Problem 57 - Chemistry and Chemical Reactivity

Question

Calculate the hydronium ion concentration and \(\mathrm{pH}\) in a \(0.20 \mathrm{M}\) solution of ammonium chloride, \(\mathrm{NH}_{4} \mathrm{Cl}\).

Step-by-Step Solution

Answer

[H₃O⁺] for 0.20 M NH₄Cl is approximately calculated using Kₐ and then pH is found from \(-\log[H₃O⁺]\).

Step 1: Understand the Problem

We have a 0.20 M solution of ammonium chloride (NH₄Cl), which is a salt formed from a weak base (NH₄OH) and a strong acid (HCl). This solution will act as an acidic solution because NH₄⁺ can donate a proton to water, forming hydronium ions, which we need to calculate.

Step 2: Write the Reaction Equation

NH₄Cl dissociates in water to produce NH₄⁺ ions. The ammonium ions then react with water to form hydronium ions: NH₄⁺ + H₂O ⇌ NH₃ + H₃O⁺. This equilibrium reaction will help us determine the H₃O⁺ concentration.

Step 3: Use the Equilibrium Expression

The equilibrium constant for the reaction is the ionization constant of NH₄⁺, represented as Kₐ. First, find Kₐ using the relation Kₐ = \( \dfrac{K_w}{K_b} \), where K_w is the ion-product constant for water \(1.0 \times 10^{-14} \) and K_b is the base ionization constant for NH₄OH. Use given or standard tables to find these constants.

Step 4: Set Up the Equilibrium Table

Create a table showing initial concentrations (initially, [NH₄⁺] = 0.20 M and [H₃O⁺] = [NH₃] = 0), the change in concentrations, and equilibrium concentrations. Let 'x' be the change in concentration to form H₃O⁺ and NH₃. Therefore, equilibrium concentrations are [NH₄⁺] = 0.20 - x, [H₃O⁺] = x, and [NH₃] = x.

Step 5: Solve the Equilibrium Equation

Using the Kₐ expression for NH₄⁺: \( Kₐ = \dfrac{[NH₃][H₃O⁺]}{[NH₄⁺]} \) Substitute the equilibrium concentrations into the expression: \( Kₐ = \dfrac{x^2}{0.20-x} \). Assuming 0.20-x ≈ 0.20 (because x is small), solve for x to find [H₃O⁺].

Step 6: Calculate pH

Once the [H₃O⁺] is determined, use the formula for pH: \( ext{pH} = -\log[H₃O⁺] \). Evaluate the logarithm to find the pH of the solution.

Key Concepts

Hydronium Ion Concentration

When working with solutions like ammonium chloride, it is crucial to calculate the hydronium ion concentration. Ammonium chloride, with a concentration of 0.20 M, dissociates fully in water to form ammonium ions (NH₄⁺). These ammonium ions can interact with water molecules to form hydronium ions (H₃O⁺). This reaction can be represented as:

• NH₄⁺ + H₂O ⇌ NH₃ + H₃O⁺

The hydronium ion concentration is directly involved in determining the acidity of the solution. In most cases, the higher the H₃O⁺ concentration, the more acidic the solution. By establishing the equilibrium conditions of this reaction, we can find the concentration of hydronium ions to further calculate the pH.

Equilibrium Expression

An equilibrium expression helps quantify the relationship between the concentrations of reactants and products in a chemical reaction at equilibrium. For the reaction involving NH₄⁺ and water:

• NH₄⁺ + H₂O ⇌ NH₃ + H₃O⁺

The equilibrium constant Kₐ is used to express this balance and is given by:

• \[ Kₐ = \dfrac{[NH₃][H₃O⁺]}{[NH₄⁺]} \]

Knowing the initial concentrations and the changes occurring during the reaction assists in setting up an equilibrium table. This table estimates how concentrations shift, enabling us to solve for the exact concentration of hydronium ions at equilibrium. Understanding and manipulating the equilibrium expression is key in solving for unknown concentrations, especially in establishing pH.

Ammonium Chloride Solution

Ammonium chloride (NH₄Cl) is a white crystalline solid that is highly soluble in water. When dissolved, it dissociates into ammonium ions (NH₄⁺) and chloride ions (Cl⁻). Ammonium chloride solutions tend to be acidic due to the weak base (NH₄OH) and strong acid (HCl) from which it originates. The balanced dissociation equation in water is:

• NH₄Cl → NH₄⁺ + Cl⁻

The solution then influences the pH by producing hydronium ions through the action of ammonium ions interacting with water. Understanding the nature and behavior of an ammonium chloride solution allows us to predict if a solution will be acidic, neutral, or basic.

Ionization Constant

The ionization constant, or Kₐ, is crucial when dealing with weak acids or bases and their salts like ammonium chloride. It represents the strength of the acid or base in a solution, showing how readily it donates protons. For NH₄⁺, you can find Kₐ using the formula:

• \[ Kₐ = \dfrac{K_w}{K_b} \]

Where Kₐ is the dissociation constant for NH₄⁺, K_w is the ion-product constant for water ( \(1.0 \times 10^{-14} \)), and K_b is the base ionization constant for NH₄OH. This relationship translates the basicity to acidity for effective calculations in acidic solutions. Calculating and understanding the ionization constant allows chemists to anticipate how acidic or basic a solution will be, based on its components.