Consider separate aqueous solutions of \(\mathrm{HCl}\) and \(\mathrm{H}_{2} \mathrm{SO}_{4}\) with the same molar concentrations. You wish to neutralize an aqueous solution of \(\mathrm{NaOH}\). For which acid solution would you need to add more volume (in milliliters) to neutralize the base? a. the \(\mathrm{HCl}\) solution b. the \(\mathrm{H}_{2} \mathrm{SO}_{4}\) solution c. You need to know the acid concentrations to answer this question. d. You need to know the volume and concentration of the \(\mathrm{NaOH}\) solution to answer this question. e. \(c\) and \(\underline{d}\) Explain.

We first need to write down the balanced chemical reactions for the neutralization of the sodium hydroxide solution by both \(\mathrm{HCl}\) and \(\mathrm{H}_{2}\mathrm{SO}_{4}\). These are: a. \(\mathrm{HCl} + \mathrm{NaOH} \rightarrow \mathrm{NaCl} + \mathrm{H}_{2}\mathrm{O}\) b. \(\mathrm{H}_{2}\mathrm{SO}_{4} + 2\mathrm{NaOH} \rightarrow \mathrm{Na}_{2}\mathrm{SO}_{4} + 2\mathrm{H}_{2}\mathrm{O}\)

Next, we identify the stoichiometric coefficients of the reactants in the balanced equations. We will consider the coefficients of \(\mathrm{HCl}\) and \(\mathrm{H}_{2}\mathrm{SO}_{4}\) in their respective reactions: a. In the \(\mathrm{HCl}\) reaction, there is a 1:1 ratio between \(\mathrm{HCl}\) and \(\mathrm{NaOH}\). It means one mole of \(\mathrm{HCl}\) reacts with one mole of \(\mathrm{NaOH}\). b. In the \(\mathrm{H}_{2}\mathrm{SO}_{4}\) reaction, there is a 1:2 ratio between \(\mathrm{H}_{2}\mathrm{SO}_{4}\) and \(\mathrm{NaOH}\). It means one mole of \(\mathrm{H}_{2}\mathrm{SO}_{4}\) reacts with two moles of \(\mathrm{NaOH}\).

Since the molar concentrations of \(\mathrm{HCl}\) and \(\mathrm{H}_{2}\mathrm{SO}_{4}\) solutions are the same, we can use the stoichiometric coefficients to compare the volumes required to neutralize the base: a. In the case of \(\mathrm{HCl}\), one mole of acid is needed to neutralize one mole of base. b. In the case of \(\mathrm{H}_{2}\mathrm{SO}_{4}\), one mole of acid is needed to neutralize two moles of base (twice as much base). Using this information, we can conclude that we would need to add more volume of the \(\mathrm{HCl}\) solution to neutralize the base, as it reacts with the base in a 1:1 ratio. On the other hand, since \(\mathrm{H}_{2}\mathrm{SO}_{4}\) reacts with twice as much base as \(\mathrm{HCl}\) in a 1:2 ratio, less volume of \(\mathrm{H}_{2}\mathrm{SO}_{4}\) solution is required to neutralize the same amount of base. So, the correct answer is a. the \(\mathrm{HCl}\) solution.