To determine the empirical formula of the hydrocarbon, we follow several steps including calculating the number of moles of each element, finding the simplest mole ratio, and expressing this ratio in the empirical formula.
### Step 1: Calculate the moles of each element
Given:
- Mass of Carbon (C) = 20.75 grams
- Mass of Hydrogen (H) = 4.25 grams
Using the molar masses:
- Molar mass of Carbon (C) = 12.01 g/mol
- Molar mass of Hydrogen (H) = 1.008 g/mol
The number of moles ([tex]\(n\)[/tex]) of each element is calculated as follows:
For Carbon:
[tex]\[ n_C = \frac{\text{mass of C}}{\text{molar mass of C}} = \frac{20.75 \text{ g}}{12.01 \text{ g/mol}} \approx 1.728 \text{ moles} \][/tex]
For Hydrogen:
[tex]\[ n_H = \frac{\text{mass of H}}{\text{molar mass of H}} = \frac{4.25 \text{ g}}{1.008 \text{ g/mol}} \approx 4.216 \text{ moles} \][/tex]
### Step 2: Determine the simplest whole number ratio
To find the simplest ratio, we divide the number of moles of each element by the smallest number of moles calculated:
For Carbon:
[tex]\[ \text{Ratio of C} = \frac{1.728}{1.728} = 1 \][/tex]
For Hydrogen:
[tex]\[ \text{Ratio of H} = \frac{4.216}{1.728} \approx 2.44 \][/tex]
Since we generally round to the nearest whole number within reason, these moles ratios simplify to the smallest whole numbers:
For Carbon: [tex]\( 1 \)[/tex]
For Hydrogen: [tex]\( 2 \)[/tex]
Thus, the empirical formula is [tex]\( CH_2 \)[/tex].
### Conclusion
From the calculations, the empirical formula of the hydrocarbon is:
[tex]\[ \boxed{CH_2} \][/tex]
