 Solution

ID: past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-2/a7/d/i/solution

Let

α

be the degree of ionization. Conservation of nuclei and charge neutrality, with no pre-existing electrons or other ions, give

c_{M} = (1 - α) c_{tot}, c_{M^{+}} = c_{e} = α c_{tot} .

Substitution into the definition of

K_{c}

gives

\frac{α ^{2}}{1 - α} = y, y = \frac{K _{c} c ^{\circ}}{c _{tot}} .

The physical root lies between zero and one:

α = \frac{y ^{2} + 4 y - y}{2} = \frac{2}{1 + 1 + 4/ y} .

The second form avoids subtracting nearly equal terms at large

y

. The limits are

α ≃ y

when

y ≪ 1

and

α ≃ 1 - 1/ y

when

y ≫ 1

. The dependence on initial number density matters:

K_{c}

alone, without

c_{tot}

and the standard-state convention, does not specify the ionized fraction.

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