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Metallic magnesium has a hexagonal close-packed structure and a
density of 1.74 g/cm3. Assume magnesium atoms to be spheres of radius r.
74.1 of the space are occupied by atoms. Calculate the volume of each
atom and the atomic radius r. (Mg = 24.31)

Show answer and solution
Packing efficiency of hcp
=6 × vol.  of 1 atomVol.  of unit cell= \frac{6\ \times \ vol.\ \ of\ 1\ atom}{Vol.\ \ of\ unit\ cell}
Volume of unit cell =Z×MNA×d= \frac{Z \times M}{N_{A} \times d}
=6×24.316.023×1023×1.74= \frac{6 \times 24.31}{6.023 \times 10^{23} \times 1.74}
=13.9179×10−23cm3= 13.9179 \times 10^{- 23}{cm}^{3}
So, volume of 1 atom
=Packing fraction×Vol. of unit cell6= \frac{Packing\ fraction \times Vol.\ of\ unit\ cell}{6}
=0.741×13.9179×10−236= \frac{0.741 \times 13.9179 \times 10^{- 23}}{6}
=1.718×10−22cm3= 1.718 \times 10^{- 22}{cm}^{3}
Volume of 1 atom =43πr3= \frac{4}{3}\pi r^{3}
Radius of 1 atom
=(r)=(Volume  of 1 atom×34π)1/3= (r) = {(\frac{Volume\ \ of\ 1\ atom \times 3}{4\pi})}^{1/3}
=(0.41055×10−23)1/3=1.60A˚.= {(0.41055 \times 10^{- 23})}^{1/3} = 1.60Å.

Source: Grass-G / CH-19 Chemistry Paper 28 Nov. Fourth.docx

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