Question

Iron crystallizes in several modifications, At about ${910}^o$C, the body centred $\alpha$-form undergoes transition to the face centred cubic $\gamma$-form. Assuming that the distance between nearest neighbours is same in the two forms at the transition temperature. Calculate the ratio of density of $\gamma$-iron to that of $\alpha$-iron at the transition temperature.

• A. $2.087$
• B. $1.087$
• C. $4.06$
• D. $5.09$

Answer 1

The correct option is B $1.087$

In $\alpha -$form

$Z_{\alpha }=2\left(BCC\right)$

$4r_{iron}=a\sqrt{3}$

$\Rightarrow a_{\alpha }=\frac{4r_{iron}}{\sqrt{3}}$ (along body diagonal)

${\rho }_{\alpha }=Density=\frac{2\times M_A\times 3\sqrt{3}}{N_A\times 64r_{iron}^3}$

In $\gamma -$form

$Z_{\gamma }=4$

$4r_{iron}=a_r\sqrt{2}$ (along face diagonal)

$\Rightarrow a_r=2\sqrt{2}{r}_{iron}$

${\rho }_{\gamma }=$ Density $=\frac{4\times M_A}{N_A\times 16\sqrt{2}{r}_{iron}^3}$

$\frac{{\rho }_{\gamma }}{{\rho }_{\alpha }}=\frac{4}{16\sqrt{2}}\times \frac{32}{3\sqrt{3}}=1.088$