What volume of \(0.123 \mathrm{M} \mathrm{NaOH}(\mathrm{aq})\) must be added to \(125 \mathrm{~mL}\) of \(0.197 \mathrm{M} \mathrm{H}_{2} \mathrm{SO}_{3}(\mathrm{aq})\) to reach (a) the first stoichiometric point; (b) the second stoichiometric point?

Understanding chemical equations is crucial when studying chemical reactions. A chemical equation is a symbolic representation of a chemical reaction; it shows the reactants that interact and the products formed, along with their molecular quantities. For instance, let's consider the neutralization reaction where sodium hydroxide (NaOH) reacts with sulfurous acid (H2SO3). The balanced chemical equation for the first neutralization step is:
\[ \text{H}_{2}\text{SO}_{3}(\text{aq}) + \text{NaOH}(\text{aq}) \rightarrow \text{NaHSO}_{3}(\text{aq}) + \text{H}_{2}\text{O}(\text{l}) \].

This represents the first stoichiometric point of the reaction. Each molecule of sulfurous acid reacts with a molecule of sodium hydroxide to produce one molecule of sodium bisulfite and water. It's essential to balance the equation so the number of atoms for each element is the same on both sides, following the conservation of mass.

Neutralization reactions are specific chemical processes where an acid and a base react to form water and a salt. They are a sub-set of acid-base reactions and often involve the transfer of a proton from the acid to the base. In essence, these reactions are stoichiometric, meaning that the moles of acid and base react in definite proportions to each other. For the exercise in question, we have a two-step neutralization process. The first step is:
\[ \text{H}_{2}\text{SO}_{3}(\text{aq}) + \text{NaOH}(\text{aq}) \rightarrow \text{NaHSO}_{3}(\text{aq}) + \text{H}_{2}\text{O}(\text{l}) \],
followed by the second step:
\[ \text{NaHSO}_{3}(\text{aq}) + \text{NaOH}(\text{aq}) \rightarrow \text{Na}_{2}\text{SO}_{3}(\text{aq}) + \text{H}_{2}\text{O}(\text{l}) \].

Each step corresponds to a stoichiometric point – the first when all H2SO3 is consumed, and the second when all resulting NaHSO3 reacts.

Molarity, represented as M, is a term used to describe the concentration of a solution. It is defined as the number of moles of solute (the substance being dissolved) per liter of solution. Thus, molarity expresses the strength of the solution in terms of how much solute is present in a given volume of solvent. To calculate molarity:
\[ \text{Molarity (M)} = \frac{\text{moles of solute}}{\text{volume of solution in liters}} \].

In the provided exercise, the molarity of NaOH informs us how many moles of NaOH are in one liter of solution. This information is essential when trying to determine how much NaOH solution is needed to reach the first and second stoichiometric points of the neutralization reaction. By combining the concept of molarity with the balanced chemical equations for each step, students can calculate the exact volume of NaOH solution required to neutralize a given quantity of H2SO3.