Step 1: Composition of \( \mathrm{Fe_3O_4} \):
\( \mathrm{Fe_3O_4} \) is a mixed oxide of \( \mathrm{FeO} \) and \( \mathrm{Fe_2O_3} \). In this structure:
Step 2: Average Oxidation State Calculation:
The formula \( \mathrm{Fe_3O_4} \) contains three iron atoms with the following oxidation states:
\[ \text{Oxidation states:} \quad 1 \, (\mathrm{Fe}^{+2}) + 2 \, (\mathrm{Fe}^{+3}). \]
The sum of oxidation numbers for all iron atoms is:
\[ (+2) + 2(+3) = +8. \]
The average oxidation state of iron is calculated as:
\[ \frac{+8}{3}. \]
Therefore, the oxidation state of \( \mathrm{Fe} \) in \( \mathrm{Fe_3O_4} \) is \( \mathbf{+8/3} \).