A metal M crystallizes into two lattices:- face centered cubic (fcc) and body centered cubic(bcc) With unit cell edge length of 2.0 and 2.5 A° respectively. The ratio of densities of lattices fcc to bcc for the metal M is __________ (nearest integer)

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answer is 4.

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Detailed Solution

We know that

FCC	BCC
$a_{FCC} = 2.0 A^{0}$	$a_{BCC} = 2.5 A^{0}$
$Z_{FCC} = 4$	$Z_{BCC} = 2$

Density $(d) = \frac{ZM}{a^{3} N_{A}}$
$\begin{array}{l} \frac{d_{FCC}}{d_{BCC}} = \frac{\frac{z_{FCC} M}{a_{FCC}^{3} N_{A}}}{\frac{z_{BCC} M}{a_{BCC}^{3} N_{A}}} \\ = \frac{z_{FCC}}{z_{BCC}} \times \frac{a_{BCC}^{3}}{a_{FCC}^{3}} = \frac{4}{2} \times \frac{(2.5)^{3}}{(2)^{3}} \\ = \frac{15.625}{4} = 3.90 \approx 4 \end{array}$
Nearest integer = 4

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