Question

A buffered solution is made by adding 50.0 \(\mathrm{g} \mathrm{NH}_{4} \mathrm{Cl}\) to 1.00 \(\mathrm{L}\) of a \(0.75-\mathrm{M}\) solution of \(\mathrm{NH}_{3}\) . Calculate the \(\mathrm{pH}\) of the final solution. (Assume no volume change.)

Short answer

The pH of the final buffered solution is 9.14 after adding 50.0 g of NH4Cl to 1.00 L of a 0.75 M solution of NH3, calculated using the Henderson-Hasselbalch equation.

Step-by-step solution

Identify the relevant equation

The acid dissociation constant (Ka) for NH4+ reaction with water is given by: \(K_\mathrm{a}=\frac{\ce{[H+][NH3]}}{\ce{[NH4+]}}\) The Henderson-Hasselbalch equation is used to relate the pH, pKa, and concentrations of a weak acid ([NH4+]) and its conjugate base ([NH3]) in a buffer solution: \(pH=pK_\mathrm{a} + \log{\frac{[\ce{NH3}]}{[\ce{NH4+}]}}\)

Determine the concentration of NH4Cl solution

First, we need to find the concentration of NH4Cl in the solution. We have the mass (50.0g) and the volume of NH3 solution (1L). To calculate the concentration, we need to convert the mass of NH4Cl into moles and divide by the volume. Molar mass of NH4Cl (g/mol): \( M_\mathrm{NH4Cl} = 14.01 \text{(N)} + 1.01 (\text{H}) * 4 + 35.45\text{(Cl)} \) \( M_\mathrm{NH4Cl} = 53.49\, \mathrm{g/mol} \) To find the moles of NH4Cl: Moles of NH4Cl = 50.0 g / 53.49 g/mol Moles of NH4Cl = 0.935 mol Finally, we can calculate the concentration of NH4Cl: Concentration of NH4Cl (M) = moles / volume Concentration of NH4Cl = 0.935 mol / 1.00 L Concentration of NH4Cl = 0.935 M

Find the pKa using the given Ka

Next, we need to find the pKa value. The pKa value is the negative logarithm of the Ka value. Given the Ka value for NH4+ is 5.56E-10, we can find the pKa as: \(pK_\mathrm{a} = -\log(5.56 \times 10^{-10})\) \(pK_\mathrm{a} = 9.25\)

Calculate the pH using the Henderson-Hasselbalch equation

Now, we can use the Henderson-Hasselbalch equation to find the pH of the buffered solution: \(pH = pK_\mathrm{a} + \log{\frac{[\ce{NH3}]}{[\ce{NH4+}]}}\) Using the concentrations of NH3 (0.75 M) and NH4+ (0.935 M) and the pKa value (9.25), we get: \(pH = 9.25 + \log{\frac{0.75}{0.935}}\) \(pH = 9.25 - 0.112\) \(pH = 9.14\) The pH of the final solution is 9.14.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a handy tool for calculating the pH of a buffer solution. It relates the pH to the pKa and the concentrations of an acid and its conjugate base.

This equation is given by:

• \(pH = pK_\mathrm{a} + \log{\frac{[\text{Base}]}{[\text{Acid}]}}\)

In the context of our problem, ammonia (\(\text{NH}_3\)) acts as the base, and ammonium chloride (\(\text{NH}_4\text{Cl}\)) provides the acid (\(\text{NH}_4^+\)).

By using the Henderson-Hasselbalch equation, we can easily determine how acidic or basic a buffer solution is, which is crucial in many chemical and biological processes.

pH Calculation

Calculating the pH of a solution helps us understand its acidity or basicity. In this exercise, we're given the concentrations of ammonia and ammonium ions to calculate the pH using the Henderson-Hasselbalch equation.

Here's the step-by-step approach:

• Find the molarity (concentration) of \(\text{NH}_4^+\) and \(\text{NH}_3\).
• Use the Henderson-Hasselbalch equation: \(pH = 9.25 + \log{\frac{0.75}{0.935}}\).
• Solve the log ratio to find that \(pH = 9.25 - 0.112 = 9.14\).

Breaking down the steps can make the calculation more approachable and ensures accuracy in the result.

Ammonium Chloride (NH4Cl)

Ammonium chloride (\(\text{NH}_4\text{Cl}\)) is an essential component in buffer solutions. It provides ammonium ions (\(\text{NH}_4^+\)) when dissolved in water.

The role of \(\text{NH}_4^+\) in the buffer equation is crucial as it acts as the weak acid in the Henderson-Hasselbalch equation.

• The molar mass of \(\text{NH}_4\text{Cl}\) is 53.49 g/mol.
• We convert the given mass (50.0 g) to moles: \(\frac{50.0}{53.49} = 0.935\) mol.
• Thus, the concentration in 1 L of solution is 0.935 M.

This helps us calculate the ratio of base to acid needed for pH determination.

Ammonia (NH3)

Ammonia (\(\text{NH}_3\)) functions as the base in our buffer solution. It is often used due to its ability to bind protons, thereby acting as a weak base.

In our exercise, ammonia is provided as part of a 0.75 M solution.

• This base concentration is used in the Henderson-Hasselbalch equation.
• The balance between NH3 and \(\text{NH}_4^+\) dictates the solution's pH.

Understanding the role of ammonia helps visualize how buffer solutions moderate changes in pH.

Acid Dissociation Constant (Ka)

The acid dissociation constant, or \(K_\mathrm{a}\), is a measure of the strength of a weak acid. It tells us how well an acid can donate protons to a base.

For our buffer solution:

• The \(K_\mathrm{a}\) for \(\text{NH}_4^+\) is \(5.56 \times 10^{-10}\).
• The \(pK_\mathrm{a}\), the negative log of \(K_\mathrm{a}\), is 9.25.

This value is used in the Henderson-Hasselbalch equation to help calculate the pH. The pKa value allows us to understand the buffering capacity and effectiveness of the solution.