Problem 27 - General Chemistry: Principles and Modern Applications

Question

What mass of benzoic acid, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\), would you dissolve in \(350.0 \mathrm{mL}\) of water to produce a solution with a \(\mathrm{pH}=2.85 ?\) $$\begin{aligned} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}^{-} & \\ K_{\mathrm{a}}=6.3 \times 10^{-5} \end{aligned}$$

Step-by-Step Solution

Answer

Calculate the mass using the formula \( mass = moles * molar mass \) where moles was derived by multiplying the concentration by volume.

Step 1: Calculate pH

From the given pH=2.85, we can work out the concentration of hydronium ions \(H_{3}O^{+}\) using the formula: \( \[ [H_{3}O^{+}] = 10^{-pH} \] \). Plugging the given pH=2.85 into the formula we get: \( \[ [H_{3}O^{+}] = 10^{-2.85} \] \). Calculate this expression to find the hydronium ion concentration.

Step 2: Determine Acid Concentration

Given the hydronium ions concentration and the acid dissociation constant \(K_{a} = 6.3 x 10^{-5}\), we can rearrange the formula for the \(K_{a}\) given by \(K_{a} = [H_{3}O^{+}][C_{6}H_{5}COO^-]/[C_{6}H_{5}COOH]\) to calculate the ratio of benzoic acid to its conjugate base. Because the acid is only weakly dissociated, we can assume that the acid concentration \([C_{6}H_{5}COOH]\) is approximately equal to the hydronium ion concentration. This gives \([C_{6}H_{5}COOH] = [H_{3}O^{+}]\). Calculate this expression to find the concentration of benzoic acid in the solution.

Step 3: Calculate Required Mass

To find the required mass of benzoic acid, calculate the molar mass of benzoic acid first. The molar mass of benzoic acid is \(122.12 g/mol\). Knowing the concentration of benzoic acid from step 2 and the volume of the solution, we can calculate the moles of benzoic acid in the solution using the formula: \[ moles = concentration * volume \]. Subsequently, the mass can be calculated as the moles multiplied by the molar mass.

Key Concepts

pH calculation

The pH of a solution is a measure of its acidity or basicity. Specifically, pH is a logarithmic scale that measures the concentration of hydrogen ions (\( \text{H}^+ \)) in a solution. Simply put, the lower the pH, the more acidic the solution. The relationship between pH and hydrogen ion concentration is given by the formula:

• \( \text{pH} = -\log[\text{H}_3\text{O}^+] \)
• Alternatively, \([\text{H}_3\text{O}^+] = 10^{-\text{pH}}\)

To calculate the pH, insert the hydronium ion concentration into the equation and solve. In reverse, if the pH is known, as in our exercise, you can calculate the \( [\text{H}_3\text{O}^+] \) by using the second formula.

acid dissociation constant

The acid dissociation constant, commonly represented as \( K_a \), is a crucial figure in acid-base chemistry. It measures the strength of an acid, specifically how well it dissociates into ions in solution. The larger the \( K_a \), the stronger the acid, as it dissociates to a greater extent, releasing more hydrogen ions. For weak acids, like benzoic acid in our exercise, \( K_a \) gives insight into the equilibrium between the undissociated acid and its dissociated ions. For benzoic acid, the given \( K_a \) is \( 6.3 \times 10^{-5} \), indicating that it only partially ionizes in water.

benzoic acid

Benzoic acid, with the chemical formula \( \text{C}_6\text{H}_5\text{COOH} \), is a simple aromatic carboxylic acid that is relatively weak due to its partial dissociation in water. It forms a conjugate base known as benzoate (\( \text{C}_6\text{H}_5\text{COO}^- \)). It naturally exists in many plants and serves as a widely used preservative in food. However, in chemistry, benzoic acid is important for studying weak acid equilibrium and calculating pH. In a typical experiment or calculation, it is essential to know its molar mass, which is \( 122.12 \text{ g/mol} \), for making solutions with desired concentrations.

weak acid equilibrium

The concept of weak acid equilibrium is central when dealing with acids like benzoic acid. These acids do not completely dissociate in water; instead, they reach an equilibrium between the undissociated acid and their ionized form. This can be represented by the equation:

• \( \text{HC}_6\text{H}_5\text{COOH} \leftrightharpoons \text{H}_3\text{O}^+ + \text{C}_6\text{H}_5\text{COO}^- \)

Under equilibrium conditions, the concentrations of the acid, its ions, and water are related through the \( K_a \) constant. In calculations, assuming that initial concentrations change negligibly can simplify the process. This approximation allows us to use the concentration of hydrogen ions as a close estimate for the concentration of undissociated weak acid, streamlining the calculation process when combined with information such as pH and \( K_a \)