Problem 70 - Chemistry An Atoms First Approach

Question

A typical aspirin tablet contains 325 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, \(K_{\mathrm{a}}=3.3 \times 10^{-4}\)

Step-by-Step Solution

Answer

The pH of the aspirin solution prepared by dissolving two tablets in 237 mL of water is 2.65.

Step 1: Calculate moles of acetylsalicylic acid in two tablets

Each tablet contains 325 mg of acetylsalicylic acid. To determine the number of moles, first convert the mass of acetylsalicylic acid to grams. Molar mass of acetylsalicylic acid, HC9H7O4 = 1(12.01) + 9(12.01) + 7(1.01) + 4(16.00) = 180.16 g/mol Mass of acetylsalicylic acid in two tablets = 2 × 325 mg = 650 mg = 0.65 g Now, calculate the moles of acetylsalicylic acid: Moles = (Mass of acetylsalicylic acid) / (Molar mass) Moles = 0.65 g / 180.16 g/mol = 0.00361 mol

Step 2: Calculate concentration of acetylsalicylic acid in the solution

The volume of the solution is 237 mL. Convert this to liters: Volume = 237 mL = 0.237 L Now, calculate the concentration of acetylsalicylic acid in the solution: Concentration = (Moles of acetylsalicylic acid) / (Volume of solution) Concentration = 0.00361 mol / 0.237 L = 0.0152 M

Step 3: Determine concentration of H+ ions using Ka

For acetylsalicylic acid, the dissociation equation is: HC9H7O4 (aq) ⇌ H+ (aq) + C9H7O4- (aq) Ka = [H+] [C9H7O4-] / [HC9H7O4] Given Ka = 3.3 × 10^-4, we need to find [H+]. Since the initial concentration of [H+] is very small compared to [HC9H7O4], we can assume the change in [H+] is x and the change in [HC9H7O4] and [C9H7O4-] are also x. Therefore: Ka = x * x / (0.0152 - x) Because Ka is very small, we can assume that x is much less than 0.0152, so the equation becomes: 3.3 × 10^-4 = x^2 / 0.0152

Step 4: Calculate pH of the solution using the concentration of H+ ions

First, solve for x: x^2 = 3.3 × 10^-4 * 0.0152 x^2 = 5.016 × 10^-6 x = 2.24 × 10^-3 Now, x is the concentration of H+ ions in the solution. To find the pH, use the formula: pH = -log[H+] pH = -log(2.24 × 10^-3) = 2.65 The pH of the aspirin solution is 2.65.

Key Concepts

Molar Mass

The concept of molar mass is crucial in chemistry as it provides the mass of one mole of a given substance. To calculate the molar mass, you sum up the atomic masses of all atoms in a chemical formula. For instance, in the case of acetylsalicylic acid (\(\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4}\)), you start by identifying the quantity of each type of atom:

• Carbon (C): 9 atoms, with atomic mass of 12.01 g/mol.
• Hydrogen (H): 7 atoms, with atomic mass of 1.01 g/mol.
• Oxygen (O): 4 atoms, with atomic mass of 16.00 g/mol.

Next, multiply the atomic mass by the number of atoms and add them together:\(\text{Molar Mass} = 1(12.01) + 9(12.01) + 7(1.01) + 4(16.00) = 180.16\, \text{g/mol}.\)Understanding molar mass helps in converting between grams and moles, which is essential for chemical calculations.

Weak Acid Dissociation

Weak acids partially dissociate in solution. This means they do not completely ionize in water, unlike strong acids. In the given problem, acetylsalicylic acid is considered a weak acid. Its partial dissociation can be represented by the equation:

• \(\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4} (aq) \rightleftharpoons \mathrm{H}^+ (aq) + \mathrm{C}_{9} \mathrm{H}_{7} \mathrm{O}_{4}^- (aq)\)

In this equilibrium equation:

• \(\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4}\) is the un-ionized acid.
• \(\mathrm{H}^+\) ions indicate the acid's ability to donate protons.
• \(\mathrm{C}_{9} \mathrm{H}_{7} \mathrm{O}_{4}^-\) is the conjugate base formed due to dissociation.

The degree to which a weak acid dissociates is governed by its acid dissociation constant (Ka), which gives insight into the acid strength and helps calculate the pH of solutions.

Acid Dissociation Constant (Ka)

The acid dissociation constant, \(K_a\), is a vital measure of an acid's strength in a solution. It shows the extent of dissociation of the acid into its ions. A smaller \(K_a\) value indicates a weaker acid, which dissociates less in solution. For acetylsalicylic acid, \(K_a = 3.3 \times 10^{-4}\).In this problem, the \(K_a\) expression for the dissociation of acetylsalicylic acid is:\[K_a = \frac{[\mathrm{H}^+][\mathrm{C}_{9} \mathrm{H}_{7} \mathrm{O}_{4}^-]}{[\mathrm{HC}_{9} \mathrm{H}_{7} \mathrm{O}_{4}]}\]Using the initial concentration of the acid and applying the formula, it's possible to estimate \([\mathrm{H}^+]\) by solving the quadratic approximation given the weak dissociation characteristic:\[K_a = \frac{x^2}{c - x} \approx \frac{x^2}{c} \text{ where } c \gg x\]This simplification aids in determining the hydrogen ion concentration, essential for pH calculation.

Concentration Calculation

Concentration measures how much solute exists in a given volume of solution. It's normally expressed in molarity (M), which is moles per liter. To find concentration, you divide the number of moles of solute by the solution's volume:

• Moles of solute: This was calculated by dividing the mass of acetylsalicylic acid by its molar mass. For two tablets, this came out to be 0.00361 moles.
• Volume of solution: The total volume given was 237 mL, which converts to 0.237 L.

The concentration of acetylsalicylic acid can be calculated with:\[\text{Concentration} = \frac{\text{Moles}}{\text{Volume}} = \frac{0.00361 \text{ mol}}{0.237 \text{ L}} = 0.0152 \text{ M}\]Understanding how to determine concentration is a foundational skill, necessary for further calculations like determining the pH of the solution.