Calculate the theoretical density of magnesium, an HCP metal, given that rMg = 0.160 nm and V = LaTeX: / frac{/sqrt{3}}{2}/:a^2c

1.74 g/cm³

Explanation:

The theoretical density for the unit cell for the HCP unit cell of magnesium is given by the mass of the atoms in the unit cell divided by the volume of the unit cell.

The volume of the HCP unit cell is:

V = 3√3/2 a² c

where a = 2r (sides of the prism), and

c = height of the prism = 1.633 a

The HCP contains 6 atoms per unit cell

Therefore, we need to compute the total mass of the unit cell, and then from the above formula calculate the volume to solve this question.

Atomic mass Mg = 24.3 g/mol

Mass of 1 atom = 24.3 g / mol x (1 mol / 6.022 x 10²³ atoms)

= 4.0 x 10 ⁻²³ g

Mass / Unit Cell = 6 atoms / Unit Cell x 4.0 x 10⁻²³ g

= 2.4 x 10⁻²² g / Unit Cell

V Unit Cell = 3√3/2 a² c = 3√3/2 a² (1.633 a)

= 4.2 a³

a = 2r = 0.160 nm x 1 m/10⁹ nm x 100 cm/m = 3.2 x 10⁻⁸ cm

V Unit Cell = 4.2 (3.2 x 10⁻⁸ cm) ³ = 1.38 x 10⁻²² cm³

d = m / v = 2.4 x 10⁻²² g / 1.38 x 10⁻²² cm³ = 1.74 g/cm³