Cranberries are rich in benzoic acid. Assuming that the only source for hydronium-ions in cranberries is benzoic acid, find its concentration in canned cranberry juice, which has an average pH of 2.41. (Ka of benzoic acid is 6.46 x 10-5
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![**Topic: Benzoic Acid Concentration in Cranberry Juice**
Cranberries are rich in benzoic acid. Assuming that the only source for hydronium ions in cranberries is benzoic acid, determine its concentration in canned cranberry juice, which has an average pH of 2.41.
The \( K_a \) of benzoic acid is \( 6.46 \times 10^{-5} \).
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**Explanation:**
To find the concentration of benzoic acid in cranberry juice based on its pH, follow these steps:
1. **Calculate Hydronium Ion Concentration:**
- The pH of a solution is related to the concentration of hydronium ions (\( [H_3O^+] \)) by the formula:
\[
[H_3O^+] = 10^{-\text{pH}}
\]
2. **Use the Acid Dissociation Constant (\( K_a \)):**
- Given the equation for benzoic acid dissociation:
\[
C_6H_5COOH \leftrightarrow C_6H_5COO^- + H^+
\]
- The \( K_a \) expression is:
\[
K_a = \frac{[C_6H_5COO^-][H^+]}{[C_6H_5COOH]}
\]
3. **Assumptions:**
- Assume initial concentration of benzoic acid is \( [C_6H_5COOH]_0 \).
- At equilibrium, \( [C_6H_5COO^-] = [H^+] = x \) and \( [C_6H_5COOH] = [C_6H_5COOH]_0 - x \).
4. **Solve for Initial Concentration:**
- Rearrange the \( K_a \) expression and solve for \( [C_6H_5COOH]_0 \):
\[
6.46 \times 10^{-5} = \frac{x^2}{[C_6H_5COOH]_0 - x}
\]
5. **Conclusion:**
- Input the values into the equations to find the concentration of benzoic acid. This connection helps in understanding the acidic nature of cranberry juice and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdff4fe-7c73-4882-a39a-fd8ef8420048%2F1bb7f2ac-0e30-49d3-96ec-f76811eb5e67%2Fdxub7j8_processed.jpeg&w=3840&q=75)
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