Problem

Iron (α-Fe) crystallises in BCC. Its density is 7.87 g/$cm^{3}$ and molar mass is 55.85 g/mol. Calculate the edge length a of the unit cell in pm.

Solution

Step 1: Identify Z BCC → Z = 2.

Step 2: Rearrange density formula to find $a^{3}$ $\rho = \frac{ZM}{a^3 N_A} \implies a^3 = \frac{ZM}{\rho N_A}$

Step 3: Substitute values $a^3 = \frac{2 \times 55.85}{7.87 \times 6.022 \times 10^{23}}$

$= \frac{111.70}{4.739 \times 10^{24}}$

$= 2.357 \times 10^{-23} \text{ cm}^3$

Step 4: Take cube root $a = (2.357 \times 10^{-23})^{1/3}$

$= (23.57 \times 10^{-24})^{1/3}$

$= (23.57)^{1/3} \times 10^{-8} \text{ cm}$

$= 2.866 \times 10^{-8} \text{ cm}$

Step 5: Convert to pm $a = 2.866 \times 10^{-8} \text{ cm} = 2.866 \times 10^{-8} \times 10^{10} \text{ pm} = \boxed{287 \text{ pm}}$

Verification: Known value for α-Fe = 287 pm. ✓