Question

(a) Will \(\mathrm{Co}(\mathrm{OH})_{2}\) precipitate from solution if the pH of a \(0.020 \mathrm{M}\) solution of \(\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}\) is adjusted to \(8.5\) ? (b) Will \(\mathrm{AgIO}_{3}\) precipitate when \(20 \mathrm{~mL}\) of \(0.010 \mathrm{M} \mathrm{AgIO}\) is mixed with \(10 \mathrm{~mL}\) of \(0.015 \mathrm{M} \mathrm{NalO}_{3}\) ? \(\left(K_{4}\right.\) of \(\mathrm{AglO}_{3}\) is \(\left.3.1 \times 10^{-t}\right)\).

Short answer

(a) At pH 8.5, the concentration of OH⁻ ions is \(1.77 \times 10^{-6} \mathrm{M}\). The reaction quotient, Q, for the given conditions is \(1.03 \times 10^{-12}\), and since Q < Kₛₚ for Co(OH)₂, no precipitation of Co(OH)₂ occurs. (b) The reaction quotient, Q, for the given conditions is \(2 \times 10^{-8}\). Comparing Q with the given Kₛₚ value for AgIO₃ \((3.1 \times 10^{-8})\), we find that Q < Kₛₚ, so no precipitation of AgIO₃ occurs.

Step-by-step solution

Calculate the concentration of OH⁻ ions at pH 8.5

Use the pH formula to calculate the concentration of OH⁻ ions: pH = -log₁₀[H₃O⁺] pOH = -log₁₀[OH⁻], where pOH = 14 - pH.

Determine the solubility product (Kₛₚ) for Co(OH)₂

The balanced chemical equation for the dissolution of Co(OH)₂ is Co(OH)₂ (s) <=> Co²⁺ (aq) + 2OH⁻ (aq). We can express the solubility product (Kₛₚ) as follows: Kₛₚ = [Co²⁺][OH⁻]².

Calculate the reaction quotient (Q) at the given conditions

Q = [Co²⁺][OH⁻]² Since the cobalt ion reacts with the hydroxide ion in a 1:2 ratio, we have: Q = (0.020 M)(2 × 0.00000117)².

Compare Q with Kₛₚ to determine if Co(OH)₂ will precipitate

If Q > Kₛₚ, precipitation will occur. If Q < Kₛₚ, precipitation will not occur. Calculate Q and compare it with the given Kₛₚ value for Co(OH)₂ to predict if a precipitate will form. (b)

Calculate the initial amount of Ag⁺ and IO₃⁻ in moles

n(Ag⁺) = (volume × concentration) = (0.020 L)(0.010 mol/L) n(IO₃⁻) = (volume × concentration) = (0.010 L)(0.015 mol/L)

Calculate the new concentration of each ion when the two solutions are mixed

Find the total volume: V = 20 mL + 10 mL = 0.030 L. Concentration of Ag⁺: (n(Ag⁺) / total volume) Concentration of IO₃⁻: (n(IO₃⁻) / total volume)

Calculate the reaction quotient (Q) at the given conditions

The reaction quotient Q can be expressed from the balanced chemical reaction: Ag⁺ (aq) + IO₃⁻ (aq) <=> AgIO₃ (s) Q = [Ag⁺][IO₃⁻] Substitute the calculated concentration values and calculate Q.

Compare Q with Kₛₚ to determine if AgIO₃ will precipitate

If Q > Kₛₚ, precipitation will occur. If Q < Kₛₚ, precipitation will not occur. Compare Q with the given Kₛₚ value for AgIO₃ (3.1 × 10⁻⁸) to predict if a precipitate will form.

Key concepts

Understanding Solubility Product Constant (Ksp)

The solubility product constant, or Ksp, is a crucial concept when predicting whether a solid will dissolve or precipitate in a solution. It represents the equilibrium between a solid and its ions in a saturated solution. In chemical terms, it's formulated as the product of the molar concentrations of the resultant ions, each raised to the power of their stoichiometric coefficients. For example, for the dissolution of Co(OH)₂, the Ksp expression is:
Ksp = [Co²⁺][OH^-]².
Understanding Ksp values helps us determine the solubility of a compound under various conditions. A small Ksp indicates low solubility, meaning the compound is less likely to dissolve and more likely to precipitate. Conversely, a larger Ksp suggests higher solubility, with the compound more readily dissolving in solution.

Calculating Reaction Quotient (Q)

The reaction quotient, Q, lets us predict the direction in which a reaction will proceed at any point before reaching equilibrium. It has the same form as the expression for the equilibrium constant but uses the initial concentrations of the reactants and products. When comparing Q to the Ksp value, we can tell if a precipitate will form in a solution. If Q is greater than Ksp, the reaction will favor the formation of a solid, leading to precipitation. If Q is smaller than Ksp, the reaction will favor dissolution, and the substance will stay in solution. For example, in a situation involving Co(OH)₂, we calculate Q using the initial concentrations of Co²⁺ and OH^-.
Q = [Co²⁺][OH^-]²,
where [Co²⁺] and [OH^-] are initial concentrations, not equilibrium concentrations.

Solving for pH and pOH in Precipitation Reactions

The acidity or basicity of a solution can impact whether a compound will precipitate. pH is the measure of the acidity, while pOH measures the basicity. They are related by the equation:
pH + pOH = 14.
To find the OH^- concentration (important for predicting precipitation of hydroxides), we rearrange the pOH formula:
[OH^-] = 10^-pOH.
Calculating the pH or pOH of a solution allows us to determine the concentration of hydrogen ions (H₃O⁺) or hydroxide ions (OH^-). With these values, we can assess the likelihood of precipitation for compounds like Co(OH)₂. If the calculated pOH leads to a Q value higher than Ksp for Co(OH)₂, the solution is too basic for the solute to remain dissolved, and Co(OH)₂ will precipitate.