Q. 33

# The correct order of the packing efficiency in different types of unit cells is ________.

A. fcc < bcc < simple cubic

B. fcc > bcc > simple cubic

C. fcc < bcc > simple cubic

D. bcc < fcc > simple cubic

Answer :

Packing efficiency is defined as the total space occupied by the constituent particles in a unit cell of a crystal lattice. Packing efficiency of each unit cell can be calculated.

For fcc,

Let us consider an fcc unit cell cube with edge length *a* and the face diagonal which connects the centers of the particles at the edges and the face AC = *b.*

In ΔABC,

AC^{2} = b^{2} = BC^{2} + AB^{2}

= a^{2} + a^{2} = 2a^{2}

b = a

If *r* is the radius of the sphere, then b = 4r =

Or,

Each unit cell in in ccp has 4 spheres. Volume of each sphere is ^{3}. So, volume of 4 spheres is 4 × ^{3} and the volume of the cube is *a*^{3} or [^{3}.

Packing efficiency is then calculated as

Packing efficiency =

Substituting values, we get, P.E. = = 74%.

Now we calculate the packing efficiency of bcc.

In body-centered cubic cell, the atoms are placed at the corners of the cube with one atom in the center of the cube. The central atom will be in touch with the other two diagonally arranged atoms.

In ΔEFD, FD^{2} = EF^{2} + ED^{2}

EF = ED = *a*, which is the side of the cube. FD = *b*.

b = a

In ΔAFD, let the diagonal AF = c.

c^{2} = a^{2} + b^{2} = a^{2} + 2a^{2} = 3a^{2}.

c = a.

The length of the diagonal of the cube is equal to 4 times of the radius of the sphere as the three spheres are along the diagonal.

Therefore, a = 4r

a = and r = a.

This structure contains two spherical atoms and the volume is calculated as 2 × ^{3}

The volume of the cube is *a*^{3}. Substituting *a* gives []^{3}.

Calculating packing efficiency,

Packing efficiency = = 68%

Now we calculate the packing efficiency of simple cubic cell.

In this arrangement, the atoms are located only at the corners of the cube and the edges of the spheres touch each other. The side of the cube, *a*, is equal to twice the radii *r* of the sphere due to this structure. This can be written as *a = 2r*.

Volume of the cube is (side)^{3} i.e. a^{3} = 8r^{3}

A simple cubic cell contains only one atom, so the volume will be

Packing efficiency is calculated as

Packing efficiency =

= = 52.4%.

From these results, it is concluded that the correct answer is (ii).

Rate this question :

An element ‘*X*’ (At. mass = 40 g mol^{–1}) having f.c.c. structure, has unit cell edge length of 400 pm. Calculate the density of ‘*X*’ and the number of unit cells in 4 g of ‘*X*’. (*N _{A}* = 6.022 × 10

^{23}mol

^{–1})

Note : In the following questions a statement of assertion followed by a statement of reason is given. Choose the correct answer out of the following choices.

Chemistry - ExemplarThe correct order of the packing efficiency in different types of unit cells is ________.

Chemistry - ExemplarThe edge lengths of the unit cells in terms of the radius of spheres constituting fcc, bcc and simple cubic unit cell are respectively________.

Chemistry - ExemplarA cubic solid is made of two elements P and Q. Atoms of Q are at the corners of the cube and P at the body-centre. What is the formula of the compound? What are the coordination numbers of P and Q?

NCERT - Chemistry Part-I