Question

A weak monoprotic acid is titrated with \(0.100 \mathrm{M} \mathrm{NaOH}\). It requires \(25.0 \mathrm{~mL}\) of the \(\mathrm{NaOH}\) solution to reach the equivalence point. After \(12.5 \mathrm{~mL}\). of base is added, the pH of the solution is 4.16 . Estimate the \(\mathrm{p} K_{a}\) of the weak acid.

Short answer

The estimated \( \text{p} K_a \) of the weak acid is 4.16.

Step-by-step solution

Understanding the Titration

We have a weak monoprotic acid being titrated with a strong base, NaOH. The equivalence point is reached when 25.0 mL of 0.100 M NaOH is added, meaning 2.5 mmol of NaOH is required to completely neutralize the acid.

Calculating Initial Moles of Acid

Since 25.0 mL of 0.100 M NaOH is needed at the equivalence point, the initial moles of the acid is equal to the moles of NaOH needed, which is 2.5 mmol.

Midpoint of Titration

At the midpoint of the titration, half of the acid has been neutralized. For this problem, this occurs at 12.5 mL of NaOH added, which is exactly half of the equivalence volume (25.0 mL). At this point, \[[ ext{HA}] = [ ext{A}^-]\]meaning the concentrations of the acid and its conjugate base are equal.

Relating pH and pK_a at Midpoint

Using the fact that at the midpoint of a titration of a weak acid with a strong base, \[ ext{pH} = ext{p} K_a\]we can therefore say that \[ ext{p} K_a = 4.16\] in this specific titration.

Key concepts

Weak Acid

A weak acid is a type of acid that only partially dissociates in a solution. This means it does not completely break down into its ions when dissolved in water. The degree of dissociation varies depending on the acid, which is why the strength of weak acids can differ.

• Weak acids have a higher pH compared to strong acids.
• Their dissociation is represented by equilibrium, unlike strong acids which dissociate completely.
• Common examples include acetic acid and citric acid.

To understand weak acids better, one can think of them as acids that take a middle ground—they aren't as aggressive as strong acids, yet they still possess acidic properties. This characteristic significantly influences how they behave during titration, as only a portion of the acid is neutralized at any given moment. This incomplete dissociation is crucial for predicting changes in pH during the titration process.

Equivalence Point

In titration, the equivalence point is the point at which the number of moles of titrant added is equal to the number of moles of the substance present in the solution being titrated. In simpler words, it’s when the reactants have reacted completely with each other.

• It is a critical point in titration as it indicates complete neutralization.
• The pH at the equivalence point does not always equal 7, especially in weak acid titrations, because the resultant solution has properties of a weak base.

For our example, the equivalence point was reached when 25.0 mL of the 0.100 M NaOH solution was added. This implies that 2.5 mmol of the NaOH had completely reacted with the weak acid in the solution.

pKa Estimation

The \( ext{pKa}\) is a crucial value that represents the strength of an acid, specifically, how easily the acid gives up its protons in solution. The smaller the \( ext{pKa}\), the stronger the acid.

• \( ext{pKa}\) is directly related to the acid dissociation constant, Ka, by the formula: \[\text{pKa} = -\log(\text{Ka})\].
• A smaller \( ext{pKa}\) means a stronger acid, while a larger \( ext{pKa}\) means a weaker one.
• During a titration, \( ext{pKa}\) provides insight into the weak acid’s behavior.

In this exercise, estimating \( ext{pKa}\) at the midpoint of a titration provides the solution’s pH, which equals the \( ext{pKa}\) of the acid. Here, the \( ext{pKa}\) was estimated to be 4.16, indicating the weak acid's relative propensity to lose its protons.

Titration Midpoint

The titration midpoint is a significant part of the titration curve where half of the weak acid has been neutralized. At this specific point, the concentration of the acid \([\text{HA}]\) is equal to the concentration of its conjugate base \([\text{A}^-]\).

• At the midpoint, the solution is a buffer system due to the presence of equal concentrations of acid and its conjugate base.
• The pH at the midpoint is equal to the \( ext{pKa}\) of the weak acid, making it a perfect opportunity to determine the \( ext{pKa}\).

In our problem, the midpoint was reached when 12.5 mL of NaOH was added, which is half of the equivalence volume. This means the pH at this stage tells us what the \( ext{pKa}\) of the weak acid is. Here, it showed \( ext{pKa}\) as 4.16, aligning perfectly with the expected value at this stage of a weak acid titration.