First slide

Crystal lattices and unit cells

Question

A metal crystallizes into two cubic phases, face-centred cubic (fcc) and body-centred cubic (bcc) whose unit cell lengths are 3.6 and 3.0 $Å$ , respectively. Calculate the ratio of densities of fcc and bcc.

Moderate

A

1.16
B

2.34
C

1.9
D

2.64

Solution

Ans. fcc unit cell length $= 3.6 Å$
Bcc unit cell length $= 3.0 Å$
Density in fcc $= \frac{n_{1} \times atomic mass}{V_{1} \times A v . n o .}$
Density in bcc $= \frac{n_{2} \times atomic mass}{V_{2} \times A v . n o .}$
$\frac{D_{f c c}}{D_{b c c}} = \frac{n_{1}}{n_{2}} \times \frac{V_{2}}{V_{1}}$
Now, $n_{1}$ for fcc = 4; Also, $V_{1} = a^{3} = {(3.6 \times 10^{- 8})}^{3}$
$n_{2}$ for bcc = 2; Also, $V_{2} = a^{3} = {(3.0 \times 10^{- 8})}^{3}$
$\frac{D_{f c c}}{D_{b c c}} = \frac{4 \times {(3.0 \times 10^{- 8})}^{3}}{2 \times {(3.6 \times 10^{- 8})}^{3}} = 1.16$

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