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Question

Consider the bcc unit cells of the solids 1 and 2 with the position of atoms as shown below. The radius of atom B is twice that of atom A. The unit cell edge length is 50% more in solid 2 than in 1. What is the approximate packing efficiency in solid 2?

A
45%
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B
65%
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C
90%
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D
75%
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Solution

The correct option is C 90%Given, rB=2 rA where, rA is radius of atom A rB is radius of atom B Packing fraction for Solid 2 is: P.F.=(Zeff×43πr3A)A+(Zeff×43πr3B)Ba3 where, a3 is volume of the unit cell of solid B For a body centred unit cell, the atoms at corner and body centred are in contact with each other by body diagonals. ∴ 2(rA+rB)=√3 a ⇒2(rA+2rA)=√3 a ⇒6rA=√3 a ⇒2√3 rA=a Voume of unit cell of solid B =(2√3 rA)3=8×3√3 r3A Zeff=1 for both A and B atoms. ⇒P.F.=1×43πr3A+43π(8r3A)8×3√3r3A=9×43π8×3√3=π2√3 Packing efficiency =π2√3×100≃90%

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