Problem 71 - Chemistry and Chemical Reactivity

Question

A Identify, from the list below, the information needed to calculate the number of atoms in \(1 \mathrm{cm}^{3}\) 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

Step-by-Step Solution

Answer

Use molar mass, density, and Avogadro's number to calculate atoms: approx. \(8.49 \times 10^{22}\) atoms in \(1 \mathrm{cm}^3\) of iron.

Step 1: Identify Essential Information

To calculate the number of atoms in \(1 \mathrm{cm}^{3}\) of iron, we need data related to the amount of matter, such as mass per volume and the characteristics of individual atoms. Thus, the required information is: (b) the molar mass of iron, (c) Avogadro's number, and (d) the density of iron.

Step 2: Determine Mass of 1 cm³ of Iron

Using the density of iron, calculate the mass of \(1 \mathrm{cm}^{3}\) of iron in grams. The formula is: \[ \text{Mass} = \text{Density} \times \text{Volume} \] Assuming a density of \(7.87 \text{ g/cm}^{3}\), the mass is \(7.87 \text{ g}\).

Step 3: Calculate Moles of Iron

Using the molar mass of iron (approximately \(55.85\text{ g/mol}\)), find the number of moles in \(7.87\text{ g}\) of iron. The formula is: \[ \text{Moles of iron} = \frac{\text{Mass}}{\text{Molar mass}} \] Substitute to get: \( \frac{7.87}{55.85} \approx 0.141 \text{ moles}\).

Step 4: Calculate Number of Atoms

Use Avogadro's number (\(6.022 \times 10^{23} \text{ atoms/mol}\)) to find the number of atoms in \(0.141 \text{ moles}\) of iron: \[ \text{Number of atoms} = \text{Moles of iron} \times \text{Avogadro's number} \] Thus, \(0.141 \times 6.022 \times 10^{23} \approx 8.49 \times 10^{22} \text{ atoms}\).

Key Concepts

Molar Mass

Molar mass is an essential concept in chemistry, referring to the mass of one mole of a substance. One mole is a fundamental unit in chemistry that represents a specific amount of particles, be they atoms, molecules, or ions. Molar mass is typically expressed in grams per mole (g/mol). For iron, the molar mass is approximately 55.85 g/mol.
To determine the number of moles present in a mass of a substance, we use the formula: \[ \text{Moles} = \frac{\text{Mass in grams}}{\text{Molar mass in g/mol}} \]This formula helps convert the quantity of the substance in grams to the number of moles, which is crucial for further calculations, especially when calculating the number of atoms or molecules. In essence, knowing the molar mass of a substance like iron allows us to connect macroscopic amounts that we can weigh to the microscopic level where chemical reactions happen.

Avogadro's Number

Avogadro's number, named after the scientist Amedeo Avogadro, is a constant that is used to describe the number of atoms, ions, or molecules in one mole of a substance. This number is a key bridge between the atomic scale and the scale of everyday laboratory measurements. The value of Avogadro's number is approximately \(6.022 \times 10^{23}\) entities/mol.

• This large number is crucial for converting between the "number of moles" and the "number of entities" (atoms or molecules).
• For example, when you determine that you have 0.141 moles of iron, you use Avogadro's number to find out how many iron atoms are present.
• By multiplying the number of moles by Avogadro's constant, you derive the total number of individual atoms in the sample.
  

Thus, Avogadro's number facilitates a deeper understanding of the amount of substance in chemical processes and reactions, making it an indispensable tool in the world of chemistry.

Density of Iron

Density is a measure of how much mass is contained in a given volume. For iron, the density is known to be approximately 7.87 g/cm³. By understanding density, we can relate the mass and volume of a piece of iron.

• The formula to calculate mass from density is: \[ \text{Mass} = \text{Density} \times \text{Volume} \]
• For a 1 cm³ volume of iron, the mass can be easily calculated using this relationship, which gives us the mass as 7.87 g.

Once the mass is known, it acts as a stepping stone to calculate how many moles of iron are present, using the molar mass, and then further calculating the number of atoms using Avogadro's number.
Thus, density connects physical properties of a substance, like mass or volume, to chemical properties that describe the amount of substance, allowing us to perform meaningful calculations in chemistry.