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A sparingly soluble salt having general formula $A_x^{p + }B_y^{q - }$ and molar solubility $S$ is in equilibrium with its saturated solution. Derive a relationship between solubility and the solubility product for such salt.
Solution
A sparingly soluble salt having general formula $\mathrm{A}_{x}^{p+} \mathrm{B}_{y}^{q} .$ Its molar solubility is $\mathrm{S} \mathrm{mol} \mathrm{L}^{-1}$. Then, $\mathrm{A}_{x}^{p+} \mathrm{B}_{y}^{q-} \square x \mathrm{~A}_{x}^{p+}(\mathrm{aq})+y \mathrm{~B}_{y}^{q-}(\mathrm{aq})$
$\mathrm{S}$ moles of $\mathrm{A}_{x} \mathrm{~B}_{y}$ dissolve to give $x$ moles of $\mathrm{A}^{p+}$ and $y$ moles of $\mathrm{B}^{q-}$ Therefore, solubility product
$\left(\mathrm{K}_{\mathrm{sp}}\right)=\left[\mathrm{A}^{p+}\right]^{x}\left[\mathrm{~B}^{q-}\right]^{y}$
$=[x \mathrm{~S}]^{x}[y \mathrm{~S}]^{y}$
$=x^{x} y^{y} \mathrm{~S}^{x+y}$
