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(C) To determine the lowest $pH$, we analyze the nature of each mixture:$(A)$ $10 \ mL$ $0.05 \ N$ $CH_{3}COOH$ $(0.5 \ mmol)$ + $5 \ mL$ $0.1 \ N$ $N

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$5 \ mL$ $0.1 \ N$ $CH_{3}COOH + 10 \ mL$ $0.05 \ N$ $CH_{3}COONa$

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(C) To determine the lowest $pH$, we analyze the nature of each mixture:$(A)$ $10 \ mL$ $0.05 \ N$ $CH_{3}COOH$ $(0.5 \ mmol)$ + $5 \ mL$ $0.1 \ N$ $NH_{4}OH$ $(0.5 \ mmol)$ forms a salt of weak acid and weak base $(CH_{3}COONH_{4})$, which is nearly neutral.$(B)$ $5 \ mL$ $0.2 \ N$ $NH_{4}Cl$ + $5 \ mL$ $0.2 \ N$ $NH_{4}OH$ forms a basic buffer $(pH > 7)$.$(C)$ $5 \ mL$ $0.1 \ N$ $CH_{3}COOH$ $(0.5 \ mmol)$ + $10 \ mL$ $0.05 \ N$ $CH_{3}COONa$ $(0.5 \ mmol)$ forms an acidic buffer where $pH = pK_{a} + \log(\frac{[Salt]}{[Acid]})$. Since $[Salt] = [Acid]$, $pH = pK_{a} \approx 4.76$.$(D)$ $5 \ mL$ $0.1 \ N$ $CH_{3}COOH$ + $5 \ mL$ $0.1 \ N$ $NaOH$ forms $CH_{3}COONa$, which undergoes anionic hydrolysis, resulting in a basic solution $(pH > 7)$.Comparing these, the acidic buffer in option $(C)$ has the lowest $pH$.

Which of the following mixtures will have the lowest $pH$ at $298 \ K$?

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