Question

The correct statement(s) for cubic close packed (CCP) three-dimensional structure is(are):
A. The number of the nearest neighbors of an atom present in the topmost layer is 12
B. The efficiency of atom packing is 74%
C. The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively
D. The unit cell edge length is $2\sqrt{2}$ times the radius of the atom

Answer 1

Hint:Three-dimensional close packing in solids is referred to as putting the second square closed packing exactly above the first. In this tight packing, the spheres are horizontally and vertically correctly balanced.

Complete step-by-step answer:Packing efficiency is the percentage ration of the total volume of a solid occupied by spherical atoms. The mathematical equation can be written as

[No. of atoms x Vol. obtained by 1 share / Total vol. of unit cell x 100%]
For calculating the packing efficiency in cubic close packing, we will assume the unit cell with the edge length of ‘a’ and face diagonals AC to let it ‘b’. when we see the ABCD face of the cube, we see the triangle of ABC in it. Assume the radius is ‘r’.
In triangle ABC,
AC2 = BC2 + AB2
Though AC = b and BC = a
Thus,
$b^{2}=a^{2}+a^{2}$
$b=\sqrt{2} a$
Also, the edge ‘b’ can be defined in terms of radius ‘r’, which is equal to
b = 4r
Therefore, according to the above equations we can write
$a=2\sqrt{2} r$
There are a total of 4 spheres in a CCP structure unit cell, the total volume occupied will be
$4\ \times 4/3\pi r3$
And the total volume of a cube is the cube of its length of the edge 3. It means a3 or is defined in terms of ‘r’, then it is
$(2\sqrt{2} r)$
Thus, packing efficiency will be written as
Packing efficiency = Vol. Occupied by 4 spheres / Total Vol. of unit cell x 100%
$4\times 4/3\pi r3/(2\sqrt{2} r)3\times 100\%$
$16/3\pi r3/(2\sqrt{2} r)3\times 100\%$
Thus, packing efficiency is calculated as 74%.
Now, let the number of close-packed spheres be N, then
The number of octahedral voids generated = N
Then number of tetrahedral voids generated = 2N
From this, we can conclude that the number of octahedral voids generated is equal to the number of close-packed spheres. The number of tetrahedral voids generated is equal to 2 times the number of closed-packed spheres.

Therefore, options (B), (C), and (D) are correct.

Note:In cubic close packing, the layers are arranged in symmetry exactly above each other. This form takes the shape of a cube and the coordination number is 12.