At what pH, is the solubility of , minimum? What is the minimum solubility?<br><br><br> — Ionic Equilibrium Chemistry Question
Question
At what pH, is the solubility of $\text{Zn(OH)}_2$, minimum? What is the minimum solubility?<br><br>$\text{Zn(OH)}_2(s) \rightleftharpoons \text{Zn}^{2+}(aq) + 2\text{OH}^-(aq); K_{sp} = 1.2 \times 10^{-17}$<br>$\text{Zn(OH)}_2(s) + 2\text{OH}^-(aq) \rightleftharpoons \text{Zn(OH)}_4^{2-}(aq); K_f = 0.12$
💡 Solution & Explanation
The total solubility is $S = [\text{Zn}^{2+}] + [\text{Zn(OH)}_4^{2-}] = \frac{1.2 \times 10^{-17}}{[\text{OH}^-]^2} + 0.12[\text{OH}^-]^2$. To find the minimum, we differentiate with respect to $[\text{OH}^-]^2$ and equate to zero, which happens when the two terms are equal: $\frac{1.2 \times 10^{-17}}{[\text{OH}^-]^2} = 0.12[\text{OH}^-]^2 \implies [\text{OH}^-]^4 = 10^{-16} \implies [\text{OH}^-] = 10^{-4} \text{ M}$. Thus $\text{pOH} = 4.0$ and $\text{pH} = 10.0$. The minimum solubility is $S = 2 \times (0.12 \times 10^{-8}) = 0.24 \times 10^{-8} = 2.4 \times 10^{-9} \text{ M}$. Therefore, correct answer is A.