Numericals on formula of compound and voids

Question

Atoms ‘X’ occupy corners and face centres of a cubic unit cell. Atoms ‘Y’ occupy edge centres and body centre. If one atom from corners,four atoms from edge centres and one atom from body centre are missing the formula of the compound would be

Moderate

Solution

‘X’ occupies 7 corners (1 atom of ‘X’ are missing from corners) and 6 face centres.

$\therefore $effective number of atoms of ‘X’ = $7\times \frac{1}{8}+6\times \frac{1}{2}=\frac{31}{8}$

‘Y’ occupies 8 edge centres (4 atoms of ‘Y’ are missing from edge centres)

$\therefore $effective number of atoms of ‘Y’ = $8\times \frac{1}{4}=\frac{8}{4}$

Ratio of atoms of ‘X’ to ‘Y’ =$\frac{31}{8}:\frac{8}{4}=31:16$

Formula of the compound = ${\mathrm{X}}_{31}{\mathrm{Y}}_{16}$

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