Question

Oxygen is required for the metabolic combustion of foods. Calculate the number of atoms in \(38.0 \mathrm{~g}\) of oxygen gas, the amount absorbed from the lungs in about 15 min when a person is at rest.

Short answer

There are approximately \(1.4298 \times 10^{24}\) oxygen atoms in 38.0 grams of oxygen gas.

Step-by-step solution

Determine the Molar Mass of Oxygen Gas

Oxygen gas (\text{O}_2) has two oxygen atoms. The atomic mass of one oxygen atom is approximately 16.00 \text{u}, so the molar mass of \text{O}_2 is \[M_{\text{O}_2} = 2 \times 16.00 \text{ g/mol} = 32.00 \text{ g/mol}\]

Calculate the Number of Moles of Oxygen Gas

Use the molar mass to convert grams of oxygen gas to moles. The formula is \[n = \frac{m}{M}\]where \(m\) is the mass in grams and \(M\) is the molar mass. Substituting the values gives \[n = \frac{38.0 \text{ g}}{32.00 \text{ g/mol}} = 1.1875 \text{ moles}\]

Calculate the Number of Molecules

Use Avogadro's number (\(6.022 \times 10^{23}\text{ molecules/mol}\)) to find the number of molecules. The formula is \[N = n \times N_A\]Substituting the values gives \[N = 1.1875 \text{ moles} \times 6.022 \times 10^{23}\text{ molecules/mol} = 7.149 \times 10^{23} \text{ molecules}\]

Calculate the Number of Atoms

Each molecule of \text{O}_2 contains 2 atoms of oxygen. Thus, the number of atoms is \[N_{\text{atoms}} = N \times 2\]Substituting the values gives \[N_{\text{atoms}} = 7.149 \times 10^{23} \text{ molecules} \times 2 = 1.4298 \times 10^{24} \text{ atoms}\]

Key concepts

Avogadro's number

Avogadro's number is a fundamental constant in chemistry. It tells us how many particles, such as atoms or molecules, are in one mole of a substance. Avogadro's number is given as:\[ N_A = 6.022 \times 10^{23} \text{ particles/mol} \]This large number helps connect the microscopic world of atoms and molecules with the macroscopic world we can measure, like grams and liters. When you know the number of moles of a substance, you can use Avogadro's number to find out the exact number of atoms or molecules in that sample.For example, if you have 1 mole of carbon atoms, there are \(6.022 \times 10^{23}\) carbon atoms. This concept is particularly useful in calculating reactions and quantities in chemistry.

Moles to atoms conversion

Converting moles to atoms can be straightforward once you grasp the basics. Here's the step-by-step approach to do it.

• Step 1: Find the number of moles of the substance. Let's say you have the number of moles as \(1.1875\).
• Step 2: Use Avogadro's number to find the number of molecules. Multiply the moles by Avogadro's number: \(1.1875 \times 6.022 \times 10^{23}\). This gives you the number of molecules, which is \(7.149 \times 10^{23}\).
• Step 3: Finally, if each molecule contains multiple atoms, multiply the number of molecules by the number of atoms in each molecule.

For an oxygen molecule (\text{O}_2), there are 2 atoms of oxygen per molecule. So, using our example, \(7.149 \times 10^{23}\text{ molecules}\) would equal \(1.4298 \times 10^{24}\text{ atoms}\).Thus, you have successfully converted the number of moles to the number of atoms!

Molar mass of O2

Understanding the molar mass is crucial for solving many chemistry problems. The molar mass of a molecule is the total mass of all the atoms within that molecule, measured in grams per mole (g/mol).For oxygen gas (\text{O}_2), it consists of two oxygen atoms. The atomic mass of a single oxygen atom is roughly 16.00 atomic mass units (u). Thus, the molar mass of \text{O}_2 is:\[M_{\text{O}_2} = 2 \times 16.00 \text{ g/mol} = 32.00 \text{ g/mol}\]This value tells you that one mole of \text{O}_2 weighs 32.00 grams. You can use this molar mass to convert between the mass of a sample and the number of moles it contains.For example, if you have 38.0 grams of \text{O}_2:

• Step 1: 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.
• Step 2: Substitute the values: \(n = \frac{38.0\text{ g}}{32.00\text{ g/mol}} = 1.1875\text{ moles}\).

By understanding the molar mass, you can bridge the gap between mass and the number of particles, making it easier to tackle various chemistry problems.