Question

Iron crystallizes in a body-centered unit cell. Its atomic radius is \(0.124 \mathrm{nm}\). Its density is \(7.86 \mathrm{~g} / \mathrm{cm}^{3}\). Using this information, estimate Avogadro's number.

Short answer

Question: Estimate Avogadro's number using the information on iron crystallizing in a body-centered unit cell with an atomic radius of 0.124 nm and a density of 7.86 g/cm³. Answer: Avogadro's number ≈ 6.06 x 10^23/mol

Step-by-step solution

Determining the edge length of the unit cell

The body-centered unit cell consists of one atom at the center and one-eighth of each atom present at the eight corners. First, we need to determine the edge length (a) of the unit cell using the given atomic radius (r). Since each atom touches along the body diagonal, we can apply Pythagorean theorem: a^2 + a^2 + a^2 = (4r)^2 Solve for a: a = √{(4r)^2 / 3} Using the r = 0.124 nm: a = √{(0.496)^2 / 3} = 0.286 nm

Calculating the volume of the unit cell

The volume (V) of the unit cell is given by: V = a^3 Plug in the value of a = 0.286 nm: V = (0.286 nm)^3 = 0.02340 nm³

Converting the volume to cm³

To use the given density value, we need to convert the volume from nm³ to cm³: 1 nm = 1 x 10^-7 cm V = 0.02340 nm³ * (1 x 10^-7 cm/nm)³ = 2.34 x 10^-23 cm³

Determining the mass of the unit cell

The mass of the unit cell is given by the product of its density (ρ) and volume (V): mass = ρV Plug in the values; ρ = 7.86 g/cm³ and V = 2.34 x 10^-23 cm³: mass = 7.86 g/cm³ * 2.34 x 10^-23 cm³ = 1.84 x 10^-22 g

Calculating the number of atoms in the unit cell

In a body-centered unit cell, there are 2 atoms present - one at the center and one-eighth of each atom at the eight corners: n_atoms = 1 + 8 * (1/8) = 2

Determining the molar mass of iron

The molar mass of iron (Fe) is 55.85 g/mol.

Calculating the number of moles in the unit cell

The number of moles in the unit cell is given by dividing the mass by the molar mass: n_moles = mass / molar_mass Plug in the values; mass = 1.84 x 10^-22 g and molar_mass = 55.85 g/mol: n_moles = 1.84 x 10^-22 g / 55.85 g/mol = 3.3 x 10^-24 mol

Estimating Avogadro's number

Since we have 2 atoms present in the unit cell, we can divide the number of atoms by the number of moles to estimate Avogadro's number: Avogadro's_number = n_atoms / n_moles Plug in the values; n_atoms = 2 and n_moles = 3.3 x 10^-24 mol: Avogadro's_number ≈ 2 / 3.3 x 10^-24 mol ≈ 6.06 x 10^23/mol