Question

In qualitative analysis, \(\mathrm{Ag}^{+}, \mathrm{Hg}_{2}^{2+}\), and \(\mathrm{Pb}^{2+}\) are separated from other cations by the addition of HCl. Calculate the concentration of \(\mathrm{Cl}^{-}\) required to just begin the precipitation of (a) \(\mathrm{AgCl}\), (b) \(\mathrm{Hg}_{2} \mathrm{Cl}_{2}\), (c) \(\mathrm{PbCl}_{2}\) in a solution having metal-ion concentrations of \(0.030 \mathrm{M}\). What fraction of the \(\mathrm{Pb}^{2+}\) remains in solution when the \(\mathrm{Ag}^{+}\) just begins to precipitate?

Short answer

Calculate the chloride ion concentration for each Ksp and find the fraction of Pb2+ remaining using equilibrium concentrations.

Step-by-step solution

Determine Solubility Product (Ksp) for Each Salt

To calculate the concentration of \(\text{Cl}^{-}\) needed to begin precipitation, start by using the solubility product constant, \(K_{sp}\), for each compound: \(\text{AgCl}\), \(\text{Hg}_2\text{Cl}_2\), and \(\text{PbCl}_2\). Typical values are \(K_{sp}(\text{AgCl}) = 1.8 \times 10^{-10}\), \(K_{sp}(\text{Hg}_2\text{Cl}_2) = 1.1 \times 10^{-18}\), and \(K_{sp}(\text{PbCl}_2) = 1.6 \times 10^{-5}\).

Key concepts

Solubility Product

The solubility product, abbreviated as \(K_{sp}\), is a constant that helps us understand the solubility of ionic compounds. It specifically represents the maximum product of the ion concentrations that can exist in a solution at equilibrium. \(K_{sp}\) is determined through experimental observations and varies for different compounds.
When a salt dissolves in water, it dissociates into its constituent ions. For example, when \(\text{AgCl}\) dissolves, it separates into \(\text{Ag}^+\) and \(\text{Cl}^-\) ions. The equation for this dissolution is represented as:
\[\text{AgCl (s)} \rightleftharpoons \text{Ag}^+ \text{(aq)} + \text{Cl}^- \text{(aq)}\]
The \(K_{sp}\) equation is:

• \(K_{sp} = [\text{Ag}^+][\text{Cl}^-]\) for \(\text{AgCl}\)

This concept is crucial in qualitative analysis as it allows us to quantitatively predict when and how salts will begin to precipitate from a solution.

Precipitation Reactions

Precipitation reactions occur when dissolved ions combine to form an insoluble solid, known as a precipitate. In a solution containing metal ions like \(\mathrm{Ag}^+, \mathrm{Hg}_{2}^{2+},\) and \(\mathrm{Pb}^{2+}\), one can add a specific anionic agent such as \(\mathrm{Cl}^-\) to selectively precipitate these ions.
To determine when a salt will start precipitating, we compare the product of the ion concentrations (known as the ionic product) to the \(K_{sp}\).

• If the ionic product exceeds the \(K_{sp}\), precipitation starts.
• If it is less, ions remain in solution.

For example, to start the precipitation of \(\text{AgCl}\), the concentration of \(\mathrm{Cl}^-\) must be increased until its product with the \(\mathrm{Ag}^+\) ion concentration equals the \(K_{sp}\) of \(\text{AgCl}\). This delicate balance is what makes precipitation reactions a key tool in isolating specific metal ions in a mixture.

Metal-Ion Concentrations

Metal-ion concentration plays a vital role in the precipitation process. In our problem, each of \(\mathrm{Ag}^+, \mathrm{Hg}_{2}^{2+},\) and \(\mathrm{Pb}^{2+}\) has an initial concentration of \(0.030 \text{ M}\).
When calculating the \(\mathrm{Cl}^-\) concentration needed for precipitation, we use these \(K_{sp}\) values and concentrations as input:

• \(\text{Set up the equation for each compound:}\)
• \(\text{AgCl: }\, K_{sp} = [\text{Ag}^+][\text{Cl}^-]\)
• \(\text{Substitute } [\text{Ag}^+] = 0.030 \text{ M}, \ K_{sp} = 1.8 \times 10^{-10}\)
• \(\text{Solve for } [\text{Cl}^-]\)

These calculations will reveal the precise \(\mathrm{Cl}^-\) concentration needed to start precipitation of each metal ion's chloride compound. Understanding these interactions helps tailor the experiment to selectively remove ions from the solution.