Question

What is the percentage efficiency of packing in BCC structure?

• A. 32
• B. 74
• C. 26
• D. 68

Hint:

Memorize the packing efficiencies for the three main cubic structures to save time: Simple Cubic (SC) = 52.4

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the standard packing efficiency (the percentage of total space occupied by particles) in a Body-Centered Cubic (BCC) crystal lattice.

Step 2: Key Formula or Approach:
Packing Efficiency is calculated as:
$$\text{Packing Efficiency} = \frac{\text{Volume occupied by atoms in a unit cell}}{\text{Total volume of the unit cell}} \times 100\%$$
For a BCC unit cell, the number of atoms per unit cell ($Z$) is 2.
The relationship between the atomic radius ($r$) and the edge length ($a$) is $4r = \sqrt{3}a$.

Step 3: Detailed Explanation:
Volume of the unit cell $= a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}}$
Volume occupied by the 2 atoms $= 2 \times \frac{4}{3}\pi r^3 = \frac{8}{3}\pi r^3$
Substitute these into the packing efficiency formula:
$$\text{Efficiency} = \frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt{3}}} \times 100\%$$
$$\text{Efficiency} = \frac{8\pi \times 3\sqrt{3}}{3 \times 64} \times 100\%$$
$$\text{Efficiency} = \frac{\sqrt{3}\pi}{8} \times 100\%$$
Plugging in $\pi \approx 3.14159$ and $\sqrt{3} \approx 1.732$:
$$\text{Efficiency} \approx 0.68 \times 100\% = 68\%$$

Step 4: Final Answer:
The packing efficiency of a BCC structure is 68