$$ \begin{array}{|c|c|} \hline \text{Substance} & K_a \\ \hline \ce{HC3H5O3(aq)} & 8.3 \times 10^{-4} \\ \hline \ce{CH3NH3+(aq)} & 2.3 \times 10^{-11} \\ \hline \end{array} $$

The acid-dissociation constants of \(\ce{HC3H5O3(aq)}\) and \(\ce{CH3NH3+(aq)} \) are given in the table above. Which of the following mixtures is a buffer with a pH of approximately 3 ?

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We are looking for a solution that is slightly acidic. Lets go through all the answer choices:

When \(\ce{CH3NH3Cl}\) is mixed with \(\ce{NaOH}\), both molecules will dissociate into ions. The solution will become alkaline due to the dissociation of \(\ce{NaOH}\).

$$ \ce{CH3NH3Cl(aq) -> CH3NH3+(aq) + Cl-(aq)} $$ $$ \ce{NaOH(aq)-> Na+(aq) + OH-(aq)} $$

When \(\ce{HC3H5O3}\) is mixed with \(\ce{NaOH}\), the hydronium ions from the dissociation of \(\ce{HC3H5O3}\) will react with the hydroxide ions from the dissociation of \(\ce{NaOH}\). There is an excess of \(\ce{HC3H5O3}\), so the solution will remain acidic.

$$ \ce{HC3H5O3(aq) <--> H+(aq) + C3H5O3-(aq)} $$ $$ \ce{NaOH(aq)-> Na+(aq) + OH-(aq)} $$ $$ \ce{H+(aq) + OH-(aq) -> H2O} $$

When \(\ce{NaC3H5O3}\) is mixed with \(\ce{NaOH}\), the solution will be alkaline due to the presence of excess hydroxide ions. \(\ce{C3H5O3-(aq)}\) also acts as a base.

$$ \ce{NaC3H5O3(aq) -> Na+(aq) + C3H5O3-(aq)} $$ $$ \ce{H+(aq) + C3H5O3-(aq) <--> HC3H5O3(aq)}$$ $$ \ce{NaOH(aq)-> Na+(aq) + OH-(aq)} $$

When \(\ce{CH3NH3Cl}\) is mixed with \(\ce{CH3NH2}\), the resulting solution will be slightly basic.

$$ \ce{CH3NH3Cl(aq) -> CH3NH3+(aq) + Cl-(aq)} $$ $$ \ce{CH3NH3+(aq) <--> H+(aq) + CH3NH2(aq)} $$

The pH can be prediced with an ICE table calculation:

$$ K_a = \frac{[\ce{H+}][\ce{CH3NH2}]}{[\ce{CH3NH3+}]} $$ $$ 2.3 \times 10^{-11} \approx \frac{(0.1)[\ce{H+}]}{(0.1)} $$ $$ \ce{H+} = 2.3 \times 10^{-11} $$ $$\text{pH} = -\log[\ce{H+}] = -\log(2.3 \times 10^{-11}) \approx 10-11 $$

Note that these calculations can be avoided if one recognizes that the \(K_a\) value is very small. \(\ce{CH3NH2}\) is therefore a stronger base than \(\ce{CH3NH3+}\) is an acid, and the resulting mixture would have a pH > 7.