Self test 3.3 please show me the steps and explian Calculating the unoccupied sp

Question

Self test 3.3 please show me the steps and explian Calculating the unoccupied space in a close-packed array Calculate the percentage of unoccupied space in a close-packed arrangement of identical spheres. Answer Because the space occupied by hard spheres is the same in the ccp and hcp arrays, we can choose the geometrically simpler structure, ccp, for the calculation. Consider Fig. 3.15. The spheres of radius r are in contact across the face of the cube and so the length of this diagonal is r + 2r + r = 4r. The side of such a cell is 8^1/2r from Pythagoras' theorem (the square of the length of the diagonal (4r)^2) equals the sum of the squares of the two sides of length a, so 2 Times a^2 = (4r)^2 giving a = 8^1/2r), so the cell volume is (8^1/2r)^3 = 8^3/2r^3. The unit cell contains 1/8 of a sphere at each corner (for 8 Times 1/8 = 1 in all) and half a sphere on each face (for 6 Times 1/2 = 3 in all), for a total of 4. Because the volume of each sphere is 4/3pir^3, the total volume occupied by the spheres themselves is 4 Times 4/3pir^3 = 16/3pir^3. The occupied fraction is therefore (16/3pir^3/(8^3/2r^3) = 16/3 pi/8^3/2, which evaluates to 0.740. The unoccupied fraction is therefore 0.260, corresponding to 26.0 per cent. Calculate the fraction of space occupied by identical spheres in a primitive cubic unit cell. That this arrangement, where each sphere has 12 nearest-neighbours, is the highest possible density of packing spheres was conjectured by Johannes Kepler in 1611; the proof was found only in 1998. A good way of showing this yourself is to get a number of identical coins and push them together on a flat surface; the most efficient arrangement for covering the area is with six coins around each coin.

Explanation / Answer

To alculate the fraction of space occupied by identical spheres in a primitive cubic unit cell divide the volume of sphere to the volume of cube followed by multiply by 100 as follows:

v(sphere)/v(cube)*100%

V(sphere)=4/3r^3
V (cube)= a ^3

and a = 2r then

V (cube)= (2r)^3

Fraction = (4/3r^3 ) /(2r)^3 =

= / 6

= 0.52381 or 52.381%

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