To determine the number of moles of [tex]\( SnO_2 \)[/tex] needed to produce 500.0 grams of tin ([tex]\( Sn \)[/tex]), we'll follow these steps:
1. Calculate the moles of tin ([tex]\( Sn \)[/tex]) produced:
- First, use the given mass of [tex]\( Sn \)[/tex] and the molar mass of [tex]\( Sn \)[/tex] to find the number of moles.
- The molar mass of tin ([tex]\( Sn \)[/tex]) is [tex]\( 118.71 \, \text{g/mol} \)[/tex].
[tex]\[
\text{Moles of } Sn = \frac{\text{Mass of } Sn}{\text{Molar mass of } Sn}
\][/tex]
Substituting the given values:
[tex]\[
\text{Moles of } Sn = \frac{500.0 \, \text{g}}{118.71 \, \text{g/mol}} \approx 4.21 \, \text{moles}
\][/tex]
2. Relate the moles of [tex]\( Sn \)[/tex] to the moles of [tex]\( SnO_2 \)[/tex] needed:
- From the balanced chemical equation [tex]\( SnO_2 + 2H_2 \rightarrow Sn + 2H_2O \)[/tex], we see that 1 mole of [tex]\( SnO_2 \)[/tex] produces 1 mole of [tex]\( Sn \)[/tex].
- Therefore, the moles of [tex]\( SnO_2 \)[/tex] needed is equal to the moles of [tex]\( Sn \)[/tex] produced.
[tex]\[
\text{Moles of } SnO_2 = \text{Moles of } Sn = 4.21 \, \text{moles}
\][/tex]
Therefore, the number of moles of [tex]\( SnO_2 \)[/tex] needed to produce 500.0 grams of [tex]\( Sn \)[/tex] is approximately 4.21 moles.
So, the correct answer is:
[tex]\[
\boxed{4.21}
\][/tex]
