Question

A metal crystallizes in a face-centered cubic structure and has a density of 11.9 g/cm^3. If the radius of the metal atom is 138 pm, what is the identity of the metal? 2) Vanadium crystallizes in a body-centered cubic structure and has an atomic radius of 131 pm. Determine the density of vanadium.

Answer 1

1) To determine the identity of the metal, we need to calculate the molar mass. First, we'll find the volume of the unit cell:

a = 2√2 * r
a = 2√2 * 138 pm = 389.6 pm

Now, convert the volume of the unit cell to cm³:

V = (389.6 * 10^(-10) cm)³ = 5.93 * 10^(-23) cm³

The face-centered cubic (fcc) structure has 4 atoms per unit cell. Using the density formula, we can find the molar mass:

density = mass/volume
11.9 g/cm³ = (4 * Molar mass) / (V * Avogadro's number)
Molar mass = (11.9 * 5.93 * 10^(-23) * 6.022 * 10^(23)) / 4
Molar mass ≈ 107 g/mol

The metal with a molar mass of approximately 107 g/mol is silver (Ag).

2) For vanadium, we'll calculate the density using its body-centered cubic (bcc) structure:

a = 4r / √3
a = (4 * 131 pm) / √3 = 302 pm

Now, convert the volume of the unit cell to cm³:

V = (302 * 10^(-10) cm)³ = 2.76 * 10^(-23) cm³

The bcc structure has 2 atoms per unit cell. The molar mass of vanadium is 50.94 g/mol. Using the density formula:

density = (2 * 50.94) / (2.76 * 10^(-23) * 6.022 * 10^(23))
density ≈ 6.11 g/cm³

The density of vanadium is approximately 6.11 g/cm³.