Question

A metal crystallizes with a face centered cubic lattice. The edge length of the unit cell is $408pm$. The diameter of the metal atom is:
A.$288pm$
B.$176pm$
C.$200pm$
D.None of the above

Answer 1

This question gives the knowledge about the FCC crystal lattice. Fcc crystal lattice is called as face-centered cubic crystal lattice. It contains six atoms at each face of the cube and eight atoms at the corners of the cube.
Formula used:
The formula used to determine the radius of the metal atom is as follows:
$r = \dfrac{a}{{2\sqrt 2 }}$
Where $r$ is the radius of the metal atom and $a$ is the edge length of the unit cell.

Complete step by step answer:
Fcc crystal lattice contains six atoms at each face of the cube and eight atoms at the corners of the cube. The effective number of atoms in a FCC crystal unit cell is always $4$. The coordination number of FCC lattice is $12$. The elements which possess face centered unit cells are copper, aluminum, silver and so forth. Face centered crystal lattice is a closed packed structure.
Now, we will determine the radius of the metal atom is as follows:
$\Rightarrow r = \dfrac{a}{{2\sqrt 2 }}$
Substitute the value of the edge length of the unit cell as $408pm$ in the above formula to determine the radius.
$\Rightarrow r = \dfrac{{408}}{{2\sqrt 2 }}$
On simplifying, we get
$\Rightarrow r = 144pm$
As we know, the diameter is the double of radius of the metal atom. So, the radius is multiplied by $2$ to determine the diameter.
Therefore, the diameter of the metal atom is $288pm$.

Hence, option $A$ is the correct option.

Note:
Always remember that the effective number of atoms in a face centered cubic crystal unit cell is always $4$. FCC crystal lattice contains six atoms at each face of the cube and eight atoms at the corners of the cube.