A solution is 0.025 M in Pb2+. What is the minimum concentration of Cl− required to initiate precipitation of PbCl2, given that the Ksp for PbCl2 is 1.17 × 10^−5?

To find the minimum concentration of Cl− ions required to initiate precipitation of PbCl2, we need to use the solubility product constant (Ksp) of PbCl2. The Ksp is the product of the molar concentrations of the ions in a saturated solution of a compound, each raised to the power of their respective stoichiometric coefficients.

The chemical equation for the dissociation of PbCl2 in water is: PbCl2 (s) ⇌ Pb2+ (aq) + 2Cl− (aq)

The Ksp expression for PbCl2 is: Ksp = [Pb2+][Cl−]^2

Given that Ksp for PbCl2is 1.17 × 10^−5 and the concentration of Pb2+ is 0.025 M, we can plug these values into the Ksp expression and solve for the concentration of Cl− ([Cl−]):

1.17 × 10^−5 = (0.025)[Cl−]^2

Now solve for [Cl−]^2:

[Cl−]^2 = (1.17 × 10^−5) / (0.025)

Calculate the right-hand side of the equation:

[Cl−]^2 = 4.68 × 10^−4

Next, take the square root on both sides to solve for [Cl−]:

[Cl−] = √(4.68 × 10^−4)

[Cl−] = 0.0216 M (approximately)

Therefore, the minimum concentration of Cl− required to initiate precipitation of PbCl2 is approximately 0.0216 M.

Extra: The Ksp value is specific to a particular ionic compound at a given temperature and represents the level at which a solution becomes saturated with that compound. When the product of the concentrations of the ions exceeds the Ksp, the solution is supersaturated, and precipitation occurs. The Ksp value helps to predict whether a precipitate will form under certain conditions.

For the dissolution of PbCl2, the stoichiometry of the reaction shows that for every mole of PbCl2 that dissolves, 1 mole of Pb2+ and 2 moles of Cl− are produced. This stoichiometry is directly reflected in the Ksp expression where the concentration of Cl− is squared because of the 2:1 ratio of Cl− to Pb2+.

In the context of this question, a solution that already contains Pb2+ ions at a certain concentration is being investigated to determine how much Cl− must be added before PbCl2 starts to precipitate. Precipitation occurs when the ion product (concentration of Pb2+ times the square of the concentration of Cl-) exceeds the solubility product, Ksp.

The square root step is necessary because the Ksp expression for PbCl2 includes the concentration of Cl− squared due to the stoichiometry of the dissociation equation; hence the square root is taken to find the actual concentration of Cl−.

In practical applications, this type of calculation is crucial for understanding how to prevent or encourage precipitation in chemical processing, environmental science, and other fields. Understanding solubility equilibrium also helps in analyzing the purity of water sources and in developing methods to remove toxic ions from wastewater.