Question

Tungsten has an atomic radius of 0.136 nm. The density of tungsten is 19.4$${\text{gc}}{{\text{m}}^{{\text{ - 3}}}}$$. What is the crystal structure of tungsten? (the atomic weight of tungsten is 184).
A. Simple cubic
B. Body centered cubic
C. Face centered cubic
D. None of these

Answer 1

Hint: the unit cell is defined as the smallest unit which has been repeated in the crystal lattice for making the crystal . There are different types of unit cells such as primitive cubic, body centered cubic and face centered. The primitive unit cell contains the atom at all the corners of the cube. So the primitive or the simple cubic contains 1 atom in a cube , the body centered cubic contains 2 atoms in a cube and the face centered cubic contains 4 atoms in a cube.

Complete step by step answer:
So the values given in the question is of radius or the nearest neighbour distance between the atoms is 0.136nm or $$1.36 \times {10^{ - 8}}cm$$.
The atomic mass of the tungsten is 184$$gmo{l^{ - 1}}$$
The value for avogadro number is $$6.023 \times {10^{23}}$$
In the below formula z is the number of atoms in the crystal.
The formula used for calculating the density is the following:
$$\rho = \dfrac{{Z \times M}}{{{N_A} \times {a^3}}}$$
Now substituting the values in the formula we get,
19.4 =$$\dfrac{{Z \times 184}}{{6.023 \times {{10}^{23}} \times 1.36 \times {{10}^{ - 8}}}}$$
Z = $$\dfrac{{19.4 \times 6.023 \times {{10}^{23}} \times 1.36 \times {{10}^{ - 8}}}}{{184}}$$
Z = 2
So as the number of atoms in the crystal is 2 so it is a body centered cubic.

So, the correct answer is Option B.

Note: In the body centered cubic the atoms are present at all corners of the cube and also at the body or the center of the cubic. Each corner shares 1/8th part of an atom so it constitutes 1 atom and the whole 1 atom is present at the body center thus constituting two atoms in a cube. In the unit cell the number of the coordinates of the atom is defined as the number of atoms that it touches.