To determine the empirical formula, the steps involve converting the masses of each element to moles, then finding the simplest whole number ratio of moles of each element.
### Step 1: Calculate the number of moles of each element
1. Lead (Pb)
Given mass = 38.43 g
Molar mass of Pb = 207.2 g/mol
[tex]\[
\text{Moles of Pb} = \frac{38.43 \text{ g}}{207.2 \text{ g/mol}} = 0.185472972972973 \text{ moles}
\][/tex]
2. Carbon (C)
Given mass = 17.83 g
Molar mass of C = 12.01 g/mol
[tex]\[
\text{Moles of C} = \frac{17.83 \text{ g}}{12.01 \text{ g/mol}} = 1.484596169858451 \text{ moles}
\][/tex]
3. Hydrogen (H)
Given mass = 3.74 g
Molar mass of H = 1.008 g/mol
[tex]\[
\text{Moles of H} = \frac{3.74 \text{ g}}{1.008 \text{ g/mol}} = 3.7103174603174605 \text{ moles}
\][/tex]
### Step 2: Calculate the mole ratio
To find the simplest ratio, divide the number of moles of each element by the smallest number of moles calculated.
[tex]\[
\text{Smallest number of moles} = 0.185472972972973 \text{ moles (Pb)}
\][/tex]
1. Ratio for Pb
[tex]\[
\text{Ratio for Pb} = \frac{0.185472972972973 \text{ moles}}{0.185472972972973} = 1.0
\][/tex]
2. Ratio for C
[tex]\[
\text{Ratio for C} = \frac{1.484596169858451 \text{ moles}}{0.185472972972973} = 8.004380077925346
\][/tex]
3. Ratio for H
[tex]\[
\text{Ratio for H} = \frac{3.7103174603174605 \text{ moles}}{0.185472972972973} = 20.004626015554976
\][/tex]
### Step 3: Round the ratios to the nearest whole numbers
The ratios are close to 1, 8, and 20. Therefore:
- 1 for Pb
- 8 for C
- 20 for H
### Conclusion: Empirical Formula
From the ratios, the empirical formula is:
[tex]\[
\text{Empirical formula} = Pb C_8 H_{20}
\][/tex]
So, the correct choice from the given options is:
[tex]\[
\boxed{PbC_8H_{20}}
\][/tex]
