- Identify the 1:2 stoichiometric ratio of \(\text{Fe}_2\text{O}_3\) to \(\text{Fe}\) from the balanced equation.
- Multiply the given \(2.50\,\text{mol}\) of \(\text{Fe}_2\text{O}_3\) by the ratio of 2.
- Apply significant figure rules to obtain the final value of 5.00.
- Final answer: \(\boxed{5.00}\) \(\text{mol Fe}\).
Explanation
- Stoichiometry Analysis
The balanced chemical equation is:
\[\text{Fe}_2\text{O}_3 + 2\text{Al} \rightarrow \text{Al}_2\text{O}_3 + 2\text{Fe}\]
This equation shows the stoichiometry between the reactants and the products. Specifically, the reaction indicates that one mole of iron(III) oxide (\(\text{Fe}_2\text{O}_3\)) reacts with two moles of aluminum (\(\text{Al}\)) to produce one mole of aluminum oxide (\(\text{Al}_2\text{O}_3\)) and two moles of iron (\(\text{Fe}\)). Therefore, the mole-to-mole ratio between iron(III) oxide and iron is \(1:2\).
- Calculation of Iron Production
Given the mole-to-mole ratio from the balanced equation, we can determine the amount of iron produced using the amount of iron(III) oxide provided. The ratio is expressed as:
\[\frac{\text{mol Fe}}{\text{mol Fe}_2\text{O}_3} = \frac{2}{1}\]
We are given \(2.50\,\text{mol}\) of \(\text{Fe}_2\text{O}_3\). To find the moles of \(\text{Fe}\) produced, we multiply the given moles of \(\text{Fe}_2\text{O}_3\) by the stoichiometric ratio:
\[\text{Moles of Fe} = 2.50\,\text{mol Fe}_2\text{O}_3 \times \left( \frac{2\,\text{mol Fe}}{1\,\text{mol Fe}_2\text{O}_3} \right)\]
\[\text{Moles of Fe} = 5.00\,\text{mol Fe}\]
- Final Answer and Significant Figures
The initial quantity provided, \(2.50\,\text{mol}\), contains three significant figures. Stoichiometric coefficients in a balanced chemical equation are considered exact numbers and do not limit the precision of the calculation. Thus, our final answer must be reported to three significant figures, which gives us \(5.00\,\text{mol}\). The final answer is \(\boxed{5.00}\) \(\text{mol Fe}\).
Examples
This reaction, known as the thermite reaction, is a spectacular example of a redox process where aluminum reduces metal oxides. In practical applications, this process is used by crews to weld underwater structures and rail tracks, as it generates enough heat to melt the metal, creating a strong bond. Mastering stoichiometry allows engineers to calculate exactly how much reactant is needed to produce the desired amount of molten iron for these critical repairs.