Problem 103 - Chemistry Matter and Change

Question

An aqueous solution buffered by benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5}\right.\) \(\mathrm{COOH}\) ) and sodium benzoate \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOHNa}\right)\) is 0.0500 \(\mathrm{M}\) in both compounds. Given that benzoic acid's \(K_{\mathrm{a}}\) equals \(6.4 \times 10^{-5},\) what is the pH of the solution?

Step-by-Step Solution

Answer

The pH of the buffered solution is 4.19, as calculated using the Henderson-Hasselbalch equation. Given the concentrations of benzoic acid and sodium benzoate and the value of Ka, we were able to calculate the pH.

Step 1: Write down the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is: \(pH = pK_a + log\frac{[A^-]}{[HA]}\), where pH is the pH of the solution, pKa is the negative logarithm of Ka (the acid dissociation constant), [A⁻] is the concentration of the conjugate base, and [HA] is the concentration of the weak acid.

Step 2: Calculate the pKa from the given Ka

We are given that the Ka of benzoic acid is \(6.4 \times 10^{-5}\). To calculate the pKa, we take the negative logarithm of Ka: \(pK_a = -log(K_a) = -log(6.4 \times 10^{-5})\) Using a calculator, we find: \(pK_a \approx 4.19\)

Step 3: Calculate the pH using the Henderson-Hasselbalch equation

We are given that the concentrations of benzoic acid (HA) and sodium benzoate (A⁻) are both 0.0500 M. We can now use the Henderson-Hasselbalch equation to calculate the pH: \(pH = pK_a + log\frac{[A^-]}{[HA]} = 4.19 + log\frac{0.0500}{0.0500}\) Since the concentrations of HA and A⁻ are equal, the log term is equal to 0: \(pH = 4.19 + 0\) Therefore, the pH of the solution is: \(pH = 4.19\)

Key Concepts

Henderson-Hasselbalch equation

To understand how buffers work, the Henderson-Hasselbalch equation is crucial. This equation relates the pH of a solution to the properties of its buffer components. It is given by:\[ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) \] In this equation, - \( pH \) is what we're solving for—it tells us how acidic or basic a solution is. - \( pK_a \) is the negative log of the acid dissociation constant (\( K_a \)), providing a measure of the strength of the acid. - \([A^-] \) represents the concentration of the conjugate base, and \([HA] \) is the concentration of the acid itself.The beauty of the Henderson-Hasselbalch equation is that it simplifies the process of finding pH in a buffered solution. It highlights the balance between the acid and its conjugate base, showing how this balance influences the pH. By using this equation in the given problem, we can determine that the ratio between the conjugate base and the acid has a direct effect on the solution's pH. When these concentrations are equal, the \( \log \left(\frac{[A^-]}{[HA]}\right)\) becomes zero, which simplifies the evaluation to just \( pH = pK_a \). This simplification is a hallmark feature of buffer solutions.

acid dissociation constant (Ka)

The acid dissociation constant, often symbolized as \( K_a \), is vital for understanding the strength of an acid in water. This constant expresses how well an acid dissociates—which means breaks apart—in solution, forming hydrogen ions and its conjugate base.When an acid like benzoic acid is placed in water, it partially dissociates according to the equation:\[ HA \rightleftharpoons H^+ + A^- \] Here, \( HA \) is the acid, \( H^+ \) is the hydrogen ion, and \( A^- \) is the conjugate base. The \( K_a \) value gives us a numerical indication of this process. If \( K_a \) is large, the acid dissociates well, and thus, it is a strong acid. Conversely, a smaller \( K_a \) indicates a weaker acid.For benzoic acid, the \( K_a \) is \( 6.4 \times 10^{-5} \), which is relatively small, indicating it's not a strong acid. This makes it suitable for buffering because it doesn't fully dissociate. The \( pK_a \) value, which is derived by taking the negative logarithm of \( K_a \), provides a practical way to compare acid strengths and is directly used in the Henderson-Hasselbalch equation to find pH.

buffer solutions

Buffer solutions are special mixtures that resist changes in pH when small amounts of acid or base are added. They achieve this remarkable property through the presence of components like a weak acid and its conjugate base. In the given problem, benzoic acid acts as the weak acid, while sodium benzoate acts as its conjugate base.The primary roles of buffer solutions include:- Maintaining stable pH levels in many chemical processes or biological environments. - Providing control over pH conditions in experiments and industrial processes.When an acid (H\(^+\)) is added to a buffer, the conjugate base present in the buffer solution reacts with the added hydrogen ions, minimizing the pH change. Conversely, if a base (OH\(^-\)) is added, the weak acid component donates hydrogen ions to neutralize the base. The reaction can be understood as:- Adding acid: \( A^- + H^+ \rightarrow HA \) - Adding base: \( HA + OH^- \rightarrow A^- + H_2O \)Thus, buffers are crucial in systems where maintaining a specific and stable pH is necessary, as they can absorb or release hydrogen ions without significant changes in the overall pH of the solution.