53. A sparingly soluble salt having general formula ${\mathrm{A}}_{\mathrm{x}}^{\mathrm{p}+}{\mathrm{B}}_{\mathrm{y}}^{\mathrm{q}-}$ and molar solubility S is in equilibrium with its saturated solution. Derive a relationship between the solubility and solubility product for such salt.

A sparingly soluble salt having general formula ${\mathrm{A}}_{\mathrm{x}}^{\mathrm{p}+}{\mathrm{B}}_{\mathrm{y}}^{\mathrm{q}-}$. Its molar solubility is S mol L-1.
Then, ${\mathrm{A}}_{\mathrm{x}}^{\mathrm{p}+}{\mathrm{B}}_{\mathrm{y}}^{\mathrm{q}-}⇌\underset{\mathrm{x}}{{\mathrm{xA}}^{\mathrm{p}+}}\left(\mathrm{aq}\right)+\underset{\mathrm{y}}{{\mathrm{yB}}^{\mathrm{q}-}}\left(\mathrm{aq}\right)$
S moles of ${\mathrm{A}}_{\mathrm{x}}{\mathrm{B}}_{\mathrm{y}}$ dissolve to give x moles of ${\mathrm{A}}^{\mathrm{p}+}$ and y moles of ${\mathrm{B}}^{\mathrm{q}-}$
Therefore, solubility product $\left({\mathrm{K}}_{\mathrm{sp}}\right)={\left[{\mathrm{A}}^{\mathrm{p}+}\right]}^{\mathrm{x}}{\left[{\mathrm{B}}^{\mathrm{q}-}\right]}^{\mathrm{y}}$