A metal crystallizes in a face-centered cubic lattice.The atomic radius of the metal is 198 pm and the density of the metal is 6.57 g/cm³.What is the volume of the unit cell? A) 3.25 × 10⁷ pm³ B) 1.52 × 10⁹ pm³ C) 1.76 × 10⁸ pm³ D) 7.76 × 10⁶ pm³ E) 1.3 × 10⁸ pm³

Question

Quizplus · Accepted Answer

In a face-centered cubic lattice, each unit cell has four atoms. The formula unit of the crystal is therefore M4. The density (ρ) of the crystal can be calculated using the formula:
ρ = (mass of one formula unit)/(volume of one formula unit)
The mass of one formula unit can be calculated using the atomic mass (A) of the metal and Avogadro's number (NA):
mass of one formula unit = A/NA x 4
The volume of one formula unit can be calculated using the formula for the volume of a cube with edge length (a) equal to the diagonal of a face of the unit cell:
volume of one formula unit = (a/√2)3 x 1/4
Equating the two expressions:
ρ = A/NA x 4/[(a/√2)3 x 1/4]
Solving for a:
a = [4A/(NAρ√2)]1/3
Substituting the given values:
a = [4 x 197 g/mol/(6.022 x 1023/mol x 6.57 g/cm3 x √2)]1/3 = 3.61 × 10-8 cm = 1.76 × 1018 pm
The volume of the unit cell is therefore:
(1.76 × 1018 pm)3 x 1/4 = 1.76 × 1018 pm3 x 1/8 = 2.20 × 1017 pm3 = 1.76 × 10-18 cm3 = 1.76 × 10-25 m3
The correct answer is therefore (C) 1.76 × 1018 pm3.

A Metal Crystallizes in a Face-Centered Cubic Lattice