Question

Which of the following has more atoms: \(0.302 \mathrm{~g}\) of hydrogen atoms or \(14.7 \mathrm{~g}\) of chromium atoms?

Short answer

Hydrogen has more atoms.

Step-by-step solution

Find the Moles of Hydrogen Atoms

First, we need to calculate the number of moles of hydrogen using its mass. The molar mass of hydrogen is approximately \(1.01 \mathrm{~g/mol}\). Use the formula \(n = \frac{m}{M}\), where \(n\) is the number of moles, \(m\) is the mass, and \(M\) is the molar mass. Substituting the given values, we get:\[n = \frac{0.302 \mathrm{~g}}{1.01 \mathrm{~g/mol}} \approx 0.299 \text{ mol}\]

Find the Moles of Chromium Atoms

Next, we calculate the number of moles of chromium. The molar mass of chromium is approximately \(51.996 \mathrm{~g/mol}\). Again using \(n = \frac{m}{M}\), substitute the given values:\[n = \frac{14.7 \mathrm{~g}}{51.996 \mathrm{~g/mol}} \approx 0.283 \text{ mol}\]

Compare the Number of Atoms

Each mole of a substance contains Avogadro's number of atoms, \(6.022 \times 10^{23}\). Therefore, the number of atoms is directly proportional to the number of moles. For hydrogen, the number of atoms is \(0.299 \times 6.022 \times 10^{23}\), and for chromium, it is \(0.283 \times 6.022 \times 10^{23}\). Since \(0.299\) moles is greater than \(0.283\) moles, hydrogen has more atoms.

Key concepts

Molar Mass

Understanding molar mass is crucial in the world of chemistry. Molar mass is essentially the mass of a given substance expressed in grams per mole. It allows chemists to convert between the mass of a substance and the number of moles, establishing a relationship between the macroscopic amounts we see and the atomic scale quantities we study.
To calculate the molar mass, you must sum the atomic masses of all the atoms present in a molecule, utilizing the periodic table. For example, hydrogen, with an atomic mass of approximately 1.01 g/mol, means that 1 mole of hydrogen atoms weighs 1.01 grams. Chromium, on the other hand, has a molar mass of around 51.996 g/mol. Thus, to find how many moles are in a given mass, use the formula:

• \( n = \frac{m}{M} \)
• where \( n \) is the number of moles, \( m \) is the mass in grams, and \( M \) is the molar mass in grams per mole.

Knowing molar mass helps in calculating the number of moles from a given mass, as seen in exercises such as determining which substance has more atoms.

Avogadro's Number

Avogadro's Number is a fundamental constant in chemistry that connects moles to individual atoms. This large number, approximately \( 6.022 \times 10^{23} \), represents how many atoms, molecules, or particles are contained in one mole of a substance. Imagine it as a bridge linking the micro-world of atoms and molecules to the macro-world that we can observe.
To visualize its use, consider that when you calculate the number of moles using molar mass, you can further apply Avogadro's Number to get the total number of atoms or molecules involved. For instance, if you have 0.299 moles of hydrogen, you would multiply it by Avogadro's Number to find the number of hydrogen atoms:

• Number of atoms = \( 0.299 \times 6.022 \times 10^{23} \).

This concept is critical when comparing quantities of different elements and determining which sample contains more atoms.

Atomic Comparison

The comparison of atomic quantities often involves contrasting different elements. When you are trying to figure out which sample contains more atoms, you must take both the molar mass and the number of moles into account.
In our exercise, we compared 0.302 grams of hydrogen and 14.7 grams of chromium. Using their respective molar masses, we calculated moles for each: 0.299 moles of hydrogen and 0.283 moles of chromium. Despite the larger mass of chromium, the lighter atomic weight of hydrogen enables more moles and thus more atoms for a given mass.
Each mole, regardless of the element, always contains the same number of atoms, thanks to Avogadro's Number. Therefore, when you find that one sample has more moles, it automatically means that it has more atoms. This method of comparison is an essential part of understanding atomic-level differences in chemistry.