Question

Assertion :Packing efficiency of body centered cubic structure is $68\%$ Reason: $68\%$ is maximum packing efficiency any crystal can have.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is C Assertion is correct but Reason is incorrect

Packing efficiency of various crystals is given as:

Face-centered cube - $74$%

Body-centered cube - $68$%

Simple Cube - $52.4$%

In the body-centered cubic unit cell, the atoms are present at corners and body center of the cube. So, the effective number of atoms in unit cell $=2$

If the radius of the atom is $r$ and edge length of the cube is $a$. In body-centered cubic unit cell atoms along the body, diagonals touch each other.

So, $\sqrt{3}a=4r$

$\Rightarrow r=\frac{\sqrt{3}}{4}a$

So, volume occupied $=2\times \frac{4}{3}\times \pi r^3$

Total volume $=a^3={\left(\frac{4}{\sqrt{3}}r\right)}^3$

So, packing efficiency $=\frac{2\times \frac{4}{3}\pi r^3}{{\left(\frac{4}{\sqrt{3}}\right)}^3{r}^3}\times 100$%

$=\frac{\pi \times 2\sqrt{3}}{16}\times 100$% $=68$%