Hydrogen gas, \(\mathrm{H}_{2}(\mathrm{g}),\) is passed over \(\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{s})\) at \(400^{\circ} \mathrm{C} .\) Water vapor is formed together with a black residue-a compound consisting of \(72.3 \% \mathrm{Fe}\) and \(27.7 \%\) O. Write a balanced equation for this reaction.

To convert percentages by weight in a compound to mole ratios, it is necessary first to convert the weight percentages to grams. Assuming a 100g sample, then there will be 72.3g of Fe and 27.7g of O. Since mole ratios are required, these weights are then converted to moles using the molar mass of each element. The molar mass of Fe is approximately \(55.85 g/mol\) while for O, it's \(16.00 g/mol\). Thus, the number of moles of Fe is \(72.3g/55.85 g/mol \approx 1.29 mol\) and for O it's \(27.7g/16.00g/mol \approx 1.73 mol\).

Empirical formula represents the ratio of atoms in a compound, which is derived from the mole amounts of its constituent elements. Dividing each of these mole quantities by the smallest of them will give the empirical formula. Thus, as \(1.29 mol/1.29 mol \approx 1\) and \(1.73 mol/1.29 mol \approx 1.34\), we can't have fractional amounts in an empirical formula therefore we round the numbers to the closest whole number. Thus, the empirical formula for the compound is FeO.

Hydrogen reacts with iron(III) oxide to produce the compound and water. The reaction is written as follows: \(H_{2}(g) + Fe_{2}O_{3}(s) \rightarrow FeO(s) + H_{2}O(g)\). To balance this equation it is evident that there should be 3 molecules of \(H_2\) and 1 molecule of \(Fe_{2}O_{3}\) on the left hand side as one molecule of \(Fe_{2}O_{3}\) can give 2 molecules of 'Fe' necessary for 2 molecules of \(FeO\), and 3 molecules of \(H_2\) can give 3 molecules of \(H_2O\). However, 2 molecules of \(FeO\) gives 2 'O' atoms which along with the one 'O' atom from \(H_{2}O\) gives a total of 3 'O' atoms matching the left hand side. This means the balanced reaction would be: \(3H_{2}(g) + Fe_{2}O_{3}(s) \rightarrow 2FeO(s) + 3H_{2}O(g)\)