A \(25.0 \mathrm{~mL}\) sample of \(0.10 \mathrm{M} \mathrm{HCl}\) is titrated with \(0.10 \mathrm{M} \mathrm{NaOH}\). What is the \(\mathrm{pH}\) of the solution at the points where \(24.9\) and \(25.1 \mathrm{~mL}\) of \(\mathrm{NaOH}\) have been added? (a) \(3.70,10.70\) (b) \(3.30,10.30\) (c) \(3.70,10.30^{\circ}\) (d) \(3.0,11.0\)

Understanding the concept of an equivalence point in titrations is crucial when dealing with acid-base reactions. The equivalence point is the stage during titration where the amount of titrant added is just enough to completely neutralize the analyte solution. In the case of a strong acid-strong base titration, like the titration of hydrochloric acid (HCl) with sodium hydroxide (NaOH), the equivalence point occurs when the moles of acid equal the moles of base.

At this precise point, the pH will not always be neutral (pH 7) because the nature of the acid and base can affect it. However, for strong acid and strong base titrations, the pH at equivalence is expected to be near 7 due to the neutralization forming water and a salt that does not affect the pH. The step-by-step solution methodically finds the pH at points just before and just after the equivalence point, demonstrating the dramatic change in pH that occurs around this critical juncture.

Acid-base titration is a procedure used to determine the concentration of an acid or a base in a solution. It involves the slow addition of one solution of known concentration, the titrant, to a known quantity of another solution of unknown concentration, the analyte, until the reaction reaches the equivalence point. At this stage, indicators or pH meters are commonly used to detect the end of the titration.

For instance, in our original exercise, the titration of a strong acid (HCl) with a strong base (NaOH) results in a complete neutralization reaction, producing water and salt. The pH at various stages of this titration gives insight into the reaction's progress, and the step-by-step solution helps us to understand these changes. Notably, before the equivalence point, the reaction mixture is acidic because of the excess HCl; after the equivalence point, the reaction mixture becomes basic due to the excess NaOH.

The relationship between molarity and volume is fundamental to solving many problems in chemistry, especially titrations. Molarity (M) is defined as the number of moles of solute per liter of solution. This relationship can be expressed as a formula: \( M = \frac{moles}{volume(L)} \).

When performing a titration, knowing the molarity and volume of one reactant allows us to calculate the moles of that reactant. By understanding the stoichiometry of the reaction, we can then relate these moles to moles of the other reactant. In the exercise, the molarity-volume relationship is utilized to calculate the moles of HCl and NaOH present before and after the equivalence point, as seen in steps 1 and 5 of the solution. This allows the determination of the remaining concentration of ions in solution to ultimately calculate the pH.

pH and pOH are measures of the acidity and basicity of a solution, respectively. The pH is a logarithmic scale used to specify the hydrogen ion concentration (\( [H^+] \)) in a solution, while pOH deals with the hydroxide ion concentration (\( [OH^-] \)). Their formulas are given by: \( pH = -\log [H^+] \) and \( pOH = -\log [OH^-] \). The pH and pOH scales are inversely related and sum to 14 at 25°C, which is expressed as \( pH + pOH = 14 \).

In our solution steps, pH was calculated before the equivalence point by finding the concentration of \( H^+ \) ions remaining after partial neutralization (step 3), and pOH was determined after the equivalence point by the excess \( OH^- \) ions (step 6). Lastly, the pH after equivalence was then found using the relationship between pH and pOH. Understanding these calculations is indispensable for accurately determining the acidity or basicity of a solution at any point in a titration.