c123-partb-solutions

c123-partb-solutions - CHEM 123: Practice Problems B1. page...

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CHEM 123: Practice Problems page 1 of 30 B1. Simple cubic packing. The crystal lattic and unit cell for simple cubic packing are shown below. a) # spheres = () 1 8 sphere/corner 8 corners = 1 sphere b) The coordination number of each sphere is 6. To see this, focus on an “interior” sphere such as sphere “A” in the figure on the left. That sphere is in contact with 6 other spheres – the one above it, the one below, the one in front, the one behind, the one to its left and the one to its right. c) If the radius of each sphere is “ R ” then the edge length of the unit cell is a = 2 R d) The volume of the unit cell is V cell = a 3 = 8 R 3 There is one full sphere contained within the unit cell so the volume of spheres contained within the unit cell is 3 spheres 4 3 VR = π . 3 4 3 3 0.5236 6 8 spheres cell V R Packing Efficiency V R π π == = = A
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CHEM 123: Practice Problems page 2 of 30 2 B2. Body-centred cubic packing. The crystal lattice and unit cell for body-centred cubic packing are shown below. a) # spheres = () 1 8 sphere/corner 8 corners + 1 sphere at centre = 2 spheres The coordination number of each sphere is 8. To see this, focus on the sphere at the centre of the cube. (Remember that each sphere is identical – they are shaded different colours for convenience only.) The sphere in the centre touches the 8 corners and thus has a coordination number of 8. b) Let the radius of each sphere be “ R ” . (Note: In the figure above, the radius is denoted r ” rather than “ R ”.) To figure out the edge length of the cube above, we need to use two triangles. 22 2 aa F D += ( FD = face-diagonal) 22 2 aF DB D ( BD = body-diagonal) Therefore, 2 2 14 33 23 B Da a a a B D R =+ = ⇒= = c) The volume of the unit cell is V cell = a 3 = 3 3 46 4 3 R R ⎛⎞ = ⎜⎟ ⎝⎠ The volume of spheres contained within the unit cell is spheres 8 4 2 VR R = π= π . 3 8 3 3 64 3 0.6802 8 spheres cell V R Packing Efficiency V R π π === =
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CHEM 123: Practice Problems page 3 of 30 3 B3. Face-centred cubic packing. The crystal lattice and unit cell for body-centred cubic packing are shown below. a) # spheres = () 1 8 sphere/corner 8 corners + ( )( ) 1 2 sphere/face 6 faces = 1 + 3 = 4 The coordination number of each sphere is 12. To see this, focus on the face-sphere in the top face of the unit cell shown above. That particular sphere is part of two different unit cells. It touches the four face spheres below it, the four corner spheres and another four face spheres of the other unit cell. unit cell. b) Let the radius of each sphere be “ R ” . (Note: In the figure above, the radius is denoted r ” rather than “ R ”.) 22 2 aa F D += ( FD = face-diagonal) 2 a 2 = (4 R ) 2 aR = c) The volume of the unit cell is V cell = a 3 = ( ) 3 3 2 2 16 2 R R = The volume of spheres contained within the unit cell is 33 spheres 16 4 4 VR R ⎛⎞ = π= π ⎜⎟ ⎝⎠ .
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This note was uploaded on 02/25/2012 for the course CHEM 123 taught by Professor Pols during the Spring '07 term at Waterloo.

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c123-partb-solutions - CHEM 123: Practice Problems B1. page...

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