Question

A buffer solution is prepared by mixing \(75.0 \mathrm{~mL}\) of \(0.275 \mathrm{M}\) fluorobenzoic acid \(\left(\mathrm{C}_{7} \mathrm{H}_{5} \mathrm{O}_{2} \mathrm{~F}\right)\) with \(55.0 \mathrm{~mL}\) of \(0.472 \mathrm{M}\) so- dium fluorobenzoate. The \(\mathrm{p} K_{\mathrm{a}}\) of this weak acid is \(2.90\). What is the \(\mathrm{pH}\) of the buffer solution?

Short answer

The pH of the buffer solution is \(2.80\).

Step-by-step solution

Identify the given information

We are given the following information: 1. Volume of the weak acid solution (V₁): \(75.0 \mathrm{~mL}\) 2. Concentration of the weak acid solution (C₁): \(0.275 \mathrm{M}\) 3. Volume of the conjugate base solution (V₂): \(55.0 \mathrm{~mL}\) 4. Concentration of the conjugate base solution (C₂): \(0.472 \mathrm{M}\) 5. pKa of the weak acid: \(2.90\)

Calculate the moles of weak acid and conjugate base

We will now calculate the moles of weak acid and conjugate base present in their respective solutions. Moles of weak acid (n₁) = volume × concentration of weak acid n₁ = V₁ × C₁ = \(75.0 \mathrm{~mL}\) × \(0.275 \mathrm{M}\) = \(20.625 \mathrm{~mmol}\) Moles of conjugate base (n₂) = volume × concentration of conjugate base n₂ = V₂ × C₂ = \(55.0 \mathrm{~mL}\) × \(0.472 \mathrm{M}\) = \(25.96 \mathrm{~mmol}\)

Determine the mole ratio of weak acid and conjugate base

Using the moles calculated in step 2, we can now determine the mole ratio of weak acid to conjugate base, which will be important when applying the Henderson-Hasselbalch equation. Mole ratio of weak acid to conjugate base = (moles of weak acid) / (moles of conjugate base) Mole ratio = n₁ / n₂ = \(20.625 \mathrm{~mmol} / 25.96 \mathrm{~mmol}\) = \(0.795\)

Apply the Henderson-Hasselbalch equation to find the pH of the buffer

The Henderson-Hasselbalch equation is given by: pH = pKa + log(A-/HA) In our case, the mole ratio calculated in step 3 can be substituted as A-/HA (i.e., the ratio of conjugate base to weak acid). pH = pKa + log(0.795) pH = \(2.90 + \log(0.795)\) Now, let's calculate the pH: pH = \(2.90 + (-0.1002)\) pH = \(2.7998\)

Round the pH value to the appropriate decimal place

In this case, we will round the pH value to two decimal places. Final pH = \(2.80\) So, the pH of the buffer solution is \(2.80\).

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a fundamental formula used to estimate the pH of buffer solutions. It provides a straightforward relationship between the pH of a solution, the pKa (the -log of the acid dissociation constant, Ka) of the weak acid present in the buffer, and the ratio of the concentrations of the conjugate base (\text{A}^-) to the weak acid (\text{HA}). The equation is expressed as:
\[ \text{pH} = \text{pKa} + \log\left(\frac{\text{[A}^-\text{]}}{\text{[HA]}}\right) \]
In this equation, the 'log' is the base-10 logarithm. Understanding how to apply this equation is crucial for many chemistry problems, such as calculating the pH of buffer solutions after dilution or mixing different solutions.

Weak Acid and Conjugate Base

When a weak acid dissolves in water, it partially dissociates into its conjugate base and protons. Unlike strong acids, which dissociate completely, weak acids maintain an equilibrium between the undissociated acid and the ions in solution. This equilibrium is characterized by the acid dissociation constant, Ka, which provides a measure of the acid's strength. In contrast, the pKa is the negative logarithm of Ka and provides a more convenient term for working with acid strengths, as it offers a direct scale that corresponds to pH values. The relationship of a weak acid with its conjugate base is crucial in buffers, since they resist changes in pH upon the addition of small amounts of acid or base due to this equilibrium dynamic.

pKa

The pKa is a key chemical property that represents the strength of an acid. It's defined as the negative base-10 logarithm of the acid dissociation constant (Ka). In essence, the lower the value of pKa, the stronger the acid. For any weak acid, the pKa value is essential when using the Henderson-Hasselbalch equation to determine the pH of a buffer solution. Generally, pKa is used instead of Ka because it falls into a smaller and more manageable numerical range. Knowing the pKa allows chemists to predict how an acid will behave in a mixture and how it will influence the pH of a solution.

Molarity and Volume in Buffer Solutions

The molarity (M) of a solution is a measurement of the concentration of a solute in a solution, expressed as moles of solute per liter of solution. The volume (V) is simply the amount of space that the solution occupies, usually measured in liters or milliliters. For buffer solutions, the final pH depends not just on the concentrations of weak acid and conjugate base, but also on their absolute amounts, which are determined by multiplying their molarities by their volumes.
When calculating the pH of a buffer solution, the mole ratio of the weak acid to its conjugate base is particularly significant. This ratio can be found by dividing the number of moles of the weak acid by the number of moles of the conjugate base, each calculated from their respective molarity and volume. The Henderson-Hasselbalch equation uses this ratio, pKa value and allows the chemist to calculate the pH of the buffer solution.