A metal crystallises in a body centred cubic (bcc) lattice. The density of the metal is 5.0 ~g , cm ⁻³ and its molar mass is 96 ~g , mol ⁻¹ . What is the radius of the metal atom? [Given: Avogadro's constant 6 10²³ ~mol ⁻¹ ]

Step-by-step solution

The formula for the density of a unit cell is: = Z M N_ A a³ For a body centred cubic (bcc) lattice, the number of atoms per unit cell is Z = 2 . Substituting the given values: 5.0 = 2 96 6 10²³ a³ Rearranging to solve for a³ : a³ = 192 30 10²³ = 192 3 10²⁴ = 64 10⁻²⁴ ~cm ³ Taking the cube root on both sides: a = 4 10⁻⁸ ~cm Converting the edge length to picometers: a = 4 10⁻⁸ 10¹⁰ ~pm = 400 ~pm For a bcc lattice, the relationship between the atomic radius r and the edge length a is: r = 3 4 a Substituting the value