Question

A sample of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) has a mass of 52.0 \(\mathrm{g}\) a. How many carbon atoms are present? b. How many oxygen atoms are present? c. What is the mass in grams of one molecule of \(\mathrm{CO}_{2} ?\)

Short answer

a) Approximately 7.12 * 10²³ carbon atoms are present. b) Approximately 14.24 * 10²³ oxygen atoms are present. c) The mass of one CO₂ molecule is approximately 8.63 * 10⁻²³ grams.

Step-by-step solution

Calculate the number of moles of CO₂

To find the number of carbon and oxygen atoms, we will first determine the number of moles of CO₂ present. The molar mass of CO₂ is approximately 12 g/mol (C) + 2 * 16 g/mol (O) = 44 g/mol. Divide the given mass (52.0 g) by the molar mass to find the moles of CO₂: Number of moles = \( \frac{52.0\,g}{44\,g/mol} \) Number of moles = 1.182 moles of CO₂

Find the number of carbon atoms

There is one carbon atom in each CO₂ molecule. To find the total number of carbon atoms, we need to multiply the number of moles of CO₂ by Avogadro's number (6.022 * 10²³): Number of Carbon atoms = 1.182 moles * 6.022 * 10²³ atoms/mol ≈ 7.12 * 10²³ carbon atoms.

Find the number of oxygen atoms

There are two oxygen atoms in each CO₂ molecule. To find the total number of oxygen atoms, we need to multiply the number of moles of CO₂ by Avogadro's number (6.022 * 10²³) and then multiply by 2 (for the two oxygen atoms in CO₂): Number of Oxygen atoms = 1.182 moles * 6.022 * 10²³ atoms/mol * 2 ≈ 14.24 * 10²³ oxygen atoms.

Find the mass of one CO₂ molecule

First, find the mass of one mole of CO₂ by multiplying the molar mass by the number of moles: Mass of one mole of CO₂ = 1.182 moles * 44 g/mol = 52.0 g Now, divide this mass by Avogadro's number (6.022 * 10²³) to find the mass of one CO₂ molecule: Mass of one CO₂ molecule = \( \frac{52.0\,g}{6.022 * 10^{23}\, molecules} \) ≈ 8.63 * 10⁻²³ g So, the answers are: a) There are approximately 7.12 * 10²³ carbon atoms in the sample. b) There are approximately 14.24 * 10²³ oxygen atoms in the sample. c) The mass of one molecule of CO₂ is approximately 8.63 * 10⁻²³ grams.

Key concepts

Molar Mass

The molar mass is a key concept when dealing with chemical calculations. It represents the mass of one mole of a substance, typically expressed in grams per mole (g/mol). For carbon dioxide \(\text{CO}_2\), determining the molar mass involves adding the atomic masses of its constituent atoms. Carbon has an atomic mass of roughly 12 g/mol, and each oxygen atom has an atomic mass of approximately 16 g/mol. Hence, the molar mass of carbon dioxide is:

• 12 g/mol (for carbon)
• 2 × 16 g/mol (for two oxygen atoms)

This results in a total molar mass of 44 g/mol for \(\text{CO}_2\).
This calculation is crucial when converting between mass and moles in chemistry, allowing for precise stoichiometric calculations in reactions and laboratory work.

Avogadro's Number

Avogadro's number, which is approximately \(6.022 \times 10^{23}\), is a fundamental constant in chemistry. It represents the number of constituent particles—typically atoms or molecules—contained in one mole of a given substance.
This means that whether you have a mole of carbon atoms, water molecules, or carbon dioxide molecules, you will always have \(6.022 \times 10^{23}\) of the respective particle.
When solving problems involving molecules like \(\text{CO}_2\), Avogadro's number provides the bridge between the macroscopic scale of grams and the molecular scale. Therefore, to find the total number of atoms in a sample, you multiply the number of moles by Avogadro's number. This conversion allows us to understand chemical reactions at a more fundamental level.

Molecular Mass

Molecular mass refers to the sum of the atomic masses of all atoms in a molecule. It is intimately related to molar mass but used when discussing individual molecules or small amounts of substances.
For \(\text{CO}_2\), the molecular mass calculation involves adding the atomic masses of one carbon atom (approximately 12 amu) and two oxygen atoms (approximately 16 amu each). Thus, the molecular mass of carbon dioxide is 44 amu.
Understanding molecular mass is vital in calculations that involve comparing the masses of single molecules and complexes. It enables scientists and students to precisely grasp how much one molecule or a small collection of molecules would weigh, further facilitating advanced studies in chemical kinetics and reaction mechanisms.

Carbon Dioxide Calculations

Calculating properties of carbon dioxide \(\text{CO}_2\) requires understanding both the compound's structure and appropriate chemical principles.
From the mass of \(52.0\,\text{g}\) of \(\text{CO}_2\) given, we first need to calculate how many moles this corresponds to, using the molar mass discussed earlier (44 g/mol). Dividing the given mass by the molar mass gives the number of moles of \(\text{CO}_2\).

• This result is then used to determine the number of atoms using Avogadro's number, highlighting that every mole consists of \(6.022 \times 10^{23}\) molecules.
• Knowing one \(\text{CO}_2\) molecule contains one carbon atom and two oxygen atoms helps to calculate total atom counts.

Finally, the mass of a single \(\text{CO}_2\) molecule is determined by dividing the total mass by Avogadro's number, offering insight into the mass on a microscopic level. These calculations are pivotal in fields ranging from environmental science to respiratory physiology, where precise measurements of \(\text{CO}_2\) are imperative.