61. Suppose 100 mL of milk is converted into curd by *Lactobacillus*, producing lactic acid ($C_3H_6O_3$). If the final pH of the curd is 4.5, what is the approximate molar concentration of lactic acid? (Given: pKa of lactic acid = 3.86)
Key Concept: Bacterial Fermentation, Lactic Acid Production
c) 0.0046 M
[Solution Description]
pH can be calculated using the Henderson-Hasselbalch equation:
$pH = pK_a + \log \left( \frac{[A^-]}{[HA]} \right)$
At pH 4.5, assuming nearly all lactic acid is dissociated, $[A^-] \approx [HA]$.
Rearranging: $4.5 = 3.86 + \log \left( \frac{[A^-]}{[HA]} \right)$.
Solving gives $\frac{[A^-]}{[HA]} = 10^{(4.5-3.86)} = 10^{0.64} \approx 4.37$.
Since total acid concentration is mostly $[HA] + [A^-]$, let $[HA] = x$, then $[A^-] = 4.37x$.
Total concentration = $x + 4.37x = 5.37x$.
However, because the system is buffered near pKa, we estimate $[HA] \approx [A^-]$, giving $[HA] \approx 10^{-4.5}$ M (from $pH = -\log[H^+]$).
But better approximation: For weak acids, $[H^+] = \sqrt{K_a \cdot C}$.
Given $[H^+] = 10^{-4.5}$ and $K_a = 10^{-3.86}$, solving for $C$ gives $C \approx 0.0046$ M.
Your Answer is correct.
c) 0.0046 M
[Solution Description]
pH can be calculated using the Henderson-Hasselbalch equation:
$pH = pK_a + \log \left( \frac{[A^-]}{[HA]} \right)$
At pH 4.5, assuming nearly all lactic acid is dissociated, $[A^-] \approx [HA]$.
Rearranging: $4.5 = 3.86 + \log \left( \frac{[A^-]}{[HA]} \right)$.
Solving gives $\frac{[A^-]}{[HA]} = 10^{(4.5-3.86)} = 10^{0.64} \approx 4.37$.
Since total acid concentration is mostly $[HA] + [A^-]$, let $[HA] = x$, then $[A^-] = 4.37x$.
Total concentration = $x + 4.37x = 5.37x$.
However, because the system is buffered near pKa, we estimate $[HA] \approx [A^-]$, giving $[HA] \approx 10^{-4.5}$ M (from $pH = -\log[H^+]$).
But better approximation: For weak acids, $[H^+] = \sqrt{K_a \cdot C}$.
Given $[H^+] = 10^{-4.5}$ and $K_a = 10^{-3.86}$, solving for $C$ gives $C \approx 0.0046$ M.