In half molar nitric acid, the concentrations of H^{+}
and NO_{3}^{-} ions are each 0.5 mole per liter,
since strong acids are completely dissociated. In 0.5-molar acetic
acid, dissociation is incomplete, and the ion concentrations are
much less than 0.5 mole per liter.

We can find out how much less by using the acetic acid equilibrium
expression. Let the original, overall HOAc concentration be c moles
per liter before dissociation takes place, and suppose that when
equilibrium is reached, y moles per liter of this acid have dissociated.

Then c is specified by the initial conditions of the experiment,
and y is the concentration of each ion, because for every dissociated
HOAc molecule, one H^{+} ion and one OAc^{-} ion
are formed. At equilibrium:

This is a quadratic equation in y, and can be solved exactly using
the quadratic formula:

You can avoid the quadratic formula by using a little chemical
common sense instead. With a dissociation constant as small as 0.0000176,
only a small fraction of the acetic acid present will be dissociated,
so y is small. Hence we should be able to neglect y in the denominator,
because it is subtracted from the much larger c, and solve a simpler
equation instead:

For half molar acetic acid, c = 0.50, and

You can now check the validity of the approximation:

If the acid had dissociated enough so that y was a few percent
of c, this approximation would not have been valid. Our calculation
tells us that, although the overall acetic acid concentration is
0.50 mole liter^{-1}, such a small amount dissociates that
the acetate and hydrogen ions each are present at only 0.00297 mole
liter^{-1} concentration. This is why acetic acid is classed
as a weak acid.