In exactly of a saturated aqueous solution of (), exactly of solid () is aggressively added. The res — Ionic Equilibrium Chemistry Question
Question
In exactly $1 \text{ L}$ of a saturated aqueous solution of $AgCl$ ($K_{sp} = 1.6 \times 10^{-10}$), exactly $0.1 \text{ mol}$ of solid $CuCl$ ($K_{sp} = 1.0 \times 10^{-6}$) is aggressively added. The resultant concentration of $Ag^+$ in the equilibrated solution drops to $1.6 \times 10^{-x} \text{ M}$. What is the exact integer value of $x$?
💡 Solution & Explanation
The $CuCl$ is significantly more soluble and will dictate the common $[Cl^-]$ concentration. Its dissolution $CuCl \rightleftharpoons Cu^+ + Cl^-$ dominates. Let its solubility be $y$. $y^2 = 1.0 \times 10^{-6} \implies y = 10^{-3} \text{ M}$. Since $0.1 \text{ mol}$ was added to $1 \text{ L}$, it will not dissolve completely, creating a saturated $CuCl$ solution with $[Cl^-] = 10^{-3} \text{ M}$. For $AgCl$, the new $Ag^+$ concentration is determined by this common ion: $[Ag^+] = \frac{K_{sp}(AgCl)}{[Cl^-]} = \frac{1.6 \times 10^{-10}}{10^{-3}} = 1.6 \times 10^{-7} \text{ M}$. Thus, $x = 7$.