Question

Calculate the pH of a buffer solution prepared by dissolving 21.5 \(\mathrm{g}\) benzoic acid \(\left(\mathrm{HC}_{7} \mathrm{H}_{3} \mathrm{O}_{2}\right)\) and \(37.7 \mathrm{~g}\) sodium benzoate in \(200.0 \mathrm{~mL}\) of solution.

Short answer

The pH of the buffer solution prepared by dissolving 21.5g benzoic acid and 37.7g sodium benzoate in 200.0 mL of solution is approximately 4.51.

Step-by-step solution

Calculate moles of benzoic acid and sodium benzoate

To determine the moles of benzoic acid and sodium benzoate in the solution, first find their molar masses and then divide the given masses of each substance by their respective molar masses. The molar mass of benzoic acid, HC7H3O2, is: \(122.12\mathrm{g/mol}\) The molar mass of sodium benzoate, NaC7H5O2, is: \(144.11\mathrm{g/mol}\) Moles of benzoic acid: \(\frac{21.5\mathrm{g}}{122.12\mathrm{g/mol}} \approx 0.176 \mathrm{mol}\) Moles of sodium benzoate: \(\frac{37.7\mathrm{g}}{144.11\mathrm{g/mol}} \approx 0.261 \mathrm{mol}\)

Calculate molar concentrations of benzoic acid and sodium benzoate

To find the molar concentrations of benzoic acid and sodium benzoate, divide their respective moles by the volume of the solution in liters. The volume of the solution is 200.0 mL, which is equivalent to 0.200 L. Molar concentration of benzoic acid: \(\frac{0.176 \mathrm{mol}}{0.200 \mathrm{L}} = 0.880 \mathrm{M}\) Molar concentration of sodium benzoate: \(\frac{0.261 \mathrm{mol}}{0.200 \mathrm{L}} = 1.305 \mathrm{M}\)

Determine the pKa of benzoic acid

In order to use the Henderson-Hasselbalch equation, you will need to know the pKa of benzoic acid. The pKa of benzoic acid is 4.19.

Apply the Henderson-Hasselbalch equation

The pH of a buffer solution can be calculated using the Henderson-Hasselbalch equation. This equation relates the pH, pKa, and the concentrations of the acid (HA) and its conjugate base (A-) in a buffer solution as follows: pH = pKa + log\(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\) In this problem: pKa = 4.19 [HA] = 0.880 M (concentration of benzoic acid) [A-] = 1.305 M (concentration of sodium benzoate)

Calculate the pH of the buffer solution

Substitute the values from steps 3 and 2 into the Henderson-Hasselbalch equation and solve for pH. pH = 4.19 + log\(\frac{1.305\mathrm{M}}{0.880\mathrm{M}}\) pH ≈ 4.19 + 0.32 pH ≈ 4.51 The pH of the buffer solution is approximately 4.51.

Key concepts

pH calculation

Calculating the pH of a solution, especially when dealing with buffer solutions, is a fundamental skill in chemistry. A buffer solution is designed to resist changes in pH when small amounts of acid or base are added. This is especially important in various biological systems and industrial applications.

In the given exercise, we seek to calculate the pH of a buffer solution containing benzoic acid and sodium benzoate. To find the pH, we use the Henderson-Hasselbalch equation, which requires us to know the pKa of the acid and the concentrations of the acid and its conjugate base in the solution.

The final calculation requires substituting the known values of the pKa and the molar concentrations of the benzoic acid and sodium benzoate into the equation to solve for the pH. After performing these calculations, you’ll discover that the pH of the buffer solution is approximately 4.51.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a simplified way to determine the pH of a buffer solution. It relates the pH of the solution to the pKa of the weak acid and the ratio of the concentrations of its conjugate base. The equation is expressed as:

\[ \text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right) \]

Where:

• \([\text{A}^-]\) is the concentration of the conjugate base.
• \([\text{HA}]\) is the concentration of the weak acid.
• \(\text{pKa}\) is the negative log of the acid dissociation constant.

The exercise demonstrates using this equation by plugging in values for pKa, concentration of sodium benzoate \(([\text{A}^-] = 1.305 \text{ M})\), and benzoic acid \(([\text{HA}] = 0.880 \text{ M})\), leading to an accurate calculation of the buffer solution’s pH.

molar concentration

Molar concentration, often referred to simply as "concentration," is an essential concept in chemistry. It expresses the amount of a solute (substance being dissolved) present in a solution. The molar concentration is measured in moles per liter (M), which indicates the moles of solute per liter of total solution.

In our buffer solution, we calculated the molar concentrations of benzoic acid and sodium benzoate. To find these values, one must first determine the moles of each component by dividing the given mass by the molar mass, then divide these moles by the volume of the solution (in liters).

This exercise showed:

• The molar concentration of benzoic acid was calculated to be 0.880 M.
• For sodium benzoate, the concentration was 1.305 M.

This step is crucial, as these concentrations directly influence the pH when used in the Henderson-Hasselbalch equation.

benzoic acid

Benzoic acid is a common weak acid often used in food preservation, cosmetics, and as an acidulant. Chemically, it’s represented as \(\text{C}_7\text{H}_5\text{O}_2\), with a carboxylic acid group making it a useful component in buffer solutions.

In the exercise, benzoic acid acts as the weak acid in the buffer solution. We determined its molar concentration after calculating the moles of benzoic acid by dividing its mass by its molar mass. Knowing the concentration is vital, as benzoic acid pairs with sodium benzoate to form a buffer system capable of stabilizing pH around its pKa value.

The pKa of benzoic acid, which is necessary for the Henderson-Hasselbalch equation, is 4.19. This means under standard conditions, 50% of benzoic acid is dissociated into its conjugate base at a pH of 4.19.

sodium benzoate

Sodium benzoate is the sodium salt of benzoic acid, represented as \(\text{NaC}_7\text{H}_5\text{O}_2\). It’s widely used as a preservative in food due to its ability to inhibit the growth of mold, yeast, and some bacteria. In buffer solutions, it serves as the conjugate base to its weak acid counterpart, benzoic acid.

Its role in this exercise is crucial, as it forms the conjugate base in the buffer system. The concentration of sodium benzoate in the solution was calculated after determining its moles through the same process: dividing its mass by its molar mass, then by the volume of the solution in liters. With its concentration known, we could effectively use the Henderson-Hasselbalch equation to find the solution’s pH.

Thus, understanding the function and calculation of sodium benzoate’s concentration helps us comprehend its buffering action in maintaining a stable pH.