.

How can this be?

 

[zenterix]

For a little more context, the topic is what happens when we create a buffer with the same initial concentrations of weak acid and conjugate base.

In the equilibrium in the OP, we would have $\mathrm{[HA]=[A^-]}$ and so

$$\mathrm{K_a=[H_3O^+]}$$

$$\implies \mathrm{pH=-\log{(K_a)}=pK_a}$$

 

[mjc123]

Strong and weak are relative terms, not absolute. It is true that "the stronger the acid, the weaker the conjugate base", but we can't say "if one is (absolutely) strong, the other is (absolutely) weak". We can distinguish different situations:
A very strong acid (stronger than H₃O⁺) has a very weak conjugate base (essentially non-basic in water).
A very strong base (stronger than OH^-) has a very weak conjugate acid (essentially non-acidic in water).
In between these extremes we have an intermediate situation, where an acid weaker than H₃O⁺ has a conjugate base weaker than OH^-. Then in a solution of HA, only a small fraction of the HA molecules are deprotonated, while in a solution of A^-, only a small fraction of the A^- ions are protonated.
The acidity and basicity are linearly related on a logarithmic scale (pKa + pKb = 14), which means that in the mid-range, both Ka and Kb are small.