Problem 10 - Chemistry and Chemical Reactivity

Question

What mass of ammonium chloride, \(\mathrm{NH}_{4} \mathrm{Cl}\), must be added to exactly \(5.00 \times 10^{2} \mathrm{mL}\) of \(0.10 \mathrm{M} \mathrm{NH}_{3}\) solution to give a solution with a pH of 9.00?

Step-by-Step Solution

Answer

Add 1.50 grams of ammonium chloride to adjust the pH to 9.00.

Step 1: Recognize the buffer system

Identify that we are dealing with a buffer system composed of a weak base, ammonia (\(\mathrm{NH}_3\)), and its conjugate acid, ammonium ion (\(\mathrm{NH}_4^+\)), supplied by ammonium chloride (\(\mathrm{NH}_4\mathrm{Cl}\)). The addition of \(\mathrm{NH}_4\mathrm{Cl}\) will adjust the pH of the solution to the desired level.

Step 2: Use the Henderson-Hasselbalch Equation

For a buffer solution, the pH is given by the Henderson-Hasselbalch equation: \[pH = pK_a + \log \left( \frac{[\mathrm{base}]}{[\mathrm{acid}]} \right)\]Where \(pK_a\) for \(\mathrm{NH}_4^+\) is 9.25. We need to find the concentration of \(\mathrm{NH}_4^+\) that makes the pH 9.00.

Step 3: Solve for acid concentration

Rearrange the Henderson-Hasselbalch equation to solve for the concentration of \(\mathrm{NH}_4^+\):\[pH - pK_a = \log \left( \frac{[\mathrm{NH}_3]}{[\mathrm{NH}_4^+]} \right)\]\[9.00 - 9.25 = \log \left( \frac{0.10}{[\mathrm{NH}_4^+]} \right)\]This gives: \(-0.25 = \log \left( \frac{0.10}{[\mathrm{NH}_4^+]} \right)\).

Step 4: Calculate \([\mathrm{NH}_4^+]\)

Convert the logarithmic equation to an exponential form to find \([\mathrm{NH}_4^+]\):\[10^{-0.25} = \frac{0.10}{[\mathrm{NH}_4^+]}\]Solving this gives: \([\mathrm{NH}_4^+] = \frac{0.10}{10^{-0.25}} = 0.0562 \text{ M}\).

Step 5: Convert concentration to moles

Calculate the moles of \(\mathrm{NH}_4^+\) needed using the volume of the solution:\[\text{Moles of } \mathrm{NH}_4^+ = 0.0562 \text{ M} \times 0.500 \text{ L} = 0.0281 \text{ moles}\].

Step 6: Calculate mass of \(\mathrm{NH}_4\mathrm{Cl}\)

Finally, convert moles of \(\mathrm{NH}_4^+\) into mass using the molar mass of \(\mathrm{NH}_4\mathrm{Cl}\) (53.49 g/mol):\[\text{Mass of } \mathrm{NH}_4\mathrm{Cl} = 0.0281 \text{ moles} \times 53.49 \text{ g/mol} = 1.50 \text{ g}\].

Key Concepts

Henderson-Hasselbalch equation

In the world of chemistry, the Henderson-Hasselbalch equation is crucial for understanding the behavior of buffer solutions. This equation relates the pH of a solution to the pKa of the acid and the concentrations of the acid and its conjugate base.

It is expressed as:\[ pH = pK_a + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right) \]This equation helps determine the pH of a solution when you know the concentration of the weak base and its conjugate acid.

To apply this equation:

• Identify your weak acid and its conjugate base: In this case, ammonium ion \((\mathrm{NH}_4^+)\) is the acid, while ammonia \((\mathrm{NH}_3)\) is the base.
• Use the known pKa value, which is specific to each weak acid. For ammonium ion, \( pK_a \) is 9.25.
• Plug the values into the equation along with your desired pH, then solve for the unknown concentration.

The Henderson-Hasselbalch equation is a powerful tool for designing and understanding buffer systems in biological and chemical systems.

ammonium chloride

Ammonium chloride \((\mathrm{NH}_4\mathrm{Cl})\) is a common salt that plays a vital role in creating buffer solutions. In our example, when you dissolve ammonium chloride in water, it dissociates completely, releasing ammonium ions \((\mathrm{NH}_4^+)\) and chloride ions \((\mathrm{Cl}^-)\).

The ammonium ion is the conjugate acid of ammonia, an important detail in buffer systems. By adding ammonium chloride, you're increasing the concentration of \(\mathrm{NH}_4^+\) in the solution.

This change helps control the pH by reacting with any added acids or bases to minimize drastic pH shifts. For example:

• Adding an acid to the solution: The \(\mathrm{NH}_3\) in the buffer will react with the added hydrogen ions to form more \(\mathrm{NH}_4^+\), minimizing the pH change.
• Adding a base: The \(\mathrm{NH}_4^+\) will provide hydrogen ions (\(\mathrm{H}^+\)) to neutralize the added base, again stabilizing the pH.

Thus, ammonium chloride is integral in managing and maintaining the desired pH level in buffer systems.

pH calculation

Calculating the pH of a solution is an essential skill in chemistry. In buffer solutions, it often involves using the Henderson-Hasselbalch equation. However, understanding the fundamental steps is vital.

In our specific problem, the task is to calculate the mass of ammonium chloride required to achieve a certain pH. Let's break it down:

• First, use the Henderson-Hasselbalch equation to find the target concentration of the conjugate acid \([\mathrm{NH}_4^+]\). This involves subtracting the desired pH from the known \(pK_a\), which gives you the ratio of base to acid in log form.
• Next, convert the logarithmic equation to its exponential form to solve for the unknown concentration.
• With the concentration known, convert it to moles using the volume of the solution. Molarity \((\mathrm{M})\) is defined as moles per liter, so multiply the concentration by the volume of the solution in liters.
• Finally, to find the mass of ammonium chloride needed, multiply the number of moles by the molar mass of \(\mathrm{NH}_4\mathrm{Cl}\) (53.49 g/mol).

The calculated mass gives you the amount of solid ammonium chloride needed to prepare a buffer with the desired pH of 9.00.