Question

Identify, from the list below, the information needed to calculate the number of atoms in \(1.00 \mathrm{cm}^{3}\) of iron. Outline the procedure used in this calculation. (a) the structure of solid iron (b) the molar mass of iron (c) Avogadro's number (d) the density of iron (e) the temperature (f) iron's atomic number (g) the number of iron isotopes

Short answer

Use the density, molar mass of iron, and Avogadro's number.

Step-by-step solution

Identify Required Information

To calculate the number of atoms in a given volume, we need to know the density of the substance, the molar mass, and Avogadro's number. From the list, the relevant information is: (b) the molar mass of iron, (c) Avogadro's number, and (d) the density of iron. These will allow us to convert the volume of iron to mass, then to moles, and finally to atoms.

Calculate Mass From Volume

Use the density to convert the volume of iron to mass. Given the volume is \(1.00 \mathrm{cm}^{3}\), the mass \(m\) is calculated as follows:\[ m = \text{density} \times \text{volume} \]

Convert Mass to Moles

Use the molar mass of iron to convert the mass obtained in Step 2 to moles. The molar mass of iron is necessary for this conversion:\[ \, \text{Moles of Fe} = \frac{\text{Mass of Fe}}{\text{Molar Mass of Fe}} \]

Convert Moles to Atoms

Use Avogadro's number to convert the moles of iron to atoms. Avogadro's number is \(6.022 \times 10^{23}\) atoms/mole.\[ \text{Number of atoms} = \text{Moles of Fe} \times 6.022 \times 10^{23} \text{ atoms/mole} \]

Key concepts

Molar Mass

The molar mass is the mass of one mole of a substance, usually expressed in grams per mole (g/mol). For iron, the molar mass is approximately 55.85 g/mol. This value is crucial when converting the mass of iron to the number of moles. Knowing the molar mass allows us to understand how much a specific number of iron atoms weighs. For instance, if you have a sample of iron, the molar mass tells you how much each mole of that iron will weigh. This knowledge makes the conversion from mass to moles straightforward. Understanding the concept of molar mass is essential for various areas of chemistry and physics because it helps us to bridge the gap between microscopic entities, like atoms and molecules, and the measurable macroscopic world.

Avogadro's Number

Avogadro's number is a fundamental constant that defines the number of atoms, molecules, or particles in one mole of any substance. It is approximately \(6.022 \times 10^{23}\) per mole. This number is essential for converting the amount of substance in moles to the actual number of atoms or molecules. For instance, when you have 1 mole of iron, according to Avogadro's number, you possess about \(6.022 \times 10^{23}\) iron atoms. This constant links the macroscale, which we can measure, to the nanoscale, where actual chemical processes occur. Avogadro's number serves as a bridge between micro and macro world and allows chemists to perform calculations that involve large numbers of atoms.

Density of Iron

The density of iron is another key factor in our calculations. It is generally expressed in grams per cubic centimeter (g/cm³) and for iron, it is approximately 7.87 g/cm³. Understanding density is critical because it allows you to calculate the mass of iron that occupies a given volume. For example, if you know the volume of iron is 1.00 cm³, you can determine its mass using the density. The density serves as a conversion factor between volume and mass in calculations. By knowing the density, you are able to accurately calculate how much a certain volume of material will weigh, which is essential for determining how many moles or atoms are present in a sample.