Problem 45 - Chemistry: The Central Science in SI Units, Global Edition

Question

A 10.0-mL sample of \(0.250 \mathrm{M}\) acetic acid \(\left(\mathrm{CH}_{3} \mathrm{COOH}\right)\) is titrated with \(0.100 \mathrm{M}\) KOH solution. Calculate the pH after the following volumes of base have been added: (a) \(0 \mathrm{~mL},\) (b) \(12.5 \mathrm{~mL}\) (c) \(24.5 \mathrm{~mL}\) (d) \(25.0 \mathrm{~mL}\) (e) \(25.5 \mathrm{~mL}\) (f) \(30.0 \mathrm{~mL}\).

Step-by-Step Solution

Answer

The pH values after adding the specified volumes of KOH are as follows: (a) 0 mL: pH = 2.87 (b) 12.5 mL: pH = 4.73 (c) 24.5 mL: pH = 4.77 (d) 25.0 mL: pH = 5.71 (e) 25.5 mL: pH = 6.74 (f) 30.0 mL: pH = 9.77

Step 1: Write down the relevant reaction for titration.

In this titration, we will react acetic acid with KOH. The reaction that occurs is: \[\mathrm{CH}_{3}\mathrm{COOH} + \mathrm{KOH} \longrightarrow \mathrm{CH}_{3}\mathrm{COOK} + \mathrm{H}_{2}\mathrm{O}\]

Step 2: Calculate the initial moles of acetic acid and KOH.

Initial moles of acetic acid can be calculated using volume and concentration: Moles of acetic acid = Volume × Concentration Moles of acetic acid = 0.010 L × 0.250 mol/L = 0.0025 mol For KOH, the number of moles will be calculated separately for each specified volume in the problem.

Step 3: Calculate moles and concentrations for each specified volume of KOH.

For each specified volume of KOH, we will calculate the number of moles, and then find the resulting concentrations of species involved in the reaction. This will help us determine the dominant species present and the appropriate method to calculate pH.

Step 4: Calculate the pH for each specified volume of KOH.

For each specified volume, we will be using the appropriate method such as calculating pH directly, using the equilibrium constant, or considering the excess of KOH.

Step 5: Answer the problem with calculated pH values.

After following the above steps for each specified volume of KOH, we will have a pH value corresponding to each volume. We will present those pH values as a final answer.

Key Concepts

Acetic Acid

Acetic acid is a weak acid that is commonly found in vinegar, which gives it its sour taste and pungent smell. In chemical terms, its formula is \( ext{CH}_3 ext{COOH}\). Acetic acid partially ionizes in water, which means it doesn’t completely split into ions.
This characteristic affects how it reacts in acid-base reactions, making it crucial when performing titration calculations.

When acetic acid is involved in a titration with a strong base like potassium hydroxide (KOH), it reacts in a one-to-one ratio. This makes it easier to calculate the changes in moles during the reaction.

• Acetic acid's properties, including its limited ionization, are important for understanding how its pH changes during titration.
• Recognizing that acetic acid is a weak acid helps in choosing the right methods for pH calculation.

pH Calculation

Calculating the pH during a titration involves determining the concentration of ions in the solution. pH is a measure of how acidic or basic a solution is, expressed on a scale from 0 to 14, with 7 being neutral.
A lower pH means the solution is acidic, while a higher pH signifies a basic solution. In a titration context, pH changes as the titration progresses.

To calculate pH, especially for solutions with a weak acid like acetic acid, you will often look at:

• The initial concentration and volume of the acid.
• The added volume of the base, which tells you how many moles of acid or base are reacting.

For a weak acid titrated with a strong base, the methods differ depending on how far the titration has progressed.
This may involve calculating directly using the concentration of the hydrogen ions or using equilibrium constants to determine the concentrations of different species in the solution.

Acid-Base Reaction

An acid-base reaction involves the transfer of a proton from the acid to the base. In this particular titration, acetic acid donates a proton to KOH, forming water and acetate ions. This is a classic example of a neutralization reaction, where an acid and a base react to form a salt and water.

Neutralization reactions are key in titrations because they allow us to determine the concentration of an unknown acid or base by reacting it with a known concentration of its counterpart.

• The reaction between acetic acid and KOH can be simplified to: \[\text{CH}_3\text{COOH} + \text{KOH} \rightarrow \text{CH}_3\text{COOK} + \text{H}_2\text{O}\]
• Knowing the stoichiometry helps calculate changes in pH as more base is added.

Understanding the stoichiometry of this reaction is crucial, as it shows the ratio and flow of reactants to products, helping in predicting pH changes during the addition of the base.

Equilibrium Constant

Acetic acid only partially dissociates in water, meaning it establishes an equilibrium between un-ionized acetic acid molecules and the ions formed. The extent of this dissociation is expressed by the equilibrium constant, specifically the acid dissociation constant \(K_a\). For acetic acid, this constant is relatively small, indicating limited ionization compared to strong acids.

The equilibrium constant helps in calculating the pH of acetic acid solutions, especially during titration. For weak acids, the formula involves \(K_a\) and provides insights into how much of the acid remains undissociated in solution.

• The value of \(K_a\) is used in the equation to solve for the concentration of hydrogen ions: \[K_a = \frac{[\text{H}^+][\text{CH}_3\text{COO}^-]}{[\text{CH}_3\text{COOH}]}\]
• Using \(K_a\) allows accurate predictions of the pH and the extent of reaction completion for weak acids being titrated.

Understanding the equilibrium constant aids in grasping why the pH doesn’t change drastically at the start, but shifts significantly near the equivalence point where more of the acid is neutralized.