Question

Calculate the efficiency of packing in case of metal crystal for face-centred cubic.

Solution

A, B, C are the centres of the sphere. As sphere on the face centre is touching the spheres at the corners evidently $$AC = 4r$$ But from right angled triangle ABC$$AC = \sqrt{AB^2 + BC^2} + \sqrt{a^2 + a^2} = \sqrt{2}a$$$$\therefore \sqrt{2} a = 4r$$or $$a = \dfrac{4}{\sqrt 2^r}$$$$\therefore$$ volume of the unit cell$$= a^3 = \left [ \dfrac{4}{\sqrt 2}r \right ]^3 = \dfrac{32}{\sqrt 2}r^3$$Number of spheres in the unit cell$$= 8 \times \dfrac{1}{8} + 6 \times \dfrac{1}{2} = 4$$volume of 4 spheres $$= 4 \times \dfrac{4}{3} \pi r^3 = \dfrac{16}{3} \pi r^3$$$$\therefore$$ fraction occupied, i.e. packing fraction$$= \dfrac{16 \pi r^3 / 3}{32 r^3 \sqrt 2} = \dfrac{\pi \sqrt 2}{6} = 0.74$$or % occupied $$= 74\%$$Chemistry

Suggest Corrections

1

Similar questions
View More

People also searched for
View More