Calculate the \(\mathrm{pH}\) at each stage in the titration for the addition of \(0.150 \mathrm{M} \mathrm{HCl}(\mathrm{aq})\) to \(25.0 \mathrm{~mL}\) of \(0.110 \mathrm{M} \mathrm{NaOH}(\mathrm{aq})\) : (a) initially; (b) after the addition of \(5.0 \mathrm{~mL}\) of acid; (c) after the addition of a further \(5.0 \mathrm{~mL}\); (d) at the stoichiometric point; (e) after the addition of \(5.0 \mathrm{~mL}\) of acid beyond the stoichiometric point; (f) after the addition of \(10.0 \mathrm{~mL}\) of acid beyond the stoichiometric point.

Titration is a technique used to determine the concentration of a solution by reacting it with a solution of known concentration. Specifically, the titration of a strong acid with a strong base is a common procedure in chemistry used to find the concentration of one of the reactants. A strong acid, like hydrochloric acid (HCl), fully dissociates in water, as does a strong base like sodium hydroxide (NaOH). During titration, the strong acid and base react in a one-to-one molar ratio in a neutralization reaction, ultimately producing water and a salt (NaCl in this case).

In our exercise, we are adding a strong acid (HCl) to a strong base (NaOH), and the goal is to calculate the pH at different stages of the titration. The point at which the amount of acid added exactly neutralizes the amount of base present is known as the stoichiometric point or equivalence point. Here, the amount of H+ ions from the acid equals the amount of OH- ions from the base, resulting in a neutral solution typically with a pH of 7. However, if we continue to add acid beyond this point, the pH will drop, indicating the formation of an acidic solution.

The stoichiometry of neutralization reactions involves understanding the molar relationships between the acid and base during a titration. The neutralization reaction between a strong acid and a strong base is guided by the equation: \( \text{H}^+ + \text{OH}^- \rightarrow \text{H}_2\text{O} \).

Since this reaction has a 1:1 molar ratio between the hydrogen ions from the acid and the hydroxide ions from the base, we can use stoichiometry to predict the outcome of a titration at any point. For instance, initially in our exercise, we determine the moles of NaOH present and then calculate how the addition of known volumes of HCl affects the mole balance. The stoichiometric point is reached when the moles of HCl added are equal to the moles of NaOH initially present, leading to a neutral pH of 7, assuming no other reactions occur.

pH is a measure of the acidity or basicity of an aqueous solution. The pH scale ranges from 0 to 14, with 7 being neutral. Values less than 7 indicate acidity, and values greater than 7 indicate basicity. The pH can be calculated using the formula: \( \text{pH} = -\log[\text{H}^+] \).

For bases, we calculate the pOH first and then use the relationship \( \text{pH} + \text{pOH} = 14 \) to find the pH. In our exercise, to calculate the initial pH, we start with the concentration of the base (NaOH), determine the pOH, and then subtract that value from 14 to get the pH. As we add acid (HCl), the concentration of H+ ions changes, and thus we need to recalculate the pH at each step. After the stoichiometric point, when we have excess HCl, the pH is calculated directly from the concentration of the excess H+.

The relationship between molarity (concentration) and volume is fundamental in titrations. Molarity is defined as the number of moles of solute per liter of solution (\( \text{M} = \frac{\text{moles of solute}}{\text{liters of solution}} \)). This formula can be rearranged to calculate moles (\( \text{moles} = \text{M} \times \text{volume} \)).

During a titration, this molarity-volume relationship allows us to calculate the number of moles of acid or base in a given volume of solution. For example, in the provided exercise, we use the molarity of HCl and the volume added at each stage to find the moles of HCl involved in the reaction. This calculation is crucial for determining the equivalence point during titration and helps us correctly adjust the pH calculations as volumes of acid or base are added to the reaction mixture.