Question

\(\mathrm{Na}_2 \mathrm{SO}_4\) dissociates completely in water. From the information given in the table below, if \(\mathrm{Na}_2 \mathrm{SO}_4\) were added to a solution containing equal concentrations of aqueous \(\mathrm{Ca}^{2+}, \mathrm{Ag}^*\). \(\mathrm{Pb}^{2+}\), and \(\mathrm{Ba}^{2+}\) ions, which of the following solids would precipitate first? \begin{tabular}{|l|c|} \hline Compound & \(K_{2 p}\) \\ \hline \(\mathrm{CaSO}_4\) & \(6.1 \times 10^{-3}\) \\ \hline \(\mathrm{AgrO}_4\) & \(1.2 \times 10^{-5}\) \\ \hline \(\mathrm{PbSO}_4\) & \(1.3 \times 10^{-4}\) \\ \hline \(\mathrm{BaSO}_4\) & \(1.5 \times 10^{-4}\) \\ \hline \end{tabular} A. \(\mathrm{CaSO}_4\) B. \(\mathrm{Ag}_4 \mathrm{SO}_4\) C. \(\mathrm{PbSO}_4\) D. \(\mathrm{BaSO}_4\)

Short answer

B. \(\text{Ag}_2\text{SO}_4\)

Step-by-step solution

- Understand the Problem

The problem requires determining which salt will precipitate first when \(\text{Na}_2\text{SO}_4\) is added to a solution containing various ions. The salts involved are \(\text{CaSO}_4\), \(\text{Ag}_2\text{SO}_4\), \(\text{PbSO}_4\), and \(\text{BaSO}_4\), with their solubility products (K_{sp}) provided.

- Recall the Concept of Precipitation

A salt will precipitate when the product of the ion concentrations exceeds the solubility product (\(K_{sp}\)). The salt with the lowest \(K_{sp}\) will precipitate first from a solution with equivalent ion concentrations, as it reaches saturation point earliest.

- Compare the Solubility Products

The given \(K_{sp}\) values are: \[ \text{CaSO}_4: 6.1 \times 10^{-3}\ \text{Ag}_2\text{SO}_4: 1.2 \times 10^{-5}\ \text{PbSO}_4: 1.3 \times 10^{-4}\ \text{BaSO}_4: 1.5 \times 10^{-4} \] The salt with the lowest \(K_{sp}\) is \(\text{Ag}_2\text{SO}_4\).

- Determine the First Precipitate

Since \(\text{Ag}_2\text{SO}_4\) has the smallest \(K_{sp}\), \(\text{Ag}_2\text{SO}_4\) will be the first compound to precipitate when \(\text{Na}_2\text{SO}_4\) is added to the solution.

Key concepts

Ksp Values

Let's start by understanding what Ksp values are. Ksp stands for the Solubility Product Constant, which is a measure of the solubility of a compound in water. It represents the product of the molar concentrations of the ions of a compound, each raised to the power of its coefficient in the balanced equilibrium equation.
The general expression for a solubility product is given by the equation:
\[\text{Ksp} = [A^+]^m \times [B^-]^n\]
where - [A^+] is the concentration of the cation
- [B^-] is the concentration of the anion
- m and n are the respective coefficients from the balanced equation
A compound with a lower Ksp value is less soluble in water. Conversely, a compound with a higher Ksp value is more soluble. The compounds listed in the exercise each have different Ksp values, reflecting their varying solubilities.

Precipitation Reactions

When two aqueous solutions mix, a chemical reaction can occur that results in the formation of an insoluble solid called a precipitate. This happens when the product of the concentrations of the ions in the solution exceeds the compound’s Ksp value.
During a precipitation reaction, the ions in solution come together to form a new, less soluble compound. If we consider the scenario given, if you add \(\text{Na}_2\text{SO}_4\) to a solution containing \(\text{Ca}^{2+}, \text{Ag}^+, \text{Pb}^{2+}\), and \(\text{Ba}^{2+}\), the sulfates of these ions will form.
To figure out which compound precipitates first, compare their Ksp values. The compound with the lowest Ksp value will precipitate first because it will reach its saturation point more quickly.

Saturation Point

The saturation point is a crucial concept when understanding precipitation reactions. It’s the stage at which the solution contains the maximum concentration of dissolved ions, beyond which no more of the solute can dissolve. Any additional solute will precipitate out of the solution.
For example, in the given problem, as \(\text{Na}_2\text{SO}_4\) is added to a solution, it dissociates completely into \(\text{Na}^+\) and \(\text{SO}_4^{2-}\) ions. These \(\text{SO}_4^{2-}\) ions will react with, say, \(\text{Ag}^+\) ions to form \(\text{Ag}_2\text{SO}_4\) if the ion product exceeds the Ksp value for \(\text{Ag}_2\text{SO}_4\).
Using the Ksp values given, \(\text{Ag}_2\text{SO}_4\) has the lowest Ksp (\(\text{Ag}_2\text{SO}_4: 1.2 \times 10^{-5}\)). Hence, as soon as the solution hits the saturation point for \(\text{Ag}_2\text{SO}_4\), it will begin precipitating out of the solution first.