Question

A metal (atomic mass\[=50\text{ }amu\]) has a body-centered cubic crystal structure. The density of metal is \[5.96\text{ }g\text{ }c{{m}^{-3}}\]. Find the volume (\[in\text{ }c{{m}^{3}}\]) of the unit cell.
A.$13.9\times {{10}^{-24}}$
B.$55.6\times {{10}^{-24}}$
C.$27.8\times {{10}^{-24}}$
D.$6.95\times {{10}^{-24}}$

Answer 1

Hint: In the bcc structure the total of two atoms are present as the atoms present in the corners are shared by eight other atoms, and only one atom is present at the centre which is not shared by any other cell.
-The volume of a substance can be calculated in terms of density and mass as the volume is nothing but mass per unit density.

Complete step by step answer:
The body-centered cubic unit cell or bcc has atoms present at each of the eight corners of a cube along with one atom located at the center of the cube. Each of the atoms which are present at the corner is shared by eight unit cells, as it is corner to eight other unit cells. It has the coordination number of\[8\]. In a bcc unit cell, a net total of two atoms are present in one cell.

In the bcc unlike fcc or hcp arrangements, it does not allow the atoms to be closely packed together. The bcc arrangement is often the high temperature form of metals which are being close-packed at lower temperatures. The ratio of volume of atoms in a cell and the total volume taken by the cell is termed as the packing factor. The packing factor of the bcc unit cell is of \[0.68\].

Some of the examples materials in which we can observe bcc structure are, tungsten sodium, vanadium, chromium lithium, barium potassium, and, alpha-iron. Metals that have a bcc structure are generally less malleable and harder than close-packed metals like the gold. When the metal is deformed, the planes present in the atoms must slip over one another, and this is more difficult in case of bcc structure. It should be noted that there are other important mechanisms for hardening materials, such as introducing impurities or defects which make slipping more difficult.

Now if we consider the question, the number of atoms in bcc is $2$
So the Mass of unit cell \[=2\times 50=100~amu\]
We know that volume is mass per density of a substance, so
\[Volume=\dfrac{Mass}{Density}=\dfrac{100\times 1.67\times {{10}^{-24}}}{5.96}=27.8\times {{10}^{-24~}}c{{m}^{3}}\]
Therefore, the volume of the unit cell is \[27.8\times {{10}^{-24}}~c{{m}^{3}}\].

So the correct option is option C.

Note: It is necessary to keep the units of the values we are writing as the units could change the outcome of the calculated value.
-The packing efficiency of the bcc lattice is \[68%\] which is the percentage of packing factor.
-Packing efficiency, also called packing fraction is the fraction of volume in a lattice crystal structure which is occupied by constituent particles.