To determine the empirical and molecular formulas, follow these steps:
### a) Empirical Formula:
1. **Convert grams to moles:**
\[ \text{moles of } \ce{CO2} = \frac{20.548 \, \text{g}}{44.01 \, \text{g/mol}} \]
This gives you the moles of carbon because each mole of \ce{CO2} has one mole of carbon.
2. **Determine moles of carbon:**
Since the empirical formula of the arene only contains carbon and hydrogen, the moles of carbon are equal to the moles of \ce{CO2}.
3. **Calculate moles of hydrogen:**
\[ \text{moles of hydrogen} = \text{total moles} - \text{moles of carbon} \]
4. **Determine the simplest whole number ratio:**
Divide both moles of carbon and moles of hydrogen by the smaller value among them.
5. **Write the empirical formula:**
Use the ratio to write the empirical formula.
### b) Molecular Formula:
6. **Determine the molecular mass of the empirical formula:**
\[ \text{Molecular mass} = \text{Empirical formula mass} \times n \]
where \( n \) is the factor by which the molecular formula is greater than the empirical formula.
7. **Calculate \( n \):**
\[ n = \frac{\text{Molar mass given}}{\text{Empirical formula mass}} \]
8. **Write the molecular formula:**
Multiply the subscripts in the empirical formula by \( n \).
This process will give you both the empirical and molecular formulas for the arene compound.