Let's analyze the given chemical equation:
[tex]\[
4 \text{Fe} + 3 \text{O}_2 \rightarrow 2 \text{Fe}_2\text{O}_3
\][/tex]
This equation tells us that 4 moles of iron ([tex]\(\text{Fe}\)[/tex]) react with 3 moles of oxygen ([tex]\(\text{O}_2\)[/tex]) to produce 2 moles of iron(III) oxide ([tex]\(\text{Fe}_2\text{O}_3\)[/tex]).
To determine the appropriate mole ratio to use for finding the mass of iron ([tex]\(\text{Fe}\)[/tex]) from a known mass of iron(III) oxide ([tex]\(\text{Fe}_2\text{O}_3\)[/tex]), we need to focus on the coefficients of these two substances in the balanced equation.
The coefficients of Fe and Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex] are:
- Fe: 4
- Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex]: 2
Therefore, the mole ratio of Fe to Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex] is:
[tex]\[
\frac{\text{Fe}}{\text{Fe}_2\text{O}_3} = \frac{4 \text{ moles Fe}}{2 \text{ moles Fe}_2\text{O}_3}
\][/tex]
Simplifying this fraction, we get:
[tex]\[
\frac{4}{2} = 2.0
\][/tex]
So the fraction that can be used for the mole ratio to determine the mass of Fe from a known mass of Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex] is:
[tex]\[
\frac{4}{2}
\][/tex]
The other options are:
- [tex]\(\frac{3}{2}\)[/tex]: This is the ratio of O[tex]\(_2\)[/tex] to Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex], which is not relevant in this context.
- [tex]\(\frac{2}{3}\)[/tex]: This could be considered for the reverse reaction involving Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex] decomposing to Fe and O[tex]\(_2\)[/tex], but it is not the direct ratio of Fe to Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex].
- [tex]\(\frac{2}{4}\)[/tex] or [tex]\(\frac{1}{2}\)[/tex]: This represents the ratio of Fe[tex]\(_2\)[/tex]O[tex]\(_3\)[/tex] to Fe, which is the inverse of what we need.
Thus, the correct fraction for the mole ratio is:
[tex]\[
\frac{4}{2}
\][/tex]
