Let's find out the molecular formula by following these steps:
1. Identify the given information:
- Molecular weight: \( 413.44 \) g/mol
- Percent compositions:
- \( \text{Ba} \): \( 20.38\% \)
- \( \text{Ti} \): \( 11.58\% \)
- \( \text{Si} \): \( 20.38\% \)
- \( \text{O} \): \( 34.83\% \)
- Atomic weights:
- \( \text{Ba} \): \( 137.33 \) g/mol
- \( \text{Ti} \): \( 47.87 \) g/mol
- \( \text{Si} \): \( 28.09 \) g/mol
- \( \text{O} \): \( 16.00 \) g/mol
2. Calculate the moles of each element using the percent composition:
- Moles of \( \text{Ba} \):
[tex]\[
\frac{20.38 \, \text{g}}{137.33 \, \text{g/mol}} = 0.001484 \, \text{moles}
\][/tex]
- Moles of \( \text{Ti} \):
[tex]\[
\frac{11.58 \, \text{g}}{47.87 \, \text{g/mol}} = 0.002419 \, \text{moles}
\][/tex]
- Moles of \( \text{Si} \):
[tex]\[
\frac{20.38 \, \text{g}}{28.09 \, \text{g/mol}} = 0.007255 \, \text{moles}
\][/tex]
- Moles of \( \text{O} \):
[tex]\[
\frac{34.83 \, \text{g}}{16.00 \, \text{g/mol}} = 0.021769 \, \text{moles}
\][/tex]
3. Determine the smallest number of moles:
- Smallest moles: \( 0.001484 \, \text{moles of Ba} \)
4. Calculate the ratio of each element relative to the smallest number of moles:
- Ratio of \( \text{Ba} \):
[tex]\[
\frac{0.001484}{0.001484} = 1
\][/tex]
- Ratio of \( \text{Ti} \):
[tex]\[
\frac{0.002419}{0.001484} \approx 2
\][/tex]
- Ratio of \( \text{Si} \):
[tex]\[
\frac{0.007255}{0.001484} \approx 5
\][/tex]
- Ratio of \( \text{O} \):
[tex]\[
\frac{0.021769}{0.001484} \approx 15
\][/tex]
5. Summarize the ratios:
- \( \text{Ba:Ti:Si:O} = 1:2:5:15 \)
So, the molecular formula is \( \text{BaTi}_2\text{Si}_5\text{O}_{15} \). Based on the given options, none of them exactly match the formula we calculated. However, considering the options given, it appears there might have been a mistake with the provided question or choices.
Given this discrepancy, please double-check the formula and options if possible or confirm the problem details allowing us to make the best match to option A, B, C, or D. Based on our calculations, this isn’t directly reflected in any provided choices.
