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

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.