a. Iron (III) oxide reacts with hydrogen gas to form elemental iron and water, according to the balanced equation shown below. How many moles of iron will be formed from the reduction of excess iron (III) oxide by \(0.58\) moles of hydrogen gas? b. When an impure sample containing an unknown amount of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) is reacted with excess hydrogen gas, \(0.16\) moles of solid Fe are formed. How many moles of \(\mathrm{Fe}_{2} \mathrm{O}_{3}\) were in the original sample?

Chemical equations are a formal representation of chemical reactions, depicting the substances involved and their transformation. They are composed of chemical formulas that represent the reactants and products separated by an arrow indicating the direction of the reaction. A key characteristic of a balanced chemical equation is that the number of atoms for each element must be equal on both sides, ensuring that matter is conserved.

For instance, the equation provided in the exercise, \[\mathrm{Fe_2O_3(s) + 3H_2(g) \rightarrow 2Fe(s) + 3H_2O(g)}\] clearly shows the reactant iron (III) oxide (\(\mathrm{Fe_2O_3}\)) and hydrogen gas (\(\mathrm{H_2}\)) transforming into iron (\(\mathrm{Fe}\)) and water (\(\mathrm{H_2O}\)) as products. This equation is deemed balanced because the number of atoms for each element—iron, oxygen, and hydrogen—remains unchanged from the reactants side to the products side.

The mole ratio, derived from the coefficients of a balanced chemical equation, is critical for stoichiometric calculations. It establishes a proportional relationship between the quantities of reactants and products involved in a chemical reaction.

In the provided exercise, the mole ratio between hydrogen gas and iron is derived as follows: for every 3 moles of hydrogen gas used, 2 moles of iron are produced. Mathematically, this ratio is represented as \(\frac{2 \text{ moles Fe}}{3 \text{ moles H}_2}\). It's important to understand that these ratios are based on the coefficients in the balanced equation, and they allow us to convert between moles of different substances involved in the reaction.

Stoichiometric calculations enable us to use the mole ratio from the balanced chemical equation to determine the quantity of one substance, given the quantity of another. These calculations require a step-by-step approach that typically includes: writing out the balanced equation, determining the mole ratio, and then using this ratio to calculate the unknown quantity.

For example, to calculate moles of iron formed from hydrogen gas in part (a) of the exercise, we multiply the given moles of hydrogen gas by the mole ratio, which derives from the balanced equation. To find the moles of iron \(\mathrm{(Fe)}\), the calculation is as follows: \[x = \frac{2}{3} \times 0.58\]\[x = 0.387 \text{ moles of Fe}\]. This process is standard in stoichiometry, and it's pivotal to ensure that all steps and calculations adhere to the mole ratios established by the balanced equation.

A chemical reaction is a process in which reactants are transformed into products through the breaking and forming of chemical bonds. It involves a change in the chemical substances and is often accompanied by visual changes like color change, temperature change, gas formation, or precipitate formation. The reaction must be depicted by a balanced chemical equation that reflects the conservation of mass.

In the given exercise, we see a reduction-oxidation (redox) reaction taking place, where iron (III) oxide reacts with hydrogen gas—a reduction of iron oxide accompanied by the oxidation of hydrogen. This reaction not only illustrates the conversion of substances but also represents the practical application of stoichiometry in predicting the outcome of a chemical reaction, which is quintessential in fields like chemistry and engineering.