Problem 8 - General Chemistry: Principles and Modern Applications

Question

What concentration of ammonia, \(\left[\mathrm{NH}_{3}\right],\) should be present in a solution with \(\left[\mathrm{NH}_{4}^{+}\right]=0.732 \mathrm{M}\) to produce a buffer solution with \(\mathrm{pH}=9.12 ?\) For \(\mathrm{NH}_{3}\) \(K_{\mathrm{h}}=1.8 \times 10^{-5}\)

Step-by-Step Solution

Answer

The concentration of ammonia in the solution should be 240.45 M.

Step 1: Understand the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is: \( pH = pK_a + log \left(\frac{[Base]}{[Acid]}\right)\) Here, we'll modify this a bit for the given exercise where \(K_h = [H_3O^+][NH_3]/[NH_4^+]\). Taking negative logarithm on both sides we get \(pH = pK_a + log \left(\frac{[NH_3]}{[NH_4^+]}\right)\)

Step 2: Calculate the \(pK_a\)

The \(pK_a\) is calculated from the \(K_h\) by taking the negative logarithm of \(K_h\): \(pK_a = -log(K_h)\); given \(K_h = 1.8 \times 10^{-5}\), hence \(pK_a = -log(1.8 \times 10^{-5}) = 4.74\)

Step 3: Solve the Henderson Hasselbalch equation for the concentration of ammonia

Now that we have \(pH\), \(pK_a\) and \([NH_4^+]\), we can solve for \([NH_3]\) in the Henderson-Hasselbalch equation: \(9.12 = 4.74 + log \left(\frac{[NH_3]}{0.732}\right)\); Solving this gives the ammonia concentration: \([NH_3] = 0.732 \times 10^{(9.12 - 4.74)} = 0.732 \times 10^{4.38} = 240.45 \,M \)

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a simple yet powerful tool for understanding the relationship between the pH of a solution and the concentrations of its acid and base components. It's mainly used for buffer solutions, where we aim to maintain a stable pH. The equation can be expressed as: \[ pH = pK_a + \log \left(\frac{[\text{Base}]}{[\text{Acid}]}\right) \]In the context of our problem, ammonia (\(\text{NH}_3\)) serves as the base, and ammonium (\(\text{NH}_4^+\)) is the acid. This equation allows us to connect the pH of the buffer to the concentrations of \([\text{NH}_3]\) and \([\text{NH}_4^+]\). By adjusting the ratio of these components, we can achieve a desired pH.

Ammonia Concentration

Understanding ammonia concentration is crucial, especially when preparing buffer solutions. In our example, we're asked to find \([\text{NH}_3]\) that will give a specific pH. The given concentration of ammonium is 0.732 M. This means the challenge is to calculate the right amount of ammonia to add.To solve this, we rearrange the Henderson-Hasselbalch equation:- We know the target pH is 9.12.- We’ve calculated \(pK_a\) as 4.74 from the given \(K_h\).Using these values, we solve for the ammonia concentration through:\[ [\text{NH}_3] = [\text{NH}_4^+] \times 10^{(pH - pK_a)} \]This results in an ammonia concentration of 240.45 M, which ensures the buffer solution maintains the specified pH of 9.12.

pH Calculation

pH is a metric used to express the acidity or basicity of a solution. For buffer systems, predicting and controlling the pH is essential. The pH scale ranges from 0 to 14, with values below 7 being acidic, and above 7 being basic.For our exercise, the desired pH is a specific value of 9.12, indicating a slightly basic solution. By employing the Henderson-Hasselbalch equation, we combine theoretical calculations with practical needs to achieve this exact pH.
Here's how it works:

• Determine the \(pK_a\) of the acid-base pair. In our case, \(pK_a = 4.74\).
• Use the concentrations of acid (\([\text{NH}_4^+]\)) and base (\([\text{NH}_3]\)) to find their ratio.
• Insert these values into the equation to confirm the pH is properly balanced.

This process highlights the elegance of how seemingly complex chemical properties can be understood and applied with just a few calculations.