Question

The "lead" in lead pencils is actually almost pure carbon, and the mass of a period mark made by a lead pencil is about \(0.0001 \mathrm{~g}\). How many carbon atoms are in the period?

Short answer

Approximately \(5.01 \times 10^{18}\) carbon atoms.

Step-by-step solution

Find the Atomic Mass of Carbon

The atomic mass of carbon is approximately 12.01 u (unified atomic mass units), which is equivalent to 12.01 g/mol.

Use Avogadro's Number

Avogadro's Number is defined as the number of atoms or molecules in one mole of a substance. It is approximately \(6.022 \times 10^{23}\).

Calculate Number of Moles of Carbon in the Period Mark

To find the number of moles, use the formula \( \text{moles} = \frac{\text{mass (g)}}{\text{atomic mass (g/mol)}} \). Here, the mass is 0.0001 g and the atomic mass of carbon is 12.01 g/mol: \( \text{moles of carbon} = \frac{0.0001}{12.01} \approx 8.33 \times 10^{-6} \).

Find the Number of Carbon Atoms

Multiply the number of moles of carbon by Avogadro's Number to find the number of atoms: \( 8.33 \times 10^{-6} \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 5.01 \times 10^{18} \text{ atoms} \).

Key concepts

Atomic Mass of Carbon

The atomic mass of carbon is an important concept in chemistry. It is essential for calculating the number of atoms in a substance. Carbon is one of the most common elements found on Earth. It plays a fundamental role in the chemistry of life, with a number of applications ranging from lead pencils to organic compounds.
The atomic mass of carbon is approximately 12.01 atomic mass units (amu), measured as grams per mole (g/mol). This means that one mole of carbon atoms weighs about 12.01 grams. The atomic mass accounts for the various isotopes of carbon weighted by their natural abundance.
When you see 12.01 next to carbon on the periodic table, it represents a weighted average. This average considers the different isotopes of carbon, primarily carbon-12 and a small amount of carbon-13. Understanding atomic mass helps in converting mass measures into mole measures, especially when dealing with substances like the graphite in pencils.

Avogadro's Number

Avogadro's Number is a cornerstone concept in chemistry. It provides a link between the mass of a substance and the number of atoms or molecules it contains. Discovered by Amedeo Avogadro in the 19th century, this number is approximately \(6.022 \times 10^{23}\).
Avogadro's Number tells us how many particles (like atoms or molecules) are in one mole of any substance. It is a huge number because atoms and molecules are incredibly tiny. This sheer size helps in converting microscopic scales to macroscale scenarios, making measurable quantities accessible.
This number is crucial when trying to determine the number of atoms present in any mass of material. For instance, to find how many carbon atoms are present in a pencil mark, we need to multiply the moles of carbon by Avogadro's Number. Thus, it acts as a bridge between moles (a bulk property) and the count of atoms or molecules (an atomic property).

Moles Calculation

Calculating moles is an essential step in understanding how much of a chemical substance is present in a given sample. A mole is a designated quantity of substance that contains Avogadro's Number of entities, which can be atoms, molecules, or ions.
To calculate moles, use the formula:

• \( \text{moles} = \frac{\text{mass (g)}}{\text{atomic mass (g/mol)}} \)

This formula allows you to convert mass into moles by dividing the sample's mass by the atomic or molecular weight of the substance.
In the case of a pencil mark with a mass of 0.0001 grams, dividing this by the atomic mass of carbon (12.01 g/mol) gives the number of moles of carbon. The number of moles then helps us in using Avogadro's Number to find out the exact count of carbon atoms in that pencil mark, allowing complex chemical equations to be more intuitive and practical in real-world situations.