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Question: A metal has bcc structure and the edge length of its unit cell is 3.04 \[\mathop A\limits^o \]. The ...

A metal has bcc structure and the edge length of its unit cell is 3.04 Ao\mathop A\limits^o . The volume of the unit cell in cm3c{m^3} will be:
A. 1.6×1021cm31.6 \times {10^{21}}c{m^3}
B. 2.81×10−23cm32.81 \times {10^{ - 23}}c{m^3}
C. 6.02×10−23cm36.02 \times {10^{ - 23}}c{m^3}
D. 6.6×10−24cm36.6 \times {10^{ - 24}}c{m^3}

Explanation

Solution

In the body-centered cubic unit cell the volume is given as a cube of edge length which is denoted by (a)3{(a)^3}. The edge length value is given in angstrom and we need to calculate the volume in the unit of cm3c{m^3}.

Complete step by step answer: Given,
Edge length of the unit cell is 3.04 Ao\mathop A\limits^o .
The smallest repeating unit present in the cubic crystal system is known as unit cell.
The body-centered cubic cell (BCC) is the unit cell. In a body- centered cubic cell, the atoms are present in each corner of the cube and at the center of the cube.
In Body centered cubic cell, 8 corners have 1/8 contribution per atom and one body center have 1 contribution per atom.
⇒8×18=1\Rightarrow 8 \times \dfrac{1}{8} = 1atom
⇒1×1=1\Rightarrow 1 \times 1 = 1atom
Thus, the total number of atoms present in a body centered cubic unit cell is 2.
The edge length is given by a.
As it is a cubic unit cell, the volume is given as (a)3{(a)^3}.
As, the value of edge length is given in angstrom and we need to calculate the volume in cm3c{m^3}.
First convert the value in angstrom into meters.
3.04 Ao\mathop A\limits^o = 3.04×10−103.04 \times {10^{ - 10}}
Now, convert meters into centimeter
3.04×10−103.04 \times {10^{ - 10}}m = 3.04×10−83.04 \times {10^{ - 8}}cm
Substitute the value in cm into the volume formula.
⇒(3.04×10−8cm)3\Rightarrow {(3.04 \times {10^{ - 8}}cm)^3}
⇒2.81×10−23cm3\Rightarrow 2.81 \times {10^{ - 23}}c{m^3}
Thus, the volume of the unit cell in cm3c{m^3}will be 2.81×10−23cm32.81 \times {10^{ - 23}}c{m^3}.
Therefore, the correct option is B.

Note:
Make sure to convert angstrom into meters and then convert meter to centimeter. 1 Ao\mathop A\limits^o = 1×10−101 \times {10^{ - 10}}m and 1 m = 100 cm.