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A metallic element crystallizes into a lattice containing sequenceof layers of ABABAB….. Any packing of spheres leaves out voids in the lattice. What precentage by volume of this lattice is empty space.

Answer» <html><body><p></p>Solution : ABAB….. Type of packing means hexagonal close packing. A <a href="https://interviewquestions.tuteehub.com/tag/unit-1438166" style="font-weight:bold;" target="_blank" title="Click to know more about UNIT">UNIT</a> cell ( with lifted layers)is shown in the <a href="https://interviewquestions.tuteehub.com/tag/fig-460913" style="font-weight:bold;" target="_blank" title="Click to know more about FIG">FIG</a> 1.61. <br/>suppose the radius of each sphere =r <br/>Volumeof the unit cell = Base area ` xx` height (h) <br/>Base area of regular hexagon = Area of <a href="https://interviewquestions.tuteehub.com/tag/six-631160" style="font-weight:bold;" target="_blank" title="Click to know more about SIX">SIX</a> equilateral triangle s. <br/> each with side a ( i.e,2r) =` 6 xx sqrt3/4 a^(2)` <br/>Height h =2 ` xx` Distance between closest packed layers.<br/>` 2xx <a href="https://interviewquestions.tuteehub.com/tag/sqrt-1223129" style="font-weight:bold;" target="_blank" title="Click to know more about SQRT">SQRT</a>(2/3) a` <br/>voluvme of unit cell =` ( 6 xx sqrt3/4 a^(2)) ( 2xx sqrt(2/3) a)= 3sqrt2 a^(3)` <br/> putting a = 2r, volume of the unit cell` =3 sqrt 2 ( 2r)^(3)= 24 sqrt(2) r^(3)` <br/>No. of atoms in hcp per unit cell =` 12xx 1/6` (corners)` + 2 xx 1/2` ( <a href="https://interviewquestions.tuteehub.com/tag/face-982590" style="font-weight:bold;" target="_blank" title="Click to know more about FACE">FACE</a> centres)+ 3(in the body) = 6<br/>Volume of six spheres ` = 6 xx 4/3 pi r^(3) = 8 pi r ^(3)`<br/> packing fraction = `( 8 pi r^(3))/(24 sqrt2 r^(3)) = pi/( 3sqrt2)= 3.143/(3xx 1.414)= 0.74`<br/> % volume occupied = 74 % <br/>% volume empty space = 26 %<br/> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/PR_CHE_V01_XII_C01_E28_008_S01.png" width="80%"/></body></html>


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