Derive the equation of ionization constants ${K_a}$ of weak acids $HX$.

A weak acid $HX$ that is partially ionized in the aqueous solution. The equilibrium can be expressed by :

$\text { Equilibrium : }  \text { Concentration M : } \mathrm{HX}_{\text {(aq) }}+\mathrm{H}_{2} \mathrm{O}_{(l)}+\mathrm{H}_{3} \mathrm{O}_{\text {(aq) }}^{+}+\mathrm{X}_{\text {(aq) }}^{-} $

$\text {Change in conce. : } -\mathrm{C} \alpha \ \ 0 \ \ \  0 $

$ \text { Concentration at }  (\mathrm{C}-\mathrm{C} \alpha)  (0+\mathrm{C} \alpha)  (0+\mathrm{C} \alpha) $

$ \text { Equili. in molarity : }  =\mathrm{C}(1-\alpha)   \mathrm{C} \alpha   \mathrm{C} \alpha $

$   =\mathrm{C} \alpha  =\mathrm{C} \alpha$

where, $\alpha=$ Extent upto which $\mathrm{HX}$ is ionized into ions.

Suppose, initial concentration of undissociated acid $\mathrm{HX}=\mathrm{C} \alpha \mathrm{M}$

$\therefore \quad$ Ionized $\mathrm{HX}=\mathrm{C} \alpha \mathrm{M}$

At equilibrium $[\mathrm{HX}]=(\mathrm{C}-\mathrm{C} \alpha)=\mathrm{C}(1-\alpha)$

$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{X}^{-}\right]=\mathrm{C} \alpha \mathrm{M}$

The equilibrium constant for the above discussed acid-dissociation equilibrium:

$\mathrm{K}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{X}^{-}\right]}{[\mathrm{HX}]\left[\mathrm{H}_{2} \mathrm{O}\right]} \quad \therefore \mathrm{K}\left[\mathrm{H}_{2} \mathrm{O}\right]=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{X}^{-}\right]}{[\mathrm{HX}]}$

In $\left[\mathrm{H}_{2} \mathrm{O}\right]=$ constant $=\mathrm{K}^{\prime}$ So,

$\quad \mathrm{K}\left[\mathrm{H}_{2} \mathrm{O}\right]=\mathrm{K} \times \mathrm{K}^{\prime}$ new constant $\mathrm{K}_{a}$

where, $\mathrm{K}_{a}=$ Ionization constant of weak acid.

$\therefore \mathrm{K}_{a}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{X}^{-}\right]}{[\mathrm{HX}]}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}}{[\mathrm{HX}]} \ldots .($ Eq. $-{i})$

$\therefore \mathrm{K}_{a}=\frac{(\mathrm{C} \alpha)(\mathrm{C} \alpha)}{\mathrm{C}(1-\alpha)}=\frac{\mathrm{C} \alpha^{2}}{\left(1-\alpha^{2}\right)} \ldots . .( Eq.-ii )$

Above equation-$(ii)$ is used for calculation of ionization constant of weak acid $HX$.