Problem 103 - Chemistry: The Central Science

Question

The solubility-product constant for barium permanganate, \(\mathrm{Ba}\left(\mathrm{MnO}_{4}\right)_{2}\) is \(2.5 \times 10^{-10}\). Assume that solid \(\mathrm{Ba}\left(\mathrm{MnO}_{4}\right)_{2}\) is in equilibrium with a solution of \(\mathrm{KMnO}_{4}\). What concentration of \(\mathrm{KMnO}_{4}\) is required to establish a concentration of \(2.0 \times 10^{-4} \mathrm{M}\) for the \(\mathrm{Ba}^{+}\)ion in solution?

Step-by-Step Solution

Answer

A concentration of \(1.12 \times 10^{-3} M\) KMnO4 is required to establish a concentration of \(2.0 \times 10^{-4} M\) for the Ba^2+ ion in solution.

Step 1: Write the solubility equilibrium expression for barium permanganate

Barium permanganate, Ba(MnO4)2, dissociates in water as follows: \[ Ba(MnO_{4})_{2}(s) \rightleftharpoons Ba^{2+}(aq) + 2MnO_{4}^{-}(aq) \] The solubility product constant (Ksp) expression for this equilibrium is: \[ K_{sp} = [Ba^{2+}][MnO_{4}^-]^2 \]

Step 2: Use the given Ksp and concentration of Ba^2+ ions to solve for the concentration of MnO4^- ions

Plug the given values of Ksp and [Ba^2+] into the Ksp expression: \[ 2.5 \times 10^{-10} = (2.0 \times 10^{-4})[MnO_{4}^-]^2 \] \[ MnO_{4}^- \] Now, solve for the concentration of MnO4^- ions: \[ [MnO_{4}^-]^2 = \frac{2.5 \times 10^{-10}}{2.0 \times 10^{-4}} \] \[ [MnO_{4}^-]^2 = 1.25 \times 10^{-6} \] \[ [MnO_{4}^-] = \sqrt{1.25 \times 10^{-6}} \] \[ [MnO_{4}^-] = 1.12 \times 10^{-3} M \]

Step 3: Find the concentration of KMnO4 required

In the solution, one molecule of KMnO4 provides one MnO4^- ion. Therefore, the concentration of KMnO4 needed is equal to the concentration of MnO4^- ions. So, the required concentration of KMnO4 is: \[ [KMnO_{4}] = 1.12 \times 10^{-3} M \] Thus, a concentration of \(1.12 \times 10^{-3} M\) KMnO4 is required to establish a concentration of \(2.0 \times 10^{-4} M\) for the Ba^2+ ion in solution.

Key Concepts

Ion Equilibrium

The concept of ion equilibrium deals with the state of balance in which the rate of a forward reaction equals the rate of the reverse reaction for ions in a solution. In the context of solubility, ion equilibrium occurs when a solid ionic compound is dissolving in water at the same rate that its ions are precipitating to form the solid compound again. This dynamic balance is essential to understanding the dissolution of salts and the concentrations of ions in solution.

When we consider the equilibrium for barium permanganate, the balanced chemical equation is represented as:\[ Ba(MnO_{4})_{2}(s) \rightleftharpoons Ba^{2+}(aq) + 2MnO_{4}^{-}(aq) \]This indicates that solid barium permanganate dissociates into barium ions (\(Ba^{2+}\)) and permanganate ions (\(MnO_{4}^{-}\)) in the aqueous phase. The system will reach a point where the effective concentrations of these ions remain constant over time, which can be described mathematically by the solubility-product constant, Ksp.

Barium Permanganate Dissolution

Understanding how barium permanganate (\(Ba(MnO_{4})_{2}\)) dissolves in water is critical in predicting the behavior of this compound in aqueous solutions. The process of dissolution involves the breakdown of the solid ionic lattice of barium permanganate into its constituent ions. This process is guided by the principle of ion solubility and the tendency of the compound to reach a state of equilibrium in solution.

As barium permanganate dissolves, it releases barium ions (\(Ba^{2+}\)) and permanganate ions (\(MnO_{4}^{-}\)) into the solution. The dissolution can be represented by the following equilibrium:\[ Ba(MnO_{4})_{2}(s) \rightleftharpoons Ba^{2+}(aq) + 2MnO_{4}^{-}(aq) \]This dissolution is governed by the solubility-product constant, and the extent to which barium permanganate can dissolve is quantified by the Ksp value, which provides a measure of the compound's solubility under equilibrium conditions.

Ksp Calculation

The solubility-product constant (Ksp) is a crucial parameter for predicting the solubility of sparingly soluble ionic compounds. It is defined as the product of the molar concentrations of the ions in a saturated solution, each raised to the power of its stoichiometric coefficient in the equilibrium equation.

For instance, barium permanganate has the Ksp expression:\[ K_{sp} = [Ba^{2+}][MnO_{4}^-]^2 \]Given the Ksp value and the known concentration of one ion, we can calculate the concentration of the other ion. In the example provided, by substituting the Ksp value and the concentration of the \(Ba^{2+}\) ion into the expression, we can solve for the concentration of the \(MnO_{4}^{-}\) ions. This approach is invaluable when we need to understand the solubility dynamics of barium permanganate in different conditions or when we need to manipulate the composition of the solution to achieve desired concentrations.

Molar Concentration

Molar concentration, commonly referred to as molarity and symbolized by (M), represents the number of moles of solute per liter of solution. It is a fundamental measurement in chemistry, directly reflecting the strength and composition of a solution. In the dissolution of barium permanganate, knowing the molar concentration of the ions is essential to predict how the compound will behave in solution and to calculate its Ksp.

In the context of our example, establishing the molar concentration of the \(Ba^{2+}\) ion leads us directly to calculate the molar concentration of the \(MnO_{4}^{-}\) ions through the Ksp expression. Once we determine the required molar concentration of \(KMnO_{4}\) necessary to maintain the equilibrium, this information can be utilized in various applications such as chemical analysis, formulation of solutions, or even in treatment processes where specific ion concentrations are crucial.