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 \(2.0 \times 10^{-8} \mathrm{M}\) for the \(\mathrm{Ba}^{2+}\) ion in solution?

Short answer

To establish a concentration of \(2.0 \times 10^{-8}\) M for Ba²⁺ ions in solution, the required concentration of KMnO₄ is \(3.54 \times 10^{-2}\) M.

Step-by-step solution

Write the balanced chemical equation and Ksp expression

Write the balanced chemical equation for the dissolution of barium permanganate in water and the corresponding solubility-product constant (Ksp) expression. Balanced chemical equation: Ba(MnO₄)₂ (s) ⇌ Ba²⁺ (aq) + 2MnO₄⁻ (aq) Ksp expression: Ksp = [Ba²⁺] [MnO₄⁻]² Given Ksp value: \(2.5 \times 10^{-10}\)

Substitute the given ion concentrations

Substitute the given Ba²⁺ concentration value into the Ksp expression and solve for the MnO₄⁻ concentration. Ksp = [Ba²⁺] [MnO₄⁻]² Since 1 mol of Ba²⁺ releases 2 mol of MnO₄⁻, the molar ratio between Ba²⁺ and MnO₄⁻ is 1:2. So by stoichiometry, we can say that [MnO₄⁻] = 2[Ba²⁺]. Given [Ba²⁺] = \(2.0 \times 10^{-8}\) M

Solve for the MnO₄⁻ concentration

Plug in the available values to the Ksp expression and solve for the MnO₄⁻ concentration. \(2.5 \times 10^{-10}\) = \(2.0 \times 10^{-8}\) [MnO₄⁻]² [MnO₄⁻]² = \(\frac{2.5 \times 10^{-10}}{2.0 \times 10^{-8}}\) [MnO₄⁻]² = \(1.25 \times 10^{-2}\) [MnO₄⁻] = \(\sqrt{1.25 \times 10^{-2}}\) [MnO₄⁻] = \(3.54 \times 10^{-2}\) M

Calculate the concentration of KMnO₄

Since the MnO₄⁻ ion comes from the dissolution of KMnO₄, the concentration of KMnO₄ will be equal to the concentration of MnO₄⁻ ions in the solution. [KMnO₄] = [MnO₄⁻] = \(3.54 \times 10^{-2}\) M To establish a concentration of \(2.0 \times 10^{-8}\) M for Ba²⁺ ions in solution, the required concentration of KMnO₄ is \(3.54 \times 10^{-2}\) M.

Key concepts

Equilibrium Chemistry

Equilibrium chemistry is the study of chemical systems at a state where the concentrations of reactants and products do not change over time. This occurs in reversible reactions, where the forward and reverse processes happen at the same rate.

In the context of solubility, when an ionic compound like barium permanganate dissolves in water, it dissociates into its constituent ions until a dynamic equilibrium is reached. The solubility-product constant (Ksp) is an expression of this equilibrium state, reflecting the concentrations of the dissolved ions in a saturated solution.

For barium permanganate, the Ksp helps predict the extent to which the salt can dissolve to form ba2+ and MnO4- ions before precipitation begins to occur. Understanding this equilibrium is crucial when manipulating conditions, such as adding another compound to the solution, to maintain solubility or promote precipitation.

Chemical Stoichiometry

Chemical stoichiometry involves the quantitative relationship between reactants and products in a chemical reaction. It allows for the calculation of the amounts of substances required or produced by taking into account the molar ratios enforced by balanced chemical equations.

In the dissolution of barium permanganate, stoichiometry tells us that from one mole of Ba(MnO4)2 that dissociates, one mole of Ba2+ ions and two moles of MnO4- ions are produced. Thus, for every barium ion in solution, there should be twice as many permanganate ions, keeping the stoichiometric ratio of 1:2.

These ratios become especially valuable when solving for unknown concentrations. Once the concentration of one ion is known, stoichiometry enables us to find the concentration of the other ion, ensuring that we are accounting for the precise proportions in which these ions interact to form the compound.

Dissolution of Ionic Compounds

The dissolution of ionic compounds is the process by which they dissociate into their constituent ions when mixed with a solvent like water. The ionic bonds holding the compound together are overcome by the attraction between the ions and the polar water molecules.

Solubility is the maximum amount of a solute that can dissolve in a solvent at a given temperature, and it's governed by factors such as the nature of the ionic compound and the solvent, temperature, and presence of other substances in the solution. For sparingly soluble salts like Ba(MnO4)2, the dissolution creates a dynamic equilibrium between the solid phase and the dissolved ions.

By understanding the dissolution process, students can grasp how various factors affect solubility. For example, adding a compound such as KMnO4 to the solution will increase the MnO4- concentration, and through a phenomenon called the common ion effect, this can affect the solubility of Ba(MnO4)2 and shift the equilibrium towards the solid form, potentially causing more of it to precipitate out of the solution.