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A bcc lattice is made up of hollow spheres of X. Spheres of solid 'Y' are present in hollow spheres of X. The radius of 'Y' is half of the radius of 'X' . Calculate the ratio of the total volume of spheres of 'X' unoccupied by Y in a unit cell and volume of the unit cell ?

Answer» <html><body><p></p>Solution :Let the <a href="https://interviewquestions.tuteehub.com/tag/radius-1176229" style="font-weight:bold;" target="_blank" title="Click to know more about RADIUS">RADIUS</a> of <a href="https://interviewquestions.tuteehub.com/tag/hollow-1028416" style="font-weight:bold;" target="_blank" title="Click to know more about HOLLOW">HOLLOW</a> sphere X be r. <br/> As spheres X have <a href="https://interviewquestions.tuteehub.com/tag/bcc-389643" style="font-weight:bold;" target="_blank" title="Click to know more about BCC">BCC</a> structure, edge length (a)=`"4r"/sqrt3 "" (therefore "for bcc", r=sqrt3/4 a)` <br/> `therefore` <a href="https://interviewquestions.tuteehub.com/tag/volume-728707" style="font-weight:bold;" target="_blank" title="Click to know more about VOLUME">VOLUME</a> of the unit cell =`a^<a href="https://interviewquestions.tuteehub.com/tag/3-301577" style="font-weight:bold;" target="_blank" title="Click to know more about 3">3</a>=("4r"/sqrt3)^3` <br/> Radius of sphere Y=`r/2` (Given ) <br/> `therefore` Volume of sphere `X=4/3pir^3` <br/> Volume of sphere `Y=4/3 pi(r/2)^3` <br/> `therefore` Volume of X unoccupied by Y in unit cell =`2xx[4/3pir^3-4/3pi(r/2)^3]` <br/> ( `because` bcc lattice has 2 spheres per unit cell ) <br/> `=2xx4/3pir^3(1-1/8)=2xx4/3pir^3xx7/8` <br/> `therefore "Volume of X unoccupied by Y in unit cell"/"Volume of unit cell"=(2xx4/3pir^3xx7/8)/((4r)/sqrt3)^3=(7pisqrt3)/64`</body></html>


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