Question

In 1 L saturated solution of $\mathrm{AgCl}$ (${\mathrm{K}}_{\mathrm{sp}}$ $\mathrm{AgCl}$= $1.6\times {10}^{-10}$), $0.1\mathrm{mol}$of $\mathrm{CuCl}$ (${\mathrm{K}}_{\mathrm{sp}}$ $\mathrm{CuCl}$=$1.0\times {10}^{-6}$) is added. The resultant concentration of ${\mathrm{Ag}}^{+}$ in the solution is $1.6\times {10}^{-x}$. Calculate the value of $\mathrm{x}$.

• A. 3
• B. 5
• C. 7
• D. 9

Answer 1

The correct option is C

7

Explanation:

Given:

${\mathrm{K}}_{\mathrm{sp}}$ $\mathrm{AgCl}$= $1.6\times {10}^{-10}$

${\mathrm{K}}_{\mathrm{sp}}$ $\mathrm{CuCl}$= $1.0\times {10}^{-6}$

Step 1: Ionization of Copper(I) chloride $\mathrm{CuCl}$:

The solubility product of is $\mathrm{CuCl}$ =$1.0\times {10}^{-6}$

$\mathrm{CuCl}$ ionizes as

$$\begin{gathered} \mathrm{CuCl}\left(\mathrm{s}\right)\to {\mathrm{Cu}}^{+}+{\mathrm{Cl}}^{-} \\ \mathrm{Initial}\mathrm{conentration}0.1\mathrm{M}00 \\ \mathrm{At}\mathrm{equilibrium}0.1-\mathrm{x}\mathrm{x}\mathrm{x} \end{gathered}$$

Where $\mathrm{x}$ is the unknown concentration.

Step 2: Calculation of chloride ion concentration:

$$\begin{gathered} {\mathrm{K}}_{\mathrm{sp}}\left(\mathrm{CuCl}\right)=1.0\times {10}^{-6} \\ {\mathrm{K}}_{\mathrm{sp}}\left(\mathrm{CuCl}\right)=\left[{\mathrm{Cu}}^{+}\right]\left[{\mathrm{Cl}}^{-}\right] \end{gathered}$$

${\mathrm{K}}_{\mathrm{sp}}\left(\mathrm{CuCl}\right)=\left(\mathrm{x}\right)\times \left(\mathrm{x}\right)$

The chloride ion concentration is

$$\begin{gathered} \mathrm{x}=\sqrt{{\mathrm{K}}_{\mathrm{sp}\left(\mathrm{CuCl}\right)}} \\ \mathrm{x}=\sqrt{1.0\times {10}^{-6}} \\ \left[{\mathrm{Cl}}^{-}\right]=1.0\times {10}^{-3}\mathrm{M} \end{gathered}$$

Step 3: Calculation of Silver ion concentration:

Given,

${\mathrm{K}}_{\mathrm{sp}}$ $\mathrm{AgCl}$= $1.6\times {10}^{-10}$

The silver ion concentration in Silver chloride $\mathrm{AgCl}$ will be,

$$\begin{gathered} {\mathrm{K}}_{\mathrm{sp}}\left(\mathrm{AgCl}\right)=\left[{\mathrm{Ag}}^{+}\right]\left[{\mathrm{Cl}}^{-}\right] \\ \left[{\mathrm{Ag}}^{+}\right]=\frac{{\mathrm{K}}_{\mathrm{sp}}\left(\mathrm{AgCl}\right)}{\left[{\mathrm{Cl}}^{-}\right]}=\frac{1.6\times {10}^{-10}}{1.0\times {10}^{-3}} \\ \left[{\mathrm{Ag}}^{+}\right]=1.6\times {10}^{-7} \end{gathered}$$

The resultant concentration of ${\mathrm{Ag}}^{+}$ in the solution is $1.6\times {10}^{-7}$.

Hence, option (C) is the correct answer. The value of $\mathbf{x}$ is $\mathbf{7}$.