Problem 52 - Introductory Chemistry Atoms First

Question

Consider a sample containing \(12.011 \mathrm{~g}\) of naturally occurring carbon. (a) How many carbon atoms are in this sample? (b) How many dozens of carbon atoms are in this sample? (c) How many moles of carbon atoms are in this sample?

Step-by-Step Solution

Answer

(a) The sample contains approximately \(6.0225 \times 10^{23}\) carbon atoms. (b) The sample contains approximately \(5.01875 \times 10^{22}\) dozens of carbon atoms. (c) The sample contains approximately \(1.000083\) moles of carbon atoms.

Step 1: Determine the molar mass of carbon atoms

The molar mass of an element can be found on the periodic table. For carbon, the molar mass is approximately \(12.01 \mathrm{~g/mol}\). We will use this value to convert between grams of carbon and moles of carbon atoms.

Step 2: Calculate the number of moles of carbon atoms in the sample

To find the number of moles of carbon atoms in the sample, we will use the molar mass of carbon to convert from grams of carbon to moles of carbon atoms: number of moles of Carbon = \(\frac{mass}{molar~mass}\) number of moles of Carbon = \(\frac{12.011\mathrm{~g}}{12.01 \mathrm{~g/mol}} = 1.000083\mathrm{~mol}\)

Step 3: Calculate the number of carbon atoms in the sample

To calculate the number of carbon atoms, we will multiply the number of moles with Avogadro's number (\(6.022 \times 10^{23}\) atoms per mole): number of carbon atoms = \(number~of~moles \times Avogadro's~number\) number of carbon atoms = \(1.000083\mathrm{~mol} \times (6.022 \times 10^{23}\mathrm{~atoms/mol}) = 6.0225 \times 10^{23}\mathrm{~atoms}\) (a) The sample contains approximately \(6.0225 \times 10^{23}\) carbon atoms.

Step 4: Calculate the number of dozens of carbon atoms in the sample

To find the number of dozens of carbon atoms, simply divide the number of carbon atoms by 12 (since a dozen contains 12 items): number of dozens of carbon atoms = \(\frac{number~of~carbon~atoms}{12}\) number of dozens of carbon atoms = \(\frac{6.0225 \times 10^{23}\mathrm{~atoms}}{12} = 5.01875 \times 10^{22}\mathrm{~dozens}\) (b) The sample contains approximately \(5.01875 \times 10^{22}\) dozens of carbon atoms.

Step 5: State the number of moles of carbon atoms in the sample

We already found the number of moles of carbon atoms in the sample in Step 2. (c) The sample contains approximately \(1.000083\) moles of carbon atoms.

Key Concepts

Molar Mass

The concept of molar mass is crucial when dealing with chemical calculations and conversions. Molar mass is defined as the mass of one mole of a substance, usually expressed in grams per mole (g/mol). It's determined by adding up the atomic masses of all atoms in a molecule, which you can find on the periodic table. In the context of pure elements like carbon, the molar mass reflects the average atomic mass of carbon atoms due to their isotopic composition.

For carbon, which is the focus of our exercise, the molar mass is about 12.01 g/mol.

• This means that one mole of carbon atoms weighs approximately 12.01 grams.
• Molar mass helps us convert between the mass of a sample and the amount of substance, measured in moles.

Understanding molar mass allows us to bridge the gap between macroscopic amounts of material and the number of constituent particles, such as atoms. This concept is foundational in converting grams of a substance to moles, which simplifies further calculations.

Avogadro's Number

Avogadro's number is one of the most significant constants in chemistry. It represents the number of atoms, ions, or molecules contained in one mole of a substance, which is approximately 6.022 × 10^{23}. This vast number is named after the Italian scientist Amedeo Avogadro and allows us to understand the scale of particles in a mole.

When using Avogadro's number, we can make connections between moles, and the number of particles. For example, if you have one mole of carbon atoms:

• The sample comprises about 6.022 × 10^{23} carbon atoms.
• This concept is pivotal when converting moles of a substance into individual atoms or molecules.

Utilizing Avogadro's number in calculations helps you quantify the number of particles from a given mole measurement, making it a cornerstone in understanding molecular and atomic interactions.

Carbon Atoms

Carbon atoms are essential to life's chemistry and numerous industrial applications. Understanding them requires a knowledge of their atomic structure and behavior in reactions.

Carbon is a versatile element found in various forms like graphite, diamond, and fullerene, with widespread use. In the exercise, dealing with the carbon sample involves several important steps:

• Identifying carbon's molar mass as 12.01 g/mol aids in calculating moles from mass.
• Calculating atoms using Avogadro's number to translate moles to molecules/ions.

Carbon’s ability to form stable bonds with many elements, including itself, underpins its role in organic chemistry. This elemental capability explains why carbon can form an extensive variety of compounds, including those essential for life.