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Question

Select the right expression for determining the Packing fraction (PF) of NaCl unit cell (assume an ideal crystal), if the ions along an edge diagonal are absent: (Where r+ represents the radius of the cation and r represents the radius of the anion)

A
PF=d43[r3++r3]162r3
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B
PF=43π[52r3++4r3]162r3
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C
PF=43π[52r3++r3]162r3
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D
PF=43π[72r3++r3]162r3
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Solution

The correct option is B PF=43π[52r3++4r3]162r3
An edge diagonal runs from the middle of one edge - from the edge centre to the diagonally opposite edge centre.
Na+ ions occupy the edge centres and Cl ions occupy the FCC lattice.
Removing atoms along an edge diagonal, means that two edge centres and one body centre get removed. So, out of 4, 2×14 (edge centres) and 1 (body centre) are not present anymore, giving us 432=52Na+ atoms.
Na+=5/2,Cl=4
For FCC, 22a=4R and R=r here.
Volume of unit cell = a3=162r3
Occupied volume = 4πr3×52×r3+3+4πr3×4×r33=4π3×[5r3+2+4r3]
Hence, packing fraction = Occupied volume / Total volume
= 4π3×[5r3+2+4r3]162r3

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