Problem 85 - Chemistry

Question

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 \(\mathrm{Al}(\mathrm{OH})_{3} ?\)

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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

Step 3: 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

Step 4: 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.

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that explains how chemical reactions naturally reach a state where the rates of the forward and reverse reactions are equal. In this state, the concentrations of reactants and products remain constant over time. This is because the forward and reverse reactions are happening at the same rate.

Understanding chemical equilibrium involves several key ideas:

• Reversible Reactions: Equilibrium is reached when a chemical reaction can proceed in both the forward and reverse directions.
• Dynamic Process: Though the observable concentrations don't change, molecules continue to react with each other.
• Equilibrium Constant (K): This is a ratio of the concentrations of products to reactants, each raised to their respective stoichiometric coefficients.

Chemical equilibrium in solubility contexts reflects how salts dissociate in solution until saturation is achieved. Particularly, when dealing with a solubility product constant, or \( K_{sp} \), you calculate the product of the concentrations of the ionic species at equilibrium. For a compound like Al(OH)₃, this gives insights into conditions under which a solid might dissolve or precipitate out of solution.

Precipitation Reactions

Precipitation reactions occur when ions in solution combine to form an insoluble solid, known as a precipitate. This process is vital in predicting when solutes may separate out in a given solution, and it heavily depends on the solubility product constant (\( K_{sp} \)).

Whenever the product of the ion concentrations exceeds the \( K_{sp} \) of a compound, the solution is supersaturated, and precipitation occurs. Here's how this applies to the precipitation of aluminum hydroxide:

• Balancing Equations: Write the balanced equation to understand the stoichiometry, for instance, \( \text{Al(OH)}_3 \rightleftharpoons \text{Al}^{3+} + 3 \text{OH}^- \).
• Setting up ICE Tables: Using the Initial, Change, and Equilibrium method helps calculate the ion concentration at equilibrium.
• Determine Conditions: Use the \( K_{sp} \) to calculate when a specific ion concentration will cause the onset of precipitation.

By applying these principles, you can accurately predict and control precipitation reactions. For Al(OH)₃, this means finding when its ions in solution reach the solubility limit set by \( K_{sp} \). This knowledge helps in processes like water treatment, where minimizing precipitate is crucial.

pH Calculation

Calculating pH is a central concept in understanding the acidity or basicity of a solution. The pH scale, which ranges from 0 to 14, is a measure of the hydrogen ion concentration in a solution. Lower pH values indicate higher acidity, while higher pH values indicate basicity.

When dealing with solubility and precipitation, calculating pH can help determine at what point certain ions will precipitate out of solution. Here's what you need to know:

• pOH and pH Relationship: The sum of pH and pOH always equals 14 at room temperature, \( \text{pH} + \text{pOH} = 14 \).
• Hydroxide Concentration: For solutions where OH⁻ ions are of interest, the concentration can be determined from the solubility product.
• Using Logs for Calculation: Once the concentration of OH⁻ is known, \( \text{pOH} = -\log[\text{OH}^-] \) can be found, and consequently \( \text{pH} = 14 - \text{pOH} \).

In scenarios where a solution's pH causes precipitation, such as with \( \text{Al(OH)}_3 \), knowing the pH is essential for avoiding or inducing such reactions. Understanding these calculations aids in practical applications ranging from industrial processing to everyday chemistry encounters.