Let's solve this step-by-step:
1. Identify the given data and molar masses:
- Mass of iron (Fe): 44.2 grams
- Molar mass of iron (Fe): 55.845 g/mol
- Molar mass of oxygen gas (O₂): 32.00 g/mol
2. Calculate the moles of iron (Fe):
- Number of moles is given by the formula: [tex]\( \text{moles} = \frac{\text{mass}}{\text{molar mass}} \)[/tex]
- So, the moles of iron can be calculated as:
[tex]\[ \text{moles of Fe} = \frac{44.2 \text{ grams}}{55.845 \text{ g/mol}} \][/tex]
3. Determine the stoichiometric relationship from the balanced chemical equation:
- The balanced equation is:
[tex]\[ 4 Fe + 3 O_2 \rightarrow 2 Fe_2O_3 \][/tex]
- From the equation, 4 moles of Fe react with 3 moles of O₂.
4. Calculate the moles of O₂ needed:
- According to the stoichiometric ratio, for every 4 moles of Fe, 3 moles of O₂ are required:
[tex]\[ \text{moles of O}_2 = \left( \frac{3 \text{ moles O}_2}{4 \text{ moles Fe}} \right) \times \text{moles of Fe} \][/tex]
5. Calculate the mass of O₂ needed:
- Number of moles of oxygen gas can be converted to mass by multiplying by its molar mass:
[tex]\[ \text{mass of O}_2 = \text{moles of O}_2 \times 32.00 \text{ g/mol} \][/tex]
Following these steps with the initial values provided:
- Calculate moles of Fe:
[tex]\[ \text{moles of Fe} = \frac{44.2 \text{ grams}}{55.845 \text{ g/mol}} \approx 0.791 \text{ moles} \][/tex]
- Determine the moles of O₂ required:
[tex]\[ \text{moles of O}_2 = \left( \frac{3}{4} \right) \times 0.791 \approx 0.593 \text{ moles} \][/tex]
- Calculate the mass of O₂ needed:
[tex]\[ \text{mass of O}_2 = 0.593 \times 32.00 \text{ g/mol} \approx 18.99 \text{ grams} \][/tex]
Therefore, the mass of oxygen gas needed is approximately [tex]\( 19.0 \)[/tex] grams.
Among the options given:
- 19.0 g
- 101 g
- 12.7 g
- 25.3 g
The correct answer is [tex]\( 19.0 \)[/tex] grams.
