Answer
$3.72$ g of $C$
Work Step by Step
1. Calculate the molar mass of $Fe_2O_3$:
$Fe: 55.85g * 2= 111.7g $
$O: 16.00g * 3= 48.00g $
111.7g + 48.00g = 159.7g
$ \frac{1 mole (Fe_2O_3)}{ 159.7g (Fe_2O_3)}$ and $ \frac{ 159.7g (Fe_2O_3)}{1 mole (Fe_2O_3)}$
2. The balanced reaction is:
$Fe_2O_3 + 3C --\gt 2Fe + 3CO$
According to the coefficients, the ratio of $Fe_2O_3$ to $C$ is 1 to 3:
$ \frac{ 3 moles(C)}{ 1 mole(Fe_2O_3)}$ and $ \frac{ 1 mole (Fe_2O_3)}{ 3 moles(C)}$
3. Calculate the molar mass for $C$:
$C: 12.01g$
$ \frac{1 mole (C)}{ 12.01g (C)}$ and $ \frac{ 12.01g (C)}{1 mole (C)}$
4. Use the conversion factors to find the mass of $C$
$16.5g(Fe_2O_3) \times \frac{1 mole(Fe_2O_3)}{ 159.7g( Fe_2O_3)} \times \frac{ 3 moles(C)}{ 1 mole (Fe_2O_3)} \times \frac{ 12.01 g (C)}{ 1 mole (C)} = 3.72g (C)$