Question

A typical aspirin tablet contains \(325 \mathrm{mg}\) acetylsalicylic acid \(\left(\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4}\right) .\) Calculate the \(\mathrm{pH}\) of a solution that is prepared by dissolving two aspirin tablets in enough water to make one cup \((237 \mathrm{~mL})\) of solution. Assume the aspirin tablets are pure acetylsalicylic acid, \(K_{\mathrm{a}}=3.3 \times 10^{-4}\).

Short answer

The pH of the aspirin solution formed by dissolving two aspirin tablets in a cup of water is approximately 2.65.

Step-by-step solution

Determine the number of moles of acetylsalicylic acid

First, we need to determine the number of moles of acetylsalicylic acid present in the solution. To do this, we will multiply the mass of acetylsalicylic acid in one tablet by the number of tablets, and then divide by the molar mass of acetylsalicylic acid. Molar mass of acetylsalicylic acid (HC₉H₇O₄) = 180.16 g/mol Mass of acetylsalicylic acid in one tablet = 325 mg = 0.325 g Number of tablets = 2 Number of moles = (mass of acetylsalicylic acid in one tablet × number of tablets) / molar mass Number of moles = (0.325 g × 2) / 180.16 g/mol

Calculate the initial concentration of acetylsalicylic acid

Next, we need to determine the initial concentration of acetylsalicylic acid in the solution. We will divide the number of moles of acetylsalicylic acid by the volume of the solution (in liters). Number of moles = (0.325 g × 2) / 180.16 g/mol ≈ 0.00362 mol Volume of solution = 237 mL = 0.237 L Initial concentration (C₀) = number of moles / volume of solution C₀ = 0.00362 mol / 0.237 L ≈ 0.0153 M

Set up the Ka expression

Now, we will set up the Ka expression for the dissociation of acetylsalicylic acid in the solution. Ka = [H₃O⁺][C₉H₇O₄⁻] / [HC₉H₇O₄] Since we are assuming that all the acetylsalicylic acid in the aspirin tablets is converted into HC₉H₇O₄, we can write that: [H₃O⁺] = x M [C₉H₇O₄⁻] = x M [HC₉H₇O₄] = (0.0153 - x) M Substituting these values into the Ka expression: 3.3 × 10⁻⁴ = x² / (0.0153 - x)

Solve for [H₃O⁺] concentration (x)

Now, we need to solve the quadratic equation to find the value of x, which represents the concentration of H₃O⁺ ions in the solution. Since Ka is very small, we can make the assumption that x is significantly smaller than 0.0153, allowing us to simplify the equation: 3.3 × 10⁻⁴ ≈ x² / 0.0153 Solving for x (the concentration of H₃O⁺ ions): x = √(0.0153 × 3.3 × 10⁻⁴) ≈ 0.00225 M

Calculate the pH of the solution

Finally, we will use the pH equation to find the pH of the solution: pH = -log[H₃O⁺] pH = -log(0.00225) ≈ 2.65 Therefore, the pH of the aspirin solution is approximately 2.65.

Key concepts

Acetylsalicylic Acid

Acetylsalicylic acid, commonly known as aspirin, is a widely used medication known for its pain-relieving, anti-inflammatory, and antiplatelet effects. Chemically, it is a weak acid, with the formula \(\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}\), and it belongs to the category of salicylates. When acetylsalicylic acid dissolves in water, it releases \(\mathrm{H}^{+}\) ions, a process which can be used to calculate the solution's pH. This compound has particular significance in pharmaceuticals, and understanding its properties—including its behavior in different mediums—is crucial for pharmaceutical and medical studies. The balance between its un-ionized (acetylsalicylic acid) and ionized (salicylate ion and hydrogen ion) forms in solution can be represented using the acid dissociation constant.

Acid Dissociation Constant (Ka)

The acid dissociation constant \(K_{a}\) is a quantitative measure of an acid's strength in solution. It represents the equilibrium constant for the dissociation of an acid into an anion and a proton (hydrogen ion). Specifically for acetylsalicylic acid, the chemical reaction can be described as:\[\mathrm{HC}_{9}\mathrm{H}_{7}\mathrm{O}_{4}(aq) \rightleftharpoons \mathrm{H}^{+}(aq) + \mathrm{C}_{9}\mathrm{H}_{7}\mathrm{O}_{4}^{-}(aq)\]The given \(K_{a}\) value of \(3.3 \times 10^{-4}\) indicates how readily acetylsalicylic acid donates its proton in water. The lower the \(K_{a}\), the weaker the acid and the less it dissociates. When solving for pH, \(K_{a}\) is used to establish an equilibrium expression where the concentrations of the reactants and products can be related, allowing us to find the proton concentration and thus determine the solution's acidity.

Molar Mass

Molar mass is a fundamental concept in chemistry that represents the mass of one mole of a substance, expressed in grams per mole (g/mol). It can be calculated by summing the atomic masses of all the atoms in a molecule. For acetylsalicylic acid (HC₉H₇O₄), the molar mass is calculated by adding the weights of 8 carbon atoms, 8 hydrogen atoms, and 4 oxygen atoms, resulting in \(180.16\ g/mol\). The knowledge of the molar mass is critical when converting between the mass of a substance and the number of moles, as shown in the step-by-step solution. This allows us to understand the quantity of the substance in terms of its moles, which is essential for stoichiometry and determining concentration in solutions.

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between the reactants and products in a balanced chemical equation. It allows for the precise calculation of reactant and product quantities, based on their molar ratios. In the problem at hand, stoichiometry is applied in several steps: \(

\) \(
• \)Calculating the number of moles of acetylsalicylic acid used by considering the molar mass and weight of the substance.\(
• \)Determining the concentration of acetylsalicylic acid by dividing the number of moles by the volume of the solution.\(
• \)Using the equilibrium expression that relates the concentrations of the reactants and products to solve for the concentration of hydronium ions, based on the \(K_{a}\).\(
• \)Finally, calculating the pH from the hydronium ion concentration, which represents the solution's acidity level.\(

\)Through stoichiometry, we can grasp the relationship between mass and moles, and how these concepts interact to solve for other properties, such as pH, in chemical solutions.