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Calculate the efficiency of packing in case of a metal crystal for body-centred cubic

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#### Solution

**Body-centred cubic:**

It can be observed from the above figure that the atom at the centre is in contact with the other two atoms diagonally arranged.

From ΔFED, we have:

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

⇒ b^{2} = 2a^{2}

⇒ b = `sqrt2a`

Again, from ΔAFD, we have:

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

⇒ c^{2} = a^{2} + 2a^{2} (Since b^{2} = 2a^{2})

⇒ c^{2} = 3a^{2}

⇒ `c = sqrt3a`

Let the radius of the atom be *r*.

Length of the body diagonal, *c* = 4π

⇒`sqrt3a = 4r`

⇒`a =(4r)/sqrt3`

or `r = (sqrt3a)/4`

Volume of the cube `a^3 = ((4r)/sqrt3)^3`

A body-centred cubic lattice contains 2 atoms.

So, volume of the occupied cubic lattice `2pi4/3 r^3`

=`8/3pir^3`

:.Packing efficiency = `("Voulume occupied by two spheres in the unit cell")/"Total volume of unit"xx100%`

= `(8/3pir^3)/(4/(sqrt3)r)^3xx100%`

=`(8/3pir^3)/(64/(3sqrt3)r^3) xx 100%`

=68%

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