A particular sample of vinegar has a pH of 2.90. If acetic acid is the only acid that vinegar contains (Ka = 1.8 * 10^-5), calculate the concentration of acetic acid in the vinegar.

Verified step by step guidance

1. Step 1: Understand that pH is a measure of the hydrogen ion concentration \([H^+]\) in a solution. Use the formula \(pH = -\log[H^+]\) to find \([H^+]\).
2. Step 2: Rearrange the formula to solve for \([H^+]\): \([H^+] = 10^{-pH}\). Substitute the given pH value of 2.90 into this equation to calculate \([H^+]\).
3. Step 3: Recognize that acetic acid \((CH_3COOH)\) is a weak acid and partially dissociates in water according to the equation: \(CH_3COOH \rightleftharpoons CH_3COO^- + H^+\).
4. Step 4: Use the expression for the acid dissociation constant \(K_a\) for acetic acid: \(K_a = \frac{[CH_3COO^-][H^+]}{[CH_3COOH]}\). Assume \([CH_3COO^-] = [H^+]\) at equilibrium since each mole of acetic acid that dissociates produces one mole of \(H^+\) and one mole of \(CH_3COO^-\).
5. Step 5: Substitute \([H^+]\) from Step 2 and \(K_a = 1.8 \times 10^{-5}\) into the \(K_a\) expression to solve for the initial concentration of acetic acid \([CH_3COOH]\).