Question

About this time, you may be wishing you had an aspirin. Aspirin is an organic acid with a \(K_{\mathrm{a}}\) of \(3.27 \times 10^{-4}\) for the reaction $$\mathrm{HC}_{3} \mathrm{H}_{7} \mathrm{O}_{4}(\mathrm{aq})+\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftharpoons_{\mathrm{C}_{9} \mathrm{H}_{7} \mathrm{O}_{4}^{-}(\mathrm{aq})}+\mathrm{H}_{3} \mathrm{O}^{+}(\mathrm{aq})$$ If you have two tablets, each containing 0.325 g of aspirin (mixed with a neutral "binder" to hold the tablet together), and you dissolve them in a glass of water to give \(225 \mathrm{mL}\) of solution, what is the pH of the solution?

Short answer

The pH of the solution is approximately 2.64.

Step-by-step solution

Determine Moles of Aspirin

The molar mass of aspirin (C₉H₈O₄) is approximately 180.16 g/mol. Calculate the number of moles of aspirin in two tablets: \[ \text{Moles of Aspirin} = \frac{2 \times 0.325 \text{ g}}{180.16 \text{ g/mol}} \approx 0.00361 \text{ mol}. \]

Calculate Initial Concentration of Aspirin

The total volume of the solution is 225 mL, which is 0.225 L. The initial concentration of aspirin is \[ \text{Concentration of } \text{HC}_9\text{H}_7\text{O}_4 = \frac{0.00361 \text{ mol}}{0.225 \text{ L}} \approx 0.0160 \text{ M}. \]

Set Up the Equilibrium Expression

For the dissociation of aspirin, the equilibrium expression using the acid dissociation constant (\(K_a\)) is: \[ K_a = \frac{[\text{C}_9\text{H}_7\text{O}_4^-][\text{H}_3\text{O}^+]}{[\text{HC}_9\text{H}_7\text{O}_4]} = 3.27 \times 10^{-4}. \]

Use the ICE Table for Equilibrium Calculation

Create an ICE table and solve for \(x\), where \(x\) is the concentration of \([\text{H}_3\text{O}^+]\): Initial: \([\text{HC}_9\text{H}_7\text{O}_4] = 0.0160 \text{ M}, [\text{C}_9\text{H}_7\text{O}_4^-] = 0, [\text{H}_3\text{O}^+] = 0\) Change: \(-x, +x, +x\) Equilibrium: \([\text{HC}_9\text{H}_7\text{O}_4] = 0.0160 - x, [\text{C}_9\text{H}_7\text{O}_4^-] = x, [\text{H}_3\text{O}^+] = x\) Plug into the \(K_a\) expression: \[ 3.27 \times 10^{-4} = \frac{x^2}{0.0160 - x}. \]

Simplify and Solve for \(x\)

Assume that \(x\) is small compared to 0.0160, thus \(0.0160 - x \approx 0.0160\). This simplifies the equation to: \[ x^2 = 3.27 \times 10^{-4} \times 0.0160. \]Calculate \(x\): \[ x = \sqrt{3.27 \times 10^{-4} \times 0.0160} \approx 2.282 \times 10^{-3}. \]

Calculate pH of the Solution

The pH is given by the negative logarithm of the \([\text{H}_3\text{O}^+]\) concentration: \[ \text{pH} = -\log(2.282 \times 10^{-3}) \approx 2.64. \]

Key concepts

pH calculation

Let's start with understanding how to calculate the pH of a solution like aspirin dissolved in water. pH is a measure of the hydrogen ion concentration in a solution and is calculated using the formula:
\[\text{pH} = -\log [\text{H}_3\text{O}^+]\]This formula tells us that pH decreases as the concentration of hydrogen ions, \([\text{H}_3\text{O}^+]\), increases. In the case of the aspirin solution, once we know \([\text{H}_3\text{O}^+]\), we can find the pH.
For aspirin, a weak acid, the concentration of hydrogen ions is found using the acid dissociation constant, \(K_a\). When aspirin dissolves, it partially dissociates releasing \([\text{H}_3\text{O}^+]\). Once we calculated the concentration of \([\text{H}_3\text{O}^+]\) as approximately \(2.282 \times 10^{-3}\) M, we used the formula to find the pH. This gave us a pH of approximately 2.64, which indicates an acidic solution.
Calculating pH can be straightforward, as long as you know the \([\text{H}_3\text{O}^+]\) concentration. This is usually what you solve for initially in dissociation problems, especially with weak acids like aspirin.

equilibrium expression

The equilibrium expression is a crucial part for solving acid dissociation problems. When aspirin dissolves in water, it reaches an equilibrium between the undissolved aspirin and the ions it forms. The equilibrium expression for this reaction involves the concentrations of all species at equilibrium:
\[K_a = \frac{[\text{C}_9\text{H}_7\text{O}_4^-][\text{H}_3\text{O}^+]}{[\text{HC}_9\text{H}_7\text{O}_4]}\]This equation is derived from the law of mass action and describes how the aspirin acid (HC₉H₇O₄) dissociates into its conjugate base (C₉H₇O₄⁻) and hydronium ions (H₃O⁺).
To solve, we typically use an ICE table, which tracks the Initial concentrations, Changes in concentrations, and concentrations at Equilibrium. Initially, only aspirin is present, and as it dissociates, the concentrations of ions increase by \(x\), indicating their change from the initial state. By substituting into the equilibrium expression, you can solve for \(x\), the concentration of \([\text{H}_3\text{O}^+]\).
This makes equilibrium expressions essential as they link the dissociation constant with the concentrations of acids and their ions, allowing us to understand the extent of reactions in solutions.

molar mass

Molar mass is a fundamental concept for converting mass into moles, which is essential for stoichiometry in chemistry. The molar mass of a compound is the mass in grams of one mole of that compound. For aspirin, the chemical formula is \(\text{C}_9\text{H}_8\text{O}_4\) and its molar mass is approximately 180.16 g/mol.
To find the number of moles of aspirin, follow this formula:

• Moles = \(\frac{\text{mass in grams}}{\text{molar mass in g/mol}}\)

For the exercise, we had 0.325 g in each tablet, and thus for two tablets, the calculation was \(\frac{2 \times 0.325}{180.16} \approx 0.00361\) mol.
Understanding molar mass enables you to relate the mass of a substance to the number of particles it contains, which is essential for any quantitative chemical analysis. This basis is not only used for calculating initial concentration but also for determining how much reactant is available to reach equilibrium.