Question

A metal crystallizes in two cubic phases, fcc and bcc whose unit cell lengths are $3.5\mathring{A}$ and $3.0\mathring{A}$ respectively. The ratio of density of fcc and bcc is :

• A. $2$
• B. $1.26$
• C. $3.34$
• D. $1.8$

Answer 1

The correct option is B $1.26$

The formula for the calculation of density is given below:

$\text{Density}=\frac{Z\times M}{N_0\times a^3\times {10}^{-30}}$ g cm${}^{-3}$

$Z$ is the number of formula units in the unit cell.

$M$ is the formula mass of the substance.

$a$ is the length (in pm) of the unit cell.

$N_0$ is Avogadro's number.

For FCC, $Z=4,a=3.5\mathring{A}=350$ pm

$D_{fcc}=\frac{4\times M}{N_0\times (350{)}^3\times {10}^{-30}}$ g cm${}^{-3}$ $......\left(1\right)$

For BCC, $Z=2,a=3.0\mathring{A}=300$ pm

$D_{bcc}=\frac{2\times M}{N_0\times (300{)}^3\times {10}^{-30}}$ g cm${}^{-3}$ $......\left(2\right)$

Dividing equation $\left(1\right)$ by equation $\left(2\right)$, we get

$\frac{D_{fcc}}{D_{bcc}}=\frac{\frac{4\times M}{N_0\times (350{)}^3\times {10}^{-30}}}{\frac{2\times M}{N_0\times (300{)}^3\times {10}^{-30}}}=\frac{\frac{4}{{350}^3}}{\frac{2}{{300}^3}}=1.26$