Terephthalic acid is an important chemical used in the manufacture of polyesters and plasticizers. It contains only C, \(\mathrm{H}\), and \(\mathrm{O}\). Combustion of \(19.81 \mathrm{mg}\) terephthalic acid produces \(41.98 \mathrm{mg} \mathrm{CO}_{2}\) and \(6.45 \mathrm{mg} \mathrm{H}_{2} \mathrm{O}\). If \(0.250 \mathrm{~mole}\) of terephthalic acid has a mass of \(41.5 \mathrm{~g}\), determine the molecular formula for terephthalic acid.

We know the mass of CO2 and H2O produced as a result of the combustion reaction. One mole of CO2 contains one mole of Carbon, and one mole of H2O contains two moles of Hydrogen. We can use this information to find the mass of C and H in the sample and then, by difference, find the mass of oxygen. Given mass of CO2 = 41.98 mg Molar mass of CO2 = 12.01 g/mol (C) + 2(16.00 g/mol) (O) = 44.01 g/mol Given mass of H2O = 6.45 mg Molar mass of H2O = 2(1.01 g/mol) (H) + 16.00 g/mol (O) = 18.02 g/mol Now the number of moles CO2 = \( \frac{41.98 \,\text{mg}}{44.01 \frac{\text{g}}{\text{mol}}}\times \frac{1\,\text{g}}{1000\,\text{mg}} \) = 0.0010 moles Likewise, Number of moles of H2O = \( \frac{6.45 \,\text{mg}}{18.02 \frac{\text{g}}{\text{mol}}}\times \frac{1\,\text{g}}{1000\,\text{mg}} \) = 0.00036 moles Total mass of the sample = 19.81 mg Mass of C = moles × molar mass = 0.0010 moles × 12.01 g/mol = 12.0 mg Mass of H = moles × molar mass = 0.00036 moles × 2(1.01 g/mol) = 0.73 mg Mass of O (by difference) = total mass - mass of C - mass of H = 19.81 mg - 12.0 mg - 0.73 mg = 7.08 mg

Now that we have the masses of C, H, and O in the sample, we can convert them to moles. Number of moles of Carbon = \( \frac{12.0 \,\text{mg}}{12.01\, \frac{\text{g}}{\text{mol}}} \times \frac{1\,\text{g}}{1000\,\text{mg}}\) = 1.0 mmoles Number of moles of Hydrogen = \( \frac{0.73 \,\text{mg}} {1.01\,\frac{\text{g}}{\text{mol}}} \times \frac{1\,\text{g}}{1000\,\text{mg}}\) = 0.72 mmoles Number of moles of Oxygen = \( \frac{7.08 \,\text{mg}}{16.00\,\frac{\text{g}}{\text{mol}}} \times \frac{1\,\text{g}}{1000\,\text{mg}}\) = 0.44 mmoles

To find the empirical formula, we need to find the smallest whole number ratio of the elements. We can do this by dividing each element’s moles by the smallest moles value, which is 0.44 mmoles. C: \( \frac{1.0 \,\text{mmoles}}{0.44 \,\text{mmoles}}\) = 2.3 which is approximately 2 H: \( \frac{0.72 \,\text{mmoles}}{0.44 \,\text{mmoles}}\) = 1.6 which is approximately 2 O: \( \frac{0.44 \,\text{mmoles}}{0.44 \,\text{mmoles}}\) = 1 So the empirical formula is C₂H₂O.

Given that 0.250 moles of terephthalic acid has a mass of 41.5 g, we can find the molar mass of terephthalic acid: Molar mass of terephthalic acid = \(\frac{41.5 \,\text{g}}{0.250 \,\text{moles}}\) = 166 g/mol Empirical formula mass = 2(12.01 g/mol) + 2(1.01 g/mol) + 16.00 g/mol = 32.04 g/mol

Finally, we'll compare the empirical formula mass with the molar mass: Molecular formula = Empirical formula × n where n = \( \frac{\text{Molar mass}}{\text{Empirical formula mass}} \) = \( \frac{166 \,\text{g/mol}}{32.04 \,\text{g/mol}} \) = 5.18 ≈ 5 Since n is approximately 5, we can multiply the empirical formula by 5 to get the molecular formula: Molecular formula: C₂H₂O × 5 = C₁₀H₁₀O₅ The molecular formula for terephthalic acid is C₁₀H₁₀O₅.