Question

$\text{The pH of an aqueous solution containing 1 M benzoic acid (} pK_a = 4.20 \text{) and 1 M sodium benzoate is 4.5.}$
\text{The volume of benzoic acid solution in 300 mL of this buffer solution is \\_\\_\\_\\_\\_\\_ mL.}

Answer 1

Solution: To find the volume of benzoic acid in the buffer solution, we can use the Henderson-Hasselbalch equation:

$\text{pH} = pK_a + \log \left( \frac{[A^-]}{[HA]} \right)$

Where: $[A^-]$ is the concentration of the conjugate base (sodium benzoate) and $[HA]$ is the concentration of the weak acid (benzoic acid).

Given Information: pH = 4.5

$pK_a = 4.20, \quad [A^-] = 1 \, M \text{ (sodium benzoate)}$

Total volume of buffer solution = 300 mL.

Calculate the ratio of base to acid:

$4.5 = 4.20 + \log \left( \frac{[A^-]}{[HA]} \right)$

$0.30 = \log \left( \frac{1}{[HA]} \right)$

$10^{0.30} = \frac{1}{[HA]} \implies [HA] = \frac{1}{10^{0.30}} \approx 0.50 \, M$

Calculate the volume of benzoic acid: Let $V_a$ be the volume of 1M benzoic acid. The concentration in the total 300 mL buffer solution:

$[HA] = \frac{V_a}{200}$

Setting the concentrations equal gives:

$0.50 = \frac{V_a}{200} \implies V_a = 0.50 \times 200 = 100 \, mL$

Total volume of benzoic acid solution: To find the total volume of benzoic acid needed to maintain the desired pH:

$V_a = 100 \, mL \quad \text{(rounded from calculations)}$

Thus, the volume of benzoic acid solution in 300 mL of the buffer solution is: 100 mL