Problem 103 - Chemistry The Central Science In Si Units

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?

Step-by-Step Solution

Answer

The required concentration of \(\mathrm{KMnO_4}\) is \(1.12 \times 10^{-1} \mathrm{M}\).

Step 1: Write the Dissolution Equation

Consider the dissolution of barium permanganate in water. The equation for this process is: \[ \mathrm{Ba(MnO_4)_2 (s)} \rightleftharpoons \mathrm{Ba^{2+} (aq)} + 2 \mathrm{MnO_4^- (aq)} \] From this, you can see that one mole of \( \mathrm{Ba(MnO_4)_2} \) produces one mole of \( \mathrm{Ba^{2+}} \) ions and two moles of \( \mathrm{MnO_4^-} \) ions.

Step 2: Define the Solubility Equilibrium Expression

The solubility-product constant (\(K_{sp}\)) for \( \mathrm{Ba(MnO_4)_2} \) is expressed as: \[ K_{sp} = [\mathrm{Ba^{2+}}][\mathrm{MnO_4^-}]^2 \] Given \( K_{sp} = 2.5 \times 10^{-10} \), you can substitute the concentration of \( \mathrm{Ba^{2+}} \) to find \( [\mathrm{MnO_4^-}] \).

Step 3: Substitute Given Values

You know that \( [\mathrm{Ba^{2+}}] = 2.0 \times 10^{-8} \mathrm{M} \). Substitute this into the \(K_{sp}\) expression:\[ 2.5 \times 10^{-10} = (2.0 \times 10^{-8})[\mathrm{MnO_4^-}]^2 \]

Step 4: Solve for \([\mathrm{MnO_4^-}]\)

Rearrange the equation to solve for \([\mathrm{MnO_4^-}]^2\): \[ [\mathrm{MnO_4^-}]^2 = \frac{2.5 \times 10^{-10}}{2.0 \times 10^{-8}} \] Calculate: \[ [\mathrm{MnO_4^-}]^2 = 1.25 \times 10^{-2} \]

Step 5: Calculate \([\mathrm{MnO_4^-}]\)

Take the square root of both sides of the equation to find \([\mathrm{MnO_4^-}]\):\[ [\mathrm{MnO_4^-}] = \sqrt{1.25 \times 10^{-2}} \approx 1.12 \times 10^{-1} \mathrm{M} \]This is the concentration of \(\mathrm{MnO_4^-}\) in equilibrium.

Step 6: Determine Required \([\mathrm{KMnO_4}]\) Concentration

Because \(\mathrm{KMnO_4}\) is the source of \(\mathrm{MnO_4^-}\), the concentration of \(\mathrm{KMnO_4}\) needed is the same. Therefore, the concentration of \(\mathrm{KMnO_4}\) required is \(1.12 \times 10^{-1} \mathrm{M}\).

Key Concepts

Equilibrium Expression

An equilibrium expression is vital in understanding chemical reactions that have reached a stable state. In this exercise, we're looking at the dissolution of barium permanganate ([Ba(MnO₄)₂] ) into barium ( Ba²⁺) and permanganate ions ( MnO₄^-). The equilibrium expression for this reversible dissolution captures the balanced proportions of the dissolved ions at equilibrium.
To establish this, the solubility product constant ( K_sp) is used, which relates the concentrations of the ions in the saturated solution. It's written based on the stoichiometry of the dissolution reaction: K_sp = [Ba²⁺][MnO₄^-]².
Here, [Ba²⁺] and [MnO₄^-] refer to the molar concentrations of the ions in solution. The exponent on [MnO₄^-] comes from the fact that two moles of this ion are produced per mole of barium permanganate dissolved.

Ksp Calculation

The solubility product constant, K_sp, plays a critical role in predicting how much of a salt can dissolve in a solvent at equilibrium. It is particularly useful for sparingly soluble salts like barium permanganate. At equilibrium, the dissolved ions have reached a maximum concentration in a saturated solution, dictated by K_sp.
To calculate this in our example, you use the known concentration of one ion, Ba²⁺ = 2.0 × 10^-8 M. The relationship: K_sp = [Ba²⁺][MnO₄^-]² enables you to solve for the unknown [MnO₄^-].
Plugging in the known concentration and rearranging, you calculate [MnO₄^-], which reflects the balance necessary to keep the solution at equilibrium. Once achieved, even small disturbances in concentrations can shift equilibrium, showcasing the delicacy of these calculations.

Ionic Dissolution Process

Understanding the ionic dissolution process helps in visualizing how solids like barium permanganate dissociate in water to form ions. This dissociation is reversible, meaning the ions can recombine to form the solid, establishing a dynamic equilibrium.
The equation: Ba(MnO₄)₂ (s) ↔ Ba²⁺ (aq) + 2 MnO₄^- (aq) shows how one formula unit of barium permanganate dissolves to yield one Ba²⁺ ion and two MnO₄^- ions.
A clear understanding of this process is crucial because it sets the stage for equilibrium. At this point, the rate of dissolution of the solid is equal to the rate of precipitation from the solution.

• This dynamic process determines the maximum solubility of a salt in a given solvent.
• An understanding of the dissolution process is essential for working with saturated and unsaturated solutions.

The dissolution process and its balance are best described by the K_sp value, which ensures that the system remains stable unless disturbed.