Karnataka Board PUCPUC Science 2nd PUC Class 12

Calculate the Efficiency of Packing in Case of a Metal Crystal For Body-centred Cubic - Chemistry

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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:

b2 = a2 + a2

⇒ b2 =  2a2

⇒ b = `sqrt2a`

 Again, from ΔAFD, we have:

c2 = a2 + b2

⇒ c2 = a2 + 2a2     (Since b2 = 2a2)

⇒ c2 = 3a2

⇒ `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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Chapter 1: The Solid State - Exercises [Page 31]

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NCERT Chemistry Class 12
Chapter 1 The Solid State
Exercises | Q 10.2 | Page 31

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