A solution contains 0.1 M Ag ⁺ and 0.1 M Pb ²⁺ ions. Solid NaCl is added gradually to this solution without changing the volume. What is the concentration of the first precipitating metal ion left in the solution at the exact moment the second metal salt just begins to precipitate? Given: K _ sp ( AgCl ) = 1.0 10⁻¹⁰ and K _ sp ( PbCl ₂) = 1.0 10⁻⁵ .

Step-by-step solution

First, determine the concentration of Cl ⁻ required to initiate precipitation for each salt. For AgCl : [ Ag ⁺ ] [ Cl ⁻ ] = K _ sp ( AgCl ) (0.1) [ Cl ⁻ ] = 1.0 10⁻¹⁰ [ Cl ⁻ ] = 1.0 10⁻⁹ M For PbCl ₂ : [ Pb ²⁺ ] [ Cl ⁻ ]^2 = K _ sp ( PbCl ₂) (0.1) [ Cl ⁻ ]^2 = 1.0 10⁻⁵ [ Cl ⁻ ]^2 = 1.0 10⁻⁴ [ Cl ⁻ ] = 1.0 10⁻² M Since 1.0 10⁻⁹ PbCl ₂ will just begin to precipitate when the concentration of Cl ⁻ reaches 1.0 10⁻² M . At this point, the concentration of Ag ⁺ remaining in the solution is governed by the solubility prod