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ChemMedAdv 2011 • Paper 2

Resultant Silver Ion Concentration in Mixed Chloride Saturated Solution

In 1 L1\text{ L} saturated solution of AgCl\text{AgCl} [Ksp(AgCl)=1.6×10−10\text{K}_{\text{sp}}(\text{AgCl}) = 1.6 \times 10^{-10}], 0.1 mol0.1\text{ mol} of CuCl\text{CuCl} [Ksp(CuCl)=1.0×10−6\text{K}_{\text{sp}}(\text{CuCl}) = 1.0 \times 10^{-6}] is added. The resultant concentration of Ag+\text{Ag}^+ in the solution is 1.6×10−x1.6 \times 10^{-\text{x}}. The value of "x\text{x}" is

Official Numerical Answer7

Topics & Concepts

EquilibriumSolubility ProductChemical Equilibrium

Step-by-Step Solution

To determine the resultant concentration of Ag+\text{Ag}^+ in the solution, we consider the simultaneous equilibria involved:

  1. Initial State:
    A 1 L1\text{ L} saturated solution of AgCl\text{AgCl} contains:
    [Ag+]0=[Cl−]0=Ksp(AgCl)=1.6×10−10≈1.26×10−5 M[\text{Ag}^+]_0 = [\text{Cl}^-]_0 = \sqrt{K_{\text{sp}}(\text{AgCl})} = \sqrt{1.6 \times 10^{-10}} \approx 1.26 \times 10^{-5}\text{ M}

  2. Addition of CuCl\text{CuCl}:
    0.1 mol0.1\text{ mol} of CuCl\text{CuCl} is added to the solution. The solubility of CuCl\text{CuCl} alone is:
    sCuCl=Ksp(CuCl)=1.0×10−6=1.0×10−3 Ms_{\text{CuCl}} = \sqrt{K_{\text{sp}}(\text{CuCl})} = \sqrt{1.0 \times 10^{-6}} = 1.0 \times 10^{-3}\text{ M}
    Since 0.1 mol0.1\text{ mol} exceeds the solubility of CuCl\text{CuCl} in 1 L1\text{ L}, solid CuCl\text{CuCl} remains undissolved at equilibrium.

As CuCl\text{CuCl} dissolves, it produces Cl−\text{Cl}^- ions, causing the ionic product of AgCl\text{AgCl} to exceed its solubility product:
Qsp(AgCl)=[Ag+][Cl−]>Ksp(AgCl)Q_{\text{sp}}(\text{AgCl}) = [\text{Ag}^+][\text{Cl}^-] > K_{\text{sp}}(\text{AgCl})
Thus, AgCl\text{AgCl} precipitates until equilibrium is re-established. In the final equilibrium state, both solid AgCl\text{AgCl} and solid CuCl\text{CuCl} are present.

  1. Equilibrium Expressions:
    AgCl(s)⇌Ag+(aq)+Cl−(aq)withKsp(AgCl)=[Ag+][Cl−]=1.6×10−10\text{AgCl}(s) \rightleftharpoons \text{Ag}^+(aq) + \text{Cl}^-(aq) \quad \text{with} \quad K_{\text{sp}}(\text{AgCl}) = [\text{Ag}^+][\text{Cl}^-] = 1.6 \times 10^{-10}
    CuCl(s)⇌Cu+(aq)+Cl−(aq)withKsp(CuCl)=[Cu+][Cl−]=1.0×10−6\text{CuCl}(s) \rightleftharpoons \text{Cu}^+(aq) + \text{Cl}^-(aq) \quad \text{with} \quad K_{\text{sp}}(\text{CuCl}) = [\text{Cu}^+][\text{Cl}^-] = 1.0 \times 10^{-6}

From charge balance and stoichiometry:
[Cl−]=[Cu+]+[Ag+][\text{Cl}^-] = [\text{Cu}^+] + [\text{Ag}^+]
Since Ksp(CuCl)≫Ksp(AgCl)K_{\text{sp}}(\text{CuCl}) \gg K_{\text{sp}}(\text{AgCl}), the concentration of Ag+\text{Ag}^+ is negligible compared to Cu+\text{Cu}^+:
[Cl−]≈[Cu+][\text{Cl}^-] \approx [\text{Cu}^+]

Substituting this approximation into the solubility product for CuCl\text{CuCl}:
[Cl−]2≈Ksp(CuCl)=1.0×10−6[\text{Cl}^-]^2 \approx K_{\text{sp}}(\text{CuCl}) = 1.0 \times 10^{-6}
[Cl−]≈1.0×10−3 M[\text{Cl}^-] \approx 1.0 \times 10^{-3}\text{ M}

  1. Resultant Concentration of Ag+\text{Ag}^+:
    Using the KspK_{\text{sp}} expression for AgCl\text{AgCl}:
    [Ag+]=Ksp(AgCl)[Cl−]=1.6×10−101.0×10−3=1.6×10−7 M[\text{Ag}^+] = \frac{K_{\text{sp}}(\text{AgCl})}{[\text{Cl}^-]} = \frac{1.6 \times 10^{-10}}{1.0 \times 10^{-3}} = 1.6 \times 10^{-7}\text{ M}

Comparing this with [Ag+]=1.6×10−x[\text{Ag}^+] = 1.6 \times 10^{-x}, we get:
x=7x = 7

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