Question

Calculate the molar solubility of \(\mathrm{Co}(\mathrm{OH})_{3}, K_{\mathrm{sp}}=2.5 \times 10^{-43}\).

Short answer

The molar solubility of Co(OH)₃ is approximately \( 1.25 \times 10^{-11}\: \text{mol/L} \).

Step-by-step solution

1. Write the balanced chemical equation of Co(OH)₃ dissolving in water

First, we need to know how Co(OH)₃ behaves in the water when it dissolves. The balanced reaction for the dissolution of cobalt(III) hydroxide is: Co(OH)₃ (s) ⇌ Co³⁺ (aq) + 3OH⁻ (aq)

2. Express the solubility in terms of x

Now, let's assume the molar solubility of Co(OH)₃ is x mol/L. Then at equilibrium: - [Co³⁺] = x mol/L - [OH⁻] = 3x mol/L

3. Write the expression for Ksp

Now, let's write the expression for Ksp for this reaction using the molar concentrations at equilibrium: Ksp = [Co³⁺] [OH⁻]³

4. Substitute the given Ksp value and molar concentrations in the Ksp expression

Since the question provides the Ksp value of 2.5 × 10⁻⁴³, now we can substitute the values in our expression: \( 2.5 \times 10^{-43} \) = (x) (3x)³

5. Solve for x

To find the molar solubility x, we need to simplify and solve the equation for x: \( 2.5 \times 10^{-43} \) = 27x⁴ Divide both sides by 27: \( x^{4} = \frac{2.5 \times 10^{-43}}{27} \) Now, take the fourth root of both sides: \( x = \left(\frac{2.5 \times 10^{-43}}{27}\right)^{\frac{1}{4}} \)

6. Calculate the molar solubility

Finally, we can plug in the given values into our calculator to find the molar solubility: \( x = \left(\frac{2.5 \times 10^{-43}}{27}\right)^{\frac{1}{4}} \approx 1.25 \times 10^{-11}\: \text{mol/L} \) So, the molar solubility of Co(OH)₃ is approximately \( 1.25 \times 10^{-11}\: \text{mol/L} \).