Question

Sodium metal crystallizes in a body-centered cubic lattice with cell edge, $a = 4.29A^o$. What is the radius of a sodium atom?

Answer 1

Here in the question above it is mentioned that sodium metal crystallizes in body-centered cubic lattice. In the bcc structure, the unit cell has one atom at all the eight corners of the cube. And one atom at the center of the cube. Every corner atom was shared by eight adjacent unit cells.

Complete answer:
Given data contains,
Edge length, $a = 4.29{A^0}$
To find:- Radius of sodium atom
We have a predefined formula which defines the relationship between the edge length and radius for bcc lattice structure.
We can written as,
$\sqrt 3 a = 4r$
Here, in the above expression, r is denotation of radius of lattice and a is edge length.
We already have the edge length of sodium atoms. We will write the equation as,
$r = \dfrac{{\sqrt 3 a}}{4}$
We will put the value of edge length,
$r = \dfrac{{\sqrt 3 \times 4.29}}{4}$
On simplification we get,
$r = 1.857{A^0}$
Hence, the radius of sodium atom is $1.857\mathop {\text{A}}\limits^\circ$.

Additional information:
We have to know that the $NaCl$ is an ionic compound that naturally occurs as white crystals. The salt is extracted from the mineral form of halite. The structure is formed by repetition of the face centered cubic unit cell. The stoichiometry ratio of $NaCl$ is 1:1 and has a molar mass of 58.4 g/mol. Here, the compounds of sodium chloride include alkali halides, metal oxides and transition-metal compounds.
We need to remember that the rock salt structure has applications like structure and dynamics of water, crystallization of proteins and conformational behaviour of peptides and nucleic acids.

Note:
We must remember that the metal crystallizes in various different lattice forms. Such as bcc – body centered closed packing, fcc – face centered closed packing, hcp – hexagonal close packing. Sodium atoms crystallize in the bcc structure. A single unit cell contains one atom at the center of the cube and one on every eight corners of the unit cell. In total there are only two atoms in one unit cell. Since the corner is shared with eight adjacent unit cells. Bcc structure is found in atoms like – iron, chromium, tungsten, etc.