Question

Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to ______.

पर्याय

• 25

• 35

• 55

• 75

Answer 1

Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to 35.

Explanation:

In fcc, tetrahedral voids are 8 (in a unit cell).

Hence, Atom X in a unit cell (fcc lattice sites) = `8 xx 1/8 + 6 xx 1/2 = 4`

Atom X in a unit cell (in tetrahedral void) = `1/2 xx 8 = 4`

Total atoms X in one unit cell = 8

For the relation between ‘a’ and ‘r’, tetrahedral voids form at `1/4`th of the body diagonal. 

∴ `(sqrt 3 a)/4 = 2r` or `a = (8 r)/sqrt 3`

Packing efficiency = `(8 xx 4/3 pi r^3)/a^3 xx 100`

= `(8 xx 4/3 pi r^3)/(((8 r)/sqrt 3)^3) xx 100`

= `(32/3 pi r^3)/(((512 r^3)/(3sqrt 3))) xx 100`

= `(32/3 pi r^3) xx ((3sqrt 3)/(512 r^3)) xx 100`

= `(cancel (32)^1/cancel (3) pi cancel (r^3)) xx ((cancel (3)sqrt 3)/(cancel (512)_16 cancel(r^3))) xx 100`

= `((1pi)/1 ) xx ((sqrt 3)/(16 )) xx 100`

= `((pi sqrt 3)/16 ) xx  100`

= `((3.14 xx 1.73)/16 ) xx  100`

= `5.4322/16 xx 100`

= 0.34 × 100

= 34% ≈ 35%