An iron ore sample contains \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) plus other impurities. A 752 g sample of impure iron ore is heated with excess carbon, producing \(453 \mathrm{~g}\) of pure iron by the following reaction: $$ \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}(\mathrm{g}) $$ What is the mass percent of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) in the impure iron ore sample? Assume that \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) is the only source of iron and that the reaction is \(100 \%\) efficient.

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between reactants and products in a chemical reaction. It allows chemists to calculate the amounts of substances needed or produced in a chemical reaction. For example, the balanced equation \[ \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}(\mathrm{g}) \] indicates that one mole of iron(III) oxide reacts with three moles of carbon to produce two moles of iron and three moles of carbon monoxide. The coefficients in the balanced equation are essential for stoichiometric calculations as they represent the molar ratios of the substances involved.

Molar mass is a fundamental concept in stoichiometry. It is the mass of one mole of a substance, usually expressed in grams per mole (g/mol), and it allows for the conversion between the mass of a substance and the number of moles. The molar mass is calculated by summing the atomic masses of all the atoms in a molecule. For instance, the molar mass of Fe₂O₃ is found by adding together the atomic masses of two iron atoms and three oxygen atoms, resulting in \[ 2 \times 55.85 + 3 \times 16.00 = 159.70 \frac{\text{g}}{\text{mol}} \]. Knowing the molar mass is crucial when determining the mass percent composition of a compound in a mixture, as seen in the iron ore sample problem.

Balancing chemical equations is a vital skill in chemistry. It ensures that the law of conservation of mass is respected, meaning the number of atoms for each element is the same on the reactant side as on the product side of a chemical reaction. A balanced chemical equation provides the mole ratios necessary to perform stoichiometric calculations. The given reaction \[ \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{C}(s) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}(\mathrm{g}) \] is already balanced, with the reactants and products having a stoichiometric ratio of 1:3:2:3 respectively. These ratios are used to calculate how much reactant is needed to produce a certain amount of product or vice versa.

The mole concept is the cornerstone of stoichiometry, which provides a method for counting the number of atoms, molecules, or formula units in a sample of matter. One mole is defined as the amount of substance that contains as many entities as there are atoms in 12 grams of pure carbon-12, which is approximately \(6.022 \times 10^{23}\) entities (Avogadro's number). Using moles, we can relate the mass of a substance to the number of particles it contains. In the iron ore problem, by converting the mass of iron produced to moles, \[ \text{Number of moles of Fe} = \frac{453 \;\text{g}}{55.85 \;\frac{\text{g}}{\text{mol}}} \approx 8.11\;\text{moles} \], we can use the mole ratio from the balanced equation to find out how many moles of Fe₂O₃ were used, and then, by using the molar mass of Fe₂O₃, arrive at the mass of Fe₂O₃ in the sample.