To determine the empirical formula of sorbic acid, follow these steps:
1. Determine the moles of Carbon in CO₂:
Given mass of CO₂: \(2.053 \, \text{g}\)
Molar mass of CO₂ \( \left( \text{C} + 2 \text{O} \right) \):
\( 12.01 \, \text{g/mol (C)} + 2 \times 16.00 \, \text{g/mol (O)} = 44.01 \, \text{g/mol} \)
Moles of Carbon in CO₂:
[tex]\[
\text{Moles of C} = \frac{2.053 \, \text{g}}{44.01 \, \text{g/mol}} \approx 0.04665 \, \text{mol}
\][/tex]
2. Determine the moles of Hydrogen in H₂O:
Given mass of H₂O: \(0.5601 \, \text{g}\)
Molar mass of H₂O \( \left( 2 \text{H} + \text{O} \right) \):
\( 2 \times 1.008 \, \text{g/mol (H)} + 16.00 \, \text{g/mol (O)} = 18.016 \, \text{g/mol} \)
Moles of Hydrogen in H₂O:
[tex]\[
\text{Moles of H} = \frac{0.5601 \, \text{g}}{18.016 \, \text{g/mol}} \times 2 \approx 0.06211 \, \text{mol}
\][/tex]
3. Determine the moles of Oxygen:
First, calculate the mass of C and H in the sample:
Mass of Carbon:
[tex]\[
\text{Mass of C} = 0.04665 \, \text{mol} \times 12.01 \, \text{g/mol} \approx 0.560 \, \text{g}
\][/tex]
Mass of Hydrogen:
[tex]\[
\text{Mass of H} = 0.06211 \, \text{mol} \times 1.008 \, \text{g/mol} \approx 0.063 \, \text{g}
\][/tex]
Total mass of C and H:
[tex]\[
\text{Mass of C + Mass of H} = 0.560 \, \text{g} + 0.063 \, \text{g} = 0.623 \, \text{g}
\][/tex]
Given mass of sample: \(0.8715 \, \text{g}\)
Mass of Oxygen in the sample:
[tex]\[
\text{Mass of O} = 0.8715 \, \text{g} - 0.623 \, \text{g} = 0.2485 \, \text{g}
\][/tex]
Moles of Oxygen:
[tex]\[
\text{Moles of O} = \frac{0.2485 \, \text{g}}{16.00 \, \text{g/mol}} \approx 0.01553 \, \text{mol}
\][/tex]
4. Determine the molar ratio:
Compare the smallest moles to get the empirical formula:
- Moles of C: \(0.04665 \approx 3.0026\)
- Moles of H: \(0.06211 \approx 4.0022\)
- Moles of O: \(0.01553 = 1\)
(Ratios are calculated based on smallest moles of O.)
Therefore, the ratio is:
[tex]\[
\text{C} : \text{H} : \text{O} = 3.0026 : 4.0022 : 1
\][/tex]
5. Empirical formula:
Round to the nearest whole number:
[tex]\[
\text{C} : \text{H} : \text{O} = 3 : 4 : 1
\][/tex]
Hence, the empirical formula of sorbic acid is [tex]\(\text{C}_3\text{H}_4\text{O}\)[/tex], which matches choice (b).
