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Chapter 16: Problem 85

The \(K_{\mathrm{sp}}\) of \(\mathrm{Al}(\mathrm{OH})_{3}\) is $2 \times 10^{-32} .\( At what pH will a \)0.2-M\( \)\mathrm{Al}^{3+}$ solution begin to show precipitation of \(\mathrm{Al}(\mathrm{OH})_{3} ?\)

Short Answer

Expert verified
The pH at which a 0.2 M Al³⁺ solution begins to show precipitation of Al(OH)₃ is calculated by first solving for the concentration of OH⁻ ions at equilibrium, which is found to be x = (10⁻³¹)^(1/3) / 3. Then, the pOH is determined as -log(x), and finally the pH is obtained using the relationship pH + pOH = 14, giving the value for pH at which the precipitation begins.

Step by step solution

01

Write the balanced chemical equation and the expression for Kₛₚ

First, let's write the balanced chemical equation for the dissolution of Al(OH)₃ in water: Al(OH)₃ (s) ↔ Al³⁺ (aq) + 3OH⁻ (aq) Next, we write the expression for the solubility product constant (Kₛₚ): Kₛₚ = [Al³⁺][OH⁻]³ We know that Kₛₚ = 2 × 10⁻³² and the concentration of Al³⁺ ions in the solution is 0.2 M.
02

Set up the ICE table and solve for the concentration of OH⁻ ions

We will now set up the Initial, Change, Equilibrium (ICE) table to solve for the concentration of OH⁻ ions at equilibrium: - Initial concentrations: [Al³⁺] = 0.2 M, [OH⁻] = 0 - Change in concentrations: [Al³⁺] = 0, [OH⁻] = +3x - Equilibrium concentrations: [Al³⁺] = 0.2 M, [OH⁻] = 3x Now, plug into the Kₛₚ expression: 2 × 10⁻³² = (0.2)(3x)³
03

Solve for x (concentration of OH⁻ ions at equilibrium)

Let's solve the equation for x: (2 × 10⁻³²) / 0.2 = (3x)³ Divide both sides by 0.2: 10⁻³¹ = (3x)³ Take the cube root of both sides: (10⁻³¹)^(1/3) = 3x Now, divide by 3: x = (10⁻³¹)^(1/3) / 3
04

Calculate the pOH and pH of the solution

Now that we have the value of x, which represents the concentration of OH⁻ ions at equilibrium, we can calculate the pOH and pH of the solution: pOH = -log([OH⁻]) pOH = -log(x) Now, to calculate the pH, use the relationship between pH and pOH: pH + pOH = 14 pH = 14 - pOH By calculating these values, we can determine the pH at which the precipitation of Al(OH)₃ begins in the given 0.2 M Al³⁺ solution.

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Most popular questions from this chapter

The concentration of \(\mathrm{Pb}^{2+}\) in a solution saturated with \(\mathrm{PbBr}_{2}(s)\) is \(2.14 \times 10^{-2} \mathrm{M} .\) Calculate \(K_{\mathrm{sp}}\) for \(\mathrm{PbBr}_{2}.\)

What happens to the \(K_{\mathrm{sp}}\) value of a solid as the temperature of the solution changes? Consider both increasing and decreasing temperatures, and explain your answer.

What mass of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}\) must be added to 1.0 \(\mathrm{L}\) of a \(1.0-M \mathrm{HF}\) solution to begin precipitation of \(\mathrm{CaF}_{2}(s) ?\) For $\mathrm{CaF}_{2}, K_{\mathrm{sp}}= 4.0 \times 10^{-11}\( and \)K_{\mathrm{a}}\( for \)\mathrm{HF}=7.2 \times 10^{-4}$ . Assume no volume change on addition of \(\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}(s).\)

Calculate the solubility of each of the following compounds in moles per liter. Ignore any acid–base properties. a. \(P b I_{2}, K_{s p}=1.4 \times 10^{-8}\) b. \(\operatorname{CdCO}_{3}, K_{s p}=5.2 \times 10^{-12}\) c. $\operatorname{Sr}_{3}\left(\mathrm{PO}_{4}\right)_{2}, K_{s p}=1 \times 10^{-31}$

Write balanced equations for the dissolution reactions and the corresponding solubility product expressions for each of the following solids. a. \(A g C_{2} H_{3} O_{2}\) b. \(A l(O H)_{3}\) c. \(C a_{3}\left(\mathrm{PO}_{4}\right)_{2}\)
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