Question

What is the concentration of \(\mathrm{Pb}^{2+}\) when \(\mathrm{PbSO}_{4}\left(K_{s p}=1.8 \times 10^{-8}\right)\) begins to precipitate from a solution that is \(0.0045 \mathrm{M}\) in \(\mathrm{SO}_{4}^{2-}\) ? (a) \(4.0 \times 10^{-8} M\) (b) \(1.0 \times 10^{-6} \mathrm{M}\) (c) \(2.0 \times 10^{-8} M\) (d) \(4.0 \times 10^{-6} M\)

Short answer

The concentration of \(\mathrm{Pb}^{2+}\) when \(\mathrm{PbSO}_{4}\) begins to precipitate is \(4.0 \times 10^{-6} M\) (Answer d).

Step-by-step solution

Identify the Reaction

The reaction for the precipitation is: \(\mathrm{PbSO}_{4}(s) \leftrightarrow \mathrm{Pb}^{2+}(aq) + \mathrm{SO}_{4}^{2-}(aq)\). The solid \(\mathrm{PbSO}_{4}\) is in equilibrium with its ions in solution.

Write the Expression for the Solubility Product Constant (Ksp)

The expression for the solubility product constant is \(K_{sp} = [\mathrm{Pb}^{2+}][\mathrm{SO}_{4}^{2-}]\) where \([\mathrm{Pb}^{2+}]\) and \([\mathrm{SO}_{4}^{2-}]\) are the molar concentrations of lead and sulfate ions, respectively.

Calculate the Concentration of Pb2+

Given \(K_{sp} = 1.8 \times 10^{-8}\) and \([\mathrm{SO}_{4}^{2-}] = 0.0045 M\), we can calculate the concentration of \(\mathrm{Pb}^{2+}\) by substituting the known values into the Ksp expression and solving for \([\mathrm{Pb}^{2+}]\): \[K_{sp} = [\mathrm{Pb}^{2+}][\mathrm{SO}_{4}^{2-}]\] \[1.8 \times 10^{-8} = [\mathrm{Pb}^{2+}](0.0045)\] \[[\mathrm{Pb}^{2+}] = \frac{1.8 \times 10^{-8}}{0.0045}\] \[[\mathrm{Pb}^{2+}] = 4.0 \times 10^{-6} M\].

Key concepts

Precipitation Reaction

In chemistry, a precipitation reaction occurs when two aqueous solutions combine to form a solid precipitate. This type of reaction is a cornerstone of solubility principles and is vital for understanding how substances interact in solution.

When a slightly soluble compound such as lead sulfate (\r\(\mathrm{PbSO}_{4}\)) is in aqueous solution, it dissociates into its constituent ions, lead (\r\(\mathrm{Pb}^{2+}\)) and sulfate (\r\(\mathrm{SO}_{4}^{2-}\)). However, there's a limit to how much the compound can dissolve, and this limit is defined by the solubility product constant (\(K_{sp}\)). If the product of the ionic concentrations in the solution exceeds this constant, the excess ions will start to combine to form the solid precipitate of \(\mathrm{PbSO}_{4}\).

Understanding precipitation reactions is essential for a number of scientific and industrial processes, including water treatment, mineral extraction, and qualitative chemical analysis.

Equilibrium Constant Calculation

The equilibrium constant, represented by \(K_{sp}\) for solubility product, numerically defines the degree of solubility of a compound. It is a particular type of equilibrium constant that applies to the dissolution of slightly soluble salts.

When a salt is in equilibrium with its dissociated ions, the concentrations of these ions at saturation define the solubility product constant. The mathematical expression for \(K_{sp}\) is derived from the equilibrium expression which involves the concentrations of the products raised to the power of their stoichiometric coefficients, divided by the concentrations of the reactants also raised to the power of their coefficients.

For our case with lead sulfate, the equilibrium constant is given by:
\[K_{sp} = [\mathrm{Pb}^{2+}][\mathrm{SO}_{4}^{2-}]\]
This equation shows that the solubility product is the product of the molar concentrations of the lead and sulfate ions at the point of saturation. To find the unknown concentration of \(\mathrm{Pb}^{2+}\), we simply rearrange and solve this equation using the known value of the sulfate ion concentration and the equilibrium constant, \(K_{sp}\).

Ionic Concentration

Ionic concentration refers to the amount of an ion present in a certain volume of solution, usually expressed in molarity (M), which is moles per liter. The manipulation of ionic concentrations is crucial for controlling the outcome of precipitation reactions.

Understanding how the concentrations of ions in a solution affect the solubility of salts is important in predicting whether a precipitate will form when solutions are mixed. By knowing the solubility product constant (\(K_{sp}\)) and the concentration of one ion, you can calculate the maximum concentration of the other ion before a precipitate begins to form.

As demonstrated in the exercise, with a known \(K_{sp}\) and the concentration of \(\mathrm{SO}_{4}^{2-}\), you can calculate the concentration of \(\mathrm{Pb}^{2+}\) at the point when \(\mathrm{PbSO}_{4}\) starts to precipitate. This is an essential calculation in various fields, from pharmacology, where it's important for drug formulation, to environmental science, for the assessment of water pollution levels.