You take \(1.00 \mathrm{~g}\) of an aspirin tablet (a compound consisting solely of carbon, hydrogen, and oxygen), burn it in air, and collect \(2.20 \mathrm{~g} \mathrm{CO}_{2}\) and \(0.400 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}\). You know that the molar mass of aspirin is between 170 and \(190 \mathrm{~g} / \mathrm{mol}\). Reacting 1 mole of salicylic acid with 1 mole of acetic anhydride \(\left(\mathrm{C}_{4} \mathrm{H}_{6} \mathrm{O}_{3}\right)\) gives you 1 mole of aspirin and 1 mole of acetic acid \(\left(\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{O}_{2}\right.\) ). Use this information to determine the molecular formula of salicylic acid.

From the information given, upon combustion 1.00g of aspirin produces 2.20g CO2 and 0.400g H2O. We can use the moles of the products CO2 and H2O to determine the moles of carbon and hydrogen atoms in aspirin. Find the moles of CO2 and H2O produced by the combustion: \[ \text{Moles of CO2} = \frac{2.20g}{44.01 g/mol} = 0.0500 mol \] \[ \text{Moles of H2O} = \frac{0.400g}{18.02 g/mol} = 0.0222 mol \]

Now we can calculate the moles of carbon, hydrogen, and oxygen atoms in 1.00g of aspirin. As CO2 contains one carbon atom and H2O contains two hydrogen atoms, so their moles will be: \[ \text{Moles of C} = 0.0500 mol \] \[ \text{Moles of H} = 2 × 0.0222 mol = 0.0444 mol \] To find the moles of oxygen, let's subtract the grams of carbon and hydrogen from the total 1.00g of aspirin: \[ \text{Mass of C} = 0.0500 mol × 12.01 g/mol = 0.600 g \] \[ \text{Mass of H} = 0.0444 mol × 1.01 g/mol = 0.0448 g \] \[ \text{Mass of O} = 1.00 g - (0.600 g + 0.0448 g) = 0.3552 g \] Now we can find the moles of oxygen: \[ \text{Moles of O} = \frac{0.3552g}{16.00 g/mol} = 0.0222 mol \]

To find the empirical formula, divide the moles of each element by the smallest mole value: \[ \text{Mole ratio of C} = \frac{0.0500 mol}{0.0222 mol} = 2.25 \approx 2 \] \[ \text{Mole ratio of H} = \frac{0.0444 mol}{0.0222 mol} = 2.00 \approx 2 \] \[ \text{Mole ratio of O} = \frac{0.0222 mol}{0.0222 mol} = 1.00 \approx 1 \] So, the empirical formula of aspirin is C₂H₂O.

We know that the molecular weight of aspirin is between 170 and 190 g/mol. Using the empirical formula, we can determine the empirical weight: \[ (2 × 12.01 g/mol) + (2 × 1.01 g/mol) + 16.00 g/mol = 42.04 g/mol \] Now we need to find the multiple of the empirical weight that will fall in the given molecular weight range of aspirin: \[ \frac{170 g/mol}{42.04 g/mol} = 4.04 \approx 4 \] \[ \frac{190 g/mol}{42.04 g/mol} = 4.52 \approx 4 \] So, the molecular weight of aspirin is around 4 times the empirical weight (42.04 × 4 = 168.16 g/mol). Therefore, the molecular formula of aspirin is C₈H₈O₄. As the problem states, "reacting 1 mole of salicylic acid with 1 mole of acetic anhydride (C₄H₆O₃) gives you 1 mole of aspirin (C₈H₈O₄) and 1 mole of acetic acid (C₂H₄O₂)." The molecular formula of salicylic acid, which reacts with acetic anhydride, should have less carbon, hydrogen, and oxygen atoms. If we subtract the acetic anhydride molecular formula from the aspirin molecular formula, we find the molecular formula of salicylic acid. \[ C_(8)H_(8)O_(4) - C_(4)H_(6)O_(3) = C_(8-4)H_(8-6)O_(4-3) = C_(4)H_(2)O_(1) \] Finally, the molecular formula of salicylic acid is C₄H₂O.