Question

A certain acetic acid solution has \(\mathrm{pH}=2.68\). Calculate the volume of \(0.0975 M \mathrm{KOH}\) required to reach the equivalence point in the titration of \(25.0 \mathrm{~mL}\) of the acetic acid solution.

Short answer

To find the volume of 0.0975 M KOH required for titration, first, calculate the concentration of acetic acid with a given pH of 2.68: \[[\text{H}^+] = 10^{-2.68}\] Then, find the moles of acetic acid in the 25.0 mL solution: \[\text{moles of CH}_3\text{COOH} = (10^{-2.68})(0.025 \text{L})\] Since one mole of KOH reacts with one mole of CH\(_3\)COOH, the moles of KOH required for equivalence point are equal to the moles of CH\(_3\)COOH. Finally, calculate the volume of KOH needed: \[\text{volume of KOH} = \frac{\text{moles of CH}_3\text{COOH}}{0.0975 M}\]

Step-by-step solution

Calculate the concentration of acetic acid using pH

The formula to find the concentration of acetic acid using pH is: \[\text{pH} = -\log [\text{H}^+]\] We have a pH of 2.68, so we can solve for the concentration of hydrogen ions: \[[\text{H}^+] = 10^{-\text{pH}}\] \[[\text{H}^+] = 10^{-2.68}\]

Find the initial concentration of acetic acid

Since acetic acid is a weak acid, it does not ionize completely in water. However, in this case, we'll assume the initial concentration of \(\mathrm{H}^+\) ions is equal to the initial concentration of acetic acid \([\text{CH}_3\text{COOH}]\), as titration with a strong base would force the reaction to completion. Therefore, we have: \[[\text{CH}_3\text{COOH}] = [\text{H}^+]\] \[[\text{CH}_3\text{COOH}] = 10^{-2.68}\]

Using stoichiometry to find moles of acetic acid

Now that we have found the concentration of acetic acid in the solution, we can find the moles of acetic acid in the 25.0 mL solution: \[\text{moles of CH}_3\text{COOH} = (\text{concentration})\times(\text{volume})\] \[\text{moles of CH}_3\text{COOH} = (10^{-2.68})(0.025 \text{L})\]

Find moles of KOH required for equivalence point

At the equivalence point, one mole of \(\mathrm{KOH}\) will react with one mole of \(\mathrm{CH}_3\text{COOH}\). Therefore, the moles of \(\mathrm{KOH}\) required for the equivalence point will be equal to the moles of \(\mathrm{CH}_3\text{COOH}\): \[\text{moles of KOH} = \text{moles of CH}_3\text{COOH}\]

Calculate the volume of KOH needed for titration

We know the concentration of the \(\mathrm{KOH}\) solution (0.0975 M) and the moles of \(\mathrm{KOH}\) required for the equivalence point. We can now find the volume of \(\mathrm{KOH}\) needed: \[\text{volume of KOH} = \frac{\text{moles of KOH}}{\text{concentration of KOH}}\] \[\text{volume of KOH} = \frac{\text{moles of CH}_3\text{COOH}}{0.0975 M}\] Now, use the moles of CH\(_3\)COOH calculated in Step 3 to find the volume of KOH required for the titration.

Key concepts

pH calculation

Understanding the concept of pH is crucial in acid-base titrations. The pH measures the acidity or basicity of a solution.
The formula used to calculate pH is \( ext{pH} = -\log [\text{H}^+]\). This equation links the concentration of hydrogen ions \( [\text{H}^+] \) in a solution to its pH.
When you have a pH value, you can find the concentration of hydrogen ions using the inverse: \[ [\text{H}^+] = 10^{-\text{pH}} \]
For example, with a pH of 2.68, the calculation of hydrogen ion concentration becomes \[ [\text{H}^+] = 10^{-2.68} \]
Thus, finding the pH allows us to understand the strength of the acid in the solution without directly measuring ions.

equivalence point

The equivalence point in titration is a key concept, marking the stage where the amount of titrant added completely neutralizes the analyte solution.
This is where the number of moles of the acid equals the number of moles of the base. In the case of acetic acid titration with KOH, it's when every acetic acid molecule has reacted.
Reaching the equivalence point means the solution has been neutralized, often identified using indicators or pH meters.
During the titration, precise measurement ensures you know exactly when the equivalence point is reached, which is critical for calculating concentrations and volumes in the reaction.

molarity

Molarity is a method of expressing concentration, defined as the number of moles of solute per liter of solution (mol/L).
This helps in understanding how many moles of a substance are present in a given volume of solution.
In our example, we're dealing with a KOH solution with a molarity of 0.0975 M. This tells us there are 0.0975 moles of KOH in every liter of the solution.
Knowing molarity allows us to calculate how much of the titrant is needed to reach the equivalence point by relating moles to volume.

stoichiometry

Stoichiometry provides a way to calculate the proportions of reactants and products in a chemical reaction.
It uses balanced chemical equations to determine the relationship between moles of reactants and products.
In the titration of acetic acid with KOH, the stoichiometry tells us that one mole of KOH will react with one mole of \(\text{CH}_3\text{COOH}\).
This one-to-one ratio makes it straightforward to calculate the volume of KOH needed when the moles of acetic acid are known.
Using stoichiometry ensures that we have the precise amount of reactants necessary for the reaction to be complete at the equivalence point.