Question

For bcc structure, the coordination number and packing are ……….. respectively.
A. 8, $\dfrac{\pi \sqrt{3}}{8}$
B. 6, $\dfrac{\pi \sqrt{2}}{6}$
C. 8, $\dfrac{\pi \sqrt{2}}{6}$
D. None of the above

Answer 1

The three-dimensional arrangement of identical points in the space which represents how the constituent particles i.e. atoms, ions or molecules are arranged in a crystal is known by space lattice and there are different type of arrangement of these crystals which is known by fcc, bcc, simple cubic lattice and hcc.

Complete Step by step solution: BCC refers with Body centered cubic lattice which belongs entirely to one unit cell and we can say that it is not shared by any other unit cell and its contribution to the unit cell is only one and the total number of atoms present in the body centered cubic cell is 2. Coordination number is defined as the number of nearest neighbors that a particle has in a unit cell. It generally depends upon the structure of the unit cell of the crystal and in case BCC coordination number is 8. Packing efficiency is defined as the percentage of total space filled by the particles and it is represented by the formula as shown:
$Packing Efficiency=\dfrac{Volume occupied by atoms \ in unit cell(v)}{Total Volume Of The Unit Cell(V)}$
In case of bcc, edge length a = $\dfrac{4r}{\sqrt{3}}$
Volume occupied by atoms in unit cell i.e. v = $2\times volume of cube$= $2\times \dfrac{4}{3}\pi {{r}^{3}}=\dfrac{8}{3}\pi {{r}^{3}}$
Volume of unit cell (V) = ${{a}^{3}}={{(\dfrac{4r}{\sqrt{3}})}^{3}}=\dfrac{64{{r}^{3}}}{3\sqrt{3}}$
Packing efficiency = $\dfrac{v}{V}=\dfrac{\dfrac{8}{3}\pi r3}{\dfrac{64}{3\sqrt{3}}{{r}^{3}}}=\dfrac{\sqrt{3}\pi }{8}$

Hence we can say that option A is the correct answer.

Note: The unit cell of BCC has spheres in the corners of a cube and one sphere is present in the center of the cube that’s why it is known as body centered cubic lattice. In BCC there are eight spheres on corners in a cube so the total number of spheres present in a BCC unit cell is 9.