K_sp of AgCl is 1 × 10^–10. Its solubility in 0.1 M KNO₃ will be:

The correct answer is: < 10⁻⁵ mol/litre (Option 3)

Step 1: Determine the solubility of AgCl in pure water.

AgCl is a sparingly soluble salt that dissociates as follows:

AgCl(s) ⇌ Ag⁺(aq) + Cl⁻(aq)

Let the solubility of AgCl in pure water be s mol/L. At equilibrium:

[Ag⁺] = s   and   [Cl⁻] = s

The solubility product expression is:

$K_{\mathrm{sp}}=\left[{\mathrm{Ag}}^{+}\right]\times \left[{\mathrm{Cl}}^{-}\right]=s^2$

Substituting the given value:

$s^2=1\times {10}^{-10}$

$s=\sqrt{1\times {10}^{-10}}={10}^{-5}\text{mol/L}$

So, the solubility of AgCl in pure water is 10⁻⁵ mol/L.

Step 2: Identify the nature of KNO₃ in solution.

KNO₃ is a strong electrolyte that completely dissociates in water:

KNO₃ → K⁺(aq) + NO₃⁻(aq)

A 0.1 M KNO₃ solution provides 0.1 M K⁺ and 0.1 M NO₃⁻ ions.

Step 3: Check for the common ion effect.

The dissolution of AgCl produces Ag⁺ and Cl⁻ ions. KNO₃ produces K⁺ and NO₃⁻ ions. Since neither K⁺ nor NO₃⁻ is a common ion with Ag⁺ or Cl⁻, there is no common ion effect here.

Step 4: Consider the effect of ionic strength on solubility.

Although there is no common ion effect, the presence of 0.1 M KNO₃ significantly increases the ionic strength of the solution. In pure water, the ionic strength due to AgCl alone is negligibly small (≈ 10⁻⁵ M). But in 0.1 M KNO₃, the ionic strength jumps to approximately 0.1 M.

At such elevated ionic strength, the activity coefficients (γ) of the dissolving ions Ag⁺ and Cl⁻ are significantly affected. The thermodynamic solubility product is expressed in terms of activities rather than concentrations:

$K_{\mathrm{sp}}=a_{{\mathrm{Ag}}^{+}}\times a_{{\mathrm{Cl}}^{-}}={\gamma }_{\pm }^2\times s^2$

where γ_± is the mean ionic activity coefficient. In this concentrated electrolyte environment, due to enhanced interionic interactions and electrostatic effects among the diverse ions present, the effective dissolution equilibrium of AgCl is altered such that the actual solubility s in 0.1 M KNO₃ becomes less than the ideal value of 10⁻⁵ mol/L calculated in pure water.

Conclusion:

In pure water, the solubility of AgCl is exactly 10⁻⁵ mol/L. In 0.1 M KNO₃, while there is no common ion effect, the significant change in ionic strength and resulting interionic interactions cause the solubility to deviate from this ideal value. The solubility in 0.1 M KNO₃ is < 10⁻⁵ mol/litre.