Problem 41 - Chemistry: An Atoms-Focused Approach

Question

In the titration of a solution of a weak monoprotic acid with a standard solution of \(\mathrm{NaOH}\), the \(\mathrm{pH}\) halfway to the equivalence point was \(4.44 .\) In the titration of a second solution of the same acid, exactly twice as much of the standard solution of \(\mathrm{NaOH}\) was needed to reach the equivalence point. What was the \(\mathrm{pH}\) halfway to the equivalence point in this titration?

Step-by-Step Solution

Answer

Answer: The pH value halfway to the equivalence point in the second titration is 4.44.

Step 1: Recall the relationship between pH, pKa, and concentrations at the half-equivalence point

At the half-equivalence point of a titration between a weak acid and a strong base, the concentration of the acid is equal to the concentration of its conjugate base. According to the Henderson-Hasselbalch equation, we have: \(pH = pKa + \log \frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\) Since \([\mathrm{A}^-]=[\mathrm{HA}]\), the equation becomes: \(pH = pKa\)

Step 2: Determine the pKa value of the first titration

We are given the pH value halfway to the equivalence point in the first titration, which is equal to the pKa value of the weak acid. Thus, we have: \(pKa = 4.44\)

Step 3: Apply the pKa value to the second titration

In the second titration, we have twice the amount of NaOH to reach the equivalence point. However, the pKa value of the weak acid remains the same, regardless of the volume of the NaOH solution used. Therefore, the pH value halfway to the equivalence point in the second titration is also equal to the pKa value. \(pH = pKa = 4.44\) So, the pH halfway to the equivalence point in the second titration is also \(4.44\).

Key Concepts

Understanding Weak Acids

Weak acids partially dissociate in water. This means they do not release all their hydrogen ions (H⁺) into a solution. Unlike strong acids, where dissociation is complete, weak acids form an equilibrium between undissociated acid molecules and ions.

Key characteristics of weak acids include:

• They have a higher pH value compared to strong acids of the same concentration.
• The degree of ionization is less than 100%.
• They establish an equilibrium in aqueous solution between the acid (HA) and its ions (H⁺ and A⁻).

Some common examples of weak acids are acetic acid (vinegar), citric acid (found in citrus fruits), and formic acid (found in insect stings). Understanding how weak acids behave during titration is crucial, as their partial dissociation affects the pH changes observed during the process. The nature of weak acids also influences the calculation of their dissociation constant (Ka) and ultimately their pKa value.

The Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a formula used to estimate the pH of a buffer solution. It is particularly useful in the context of titration involving a weak acid and a strong base. The equation is expressed as: \[ pH = pKa + \log \frac{[\text{A}^-]}{[\text{HA}]} \] This equation helps in understanding how the concentration ratio of a base ([A⁻] - the conjugate base) and an acid ([HA]) in a solution influences its pH.

In a titration scenario, using the Henderson-Hasselbalch equation at the half-equivalence point simplifies calculations because:

• The concentration of the acid equals the concentration of its conjugate base ([A⁻] = [HA]).
• This simplifies the equation to \( pH = pKa \).

Thus, at the half-equivalence point, the pH of the solution can be directly related to the pKa of the weak acid being titrated. This relationship is crucial in determining the properties of the acid and its behavior during titration.

Understanding pKa

The term 'pKa' is a measure of the strength of an acid in solution. It is slightly more descriptive than 'Ka', the acid dissociation constant, with the primary difference being representation. pKa is expressed on a logarithmic scale: \[ pKa = -\log(Ka) \] This logarithmic nature means each integer change in pKa represents a tenfold change in acid strength.

Some critical characteristics of pKa are:

• A lower pKa value indicates a stronger acid, which dissociates more to release H⁺ ions.
• A higher pKa signifies a weaker acid, with a lesser tendency to donate protons.

In the context of titrations, knowing the pKa of a weak acid is essential. It allows chemists to predict the pH at various points in the titration process, especially at the half-equivalence point, where the pH equals the pKa. This makes pKa a fundamental parameter when analyzing acid-base reactions.