Question

How many moles of \(\mathrm{Cr}\) are in \(2.16 \mathrm{~mol}\) of \(\mathrm{Cr}_{2} \mathrm{O}_{3}\) ?

Short answer

There are 4.32 moles of \(\mathrm{Cr}\) in 2.16 mol of \(\mathrm{Cr}_2\mathrm{O}_3\).

Step-by-step solution

Understanding the Molecule

Consider the chemical formula of the compound, \(\mathrm{Cr}_2\mathrm{O}_3\). This indicates that there are 2 moles of chromium (\mathrm{Cr}) for every mole of chromium(III) oxide (\mathrm{Cr}_2\mathrm{O}_3).

Calculating Moles of Chromium

To find the number of moles of chromium, multiply the given moles of chromium(III) oxide by the ratio of moles of chromium to moles of chromium(III) oxide.\[\text{Moles of } \mathrm{Cr} = \text{Moles of } \mathrm{Cr}_2\mathrm{O}_3 \times \frac{2 \text{ moles } \mathrm{Cr}}{1 \text{ mole } \mathrm{Cr}_2\mathrm{O}_3}\]

Performing the Calculation

Plug the given amount into the equation to get the number of moles of chromium.\[\text{Moles of } \mathrm{Cr} = 2.16 \text{ mol } \mathrm{Cr}_2\mathrm{O}_3 \times \frac{2 \text{ moles } \mathrm{Cr}}{1 \text{ mole } \mathrm{Cr}_2\mathrm{O}_3} = 4.32 \text{ mol } \mathrm{Cr}\]

Key concepts

Understanding the Mole Concept

The mole concept is a fundamental aspect of chemistry that provides a bridge between the microscopic world of atoms and molecules and the macroscopic world we interact with. It's defined as the amount of substance containing as many particles—such as atoms, molecules, ions, or electrons—as there are atoms in 12 grams of carbon-12. This number is known as Avogadro's number, approximately equal to \(6.022 \times 10^{23}\) particles, and it's this large number that makes the mole concept so useful in chemistry.

When working with reactions and compounds, dealing with individual particles is impractical. Instead, we use the mole, which allows us to count particles in a way that can be weighed and measured on a scale we can easily handle. For example, the exercise asking about the moles of \(\mathrm{Cr}\) in \(2.16 \mathrm{~mol}\) of \(\mathrm{Cr}_2\mathrm{O}_3\) is probing your understanding of how many atoms of chromium are present in a macroscopic amount of chromium(III) oxide.

Deciphering the Chemical Formula

A chemical formula represents the proportions of atoms of each element in a compound. Take \(\mathrm{Cr}_2\mathrm{O}_3\), for instance—it's the chemical formula for chromium(III) oxide. The subscripts in the formula indicate the number of atoms of each element in one molecule of the compound. Here, the '2' next to \(\mathrm{Cr}\) implies there are two chromium atoms, and the '3' next to \(\mathrm{O}\) means there are three oxygen atoms within a single molecule of chromium(III) oxide.

Understanding the chemical formula is crucial because it allows us to see the ratio of elements in the compound, which directly informs stoichiometric calculations. When a problem asks how many moles of an element are in a mole of a compound, it's the chemical formula that provides the necessary information to answer the question.

Mastering Stoichiometric Calculations

Stoichiometric calculations involve quantitatively analyzing the reactants and products in a chemical reaction, commonly using the mole concept. These calculations start with a balanced chemical equation, which gives the ratio in which reactants combine and products form. With stoichiometry, we can determine the amount of substance needed or produced by factoring these ratios into our calculations.

In our example problem, we perform a stoichiometric calculation by using the given moles of \(\mathrm{Cr}_2\mathrm{O}_3\) to find the moles of \(\mathrm{Cr}\). By knowing the ratio from the chemical formula (2 moles of \(\mathrm{Cr}\) for every 1 mole of \(\mathrm{Cr}_2\mathrm{O}_3\)), we can simply multiply to get the answer. This is the essence of stoichiometry: using ratios and the mole concept to bridge the gap between the molecular scale and measurable quantities.