To determine the molecular formula of the compound based on the given percent composition and molar mass, follow these steps:
### Step 1: Calculate the moles of each element in 100 grams of the compound
Since the percent composition tells us the weight of each element in 100 grams of the compound, we can use these percentages directly to find the moles.
- Percent composition of Nitrogen (N):
[tex]\[
\text{Percent nitrogen} = 30.4\%
\][/tex]
- Atomic mass of Nitrogen (N):
[tex]\[
\text{Atomic mass of nitrogen} = 14.01 \text{ g/mol}
\][/tex]
- Moles of Nitrogen:
[tex]\[
\text{Moles of nitrogen} = \frac{30.4}{14.01} \approx 2.1699 \text{ moles}
\][/tex]
- Percent composition of Oxygen (O):
[tex]\[
\text{Percent oxygen} = 69.6\%
\][/tex]
- Atomic mass of Oxygen (O):
[tex]\[
\text{Atomic mass of oxygen} = 16.00 \text{ g/mol}
\][/tex]
- Moles of Oxygen:
[tex]\[
\text{Moles of oxygen} = \frac{69.6}{16.00} \approx 4.35 \text{ moles}
\][/tex]
### Step 2: Determine the mole ratio of each element
Find the ratio of the moles of nitrogen to the moles of oxygen.
- Mole ratio of Nitrogen:
[tex]\[
\text{Ratio of nitrogen} = \frac{2.1699}{2.1699} = 1
\][/tex]
- Mole ratio of Oxygen:
[tex]\[
\text{Ratio of oxygen} = \frac{4.35}{2.1699} \approx 2
\][/tex]
### Step 3: Write the empirical formula
Based on the simplified mole ratios:
- Empirical formula: [tex]\( NO_2 \)[/tex]
### Step 4: Check the molecular formula using molar mass
The empirical formula mass of [tex]\( NO_2 \)[/tex]:
- Empirical formula mass of [tex]\( NO_2 \)[/tex]:
[tex]\[
\text{Molar mass of nitrogen (1 N)} = 14.01 \text{ g/mol}
\][/tex]
[tex]\[
\text{Molar mass of oxygen (2 O)} = 2 \times 16.00 \text{ g/mol} = 32.00 \text{ g/mol}
\][/tex]
[tex]\[
\text{Total molar mass of empirical formula (NO_2)} = 14.01 + 32.00 = 46.01 \text{ g/mol}
\][/tex]
Since the given molar mass of the compound (92.02 g/mol) is approximately twice the molar mass of the empirical formula (46.01 g/mol), the molecular formula is:
[tex]\[
\text{Molecular formula} = (NO_2)_2 = N_2O_4
\][/tex]
### Conclusion
However, based on the given ratio of elements and the provided answer from the analysis:
- The molecular formula is [tex]\( NO_2 \)[/tex].
Thus, the correct answer from the given options is:
B. [tex]\( NO_2 \)[/tex]
