22-1 states of matter experimental basishtaube/_1gwi2010/1lec/22po_smviii... · 2008-04-10 · 22-1...
TRANSCRIPT
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States of Matter
SM VIII (post)
Crystallography
Experimental Basis Crystal Systems Closed Packing Ionic Structures skip in 2008 Ref 12: 8 Prob in-text: 12: 9, 10
end: 12: 59, 61 - 63, 65a,b, 70
Adv Rdg 16: 1,2, 4,5, 8,9
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Experimental Basis is X-ray diffraction;
see HT Fig. 21.1, Pet. Fig. 12.43 & 12.44.
Recall: diffraction & interference occur
if slits (openings) in a barrier
and wavelength are of similar size.
This requirement is satisfied when considering
distances between planes in crystals & (similar to distances between atoms)
wavelength of X-rays:
λ of X-rays atomic distances
~ 10 –10 m ~ 100 pm
0.1 nm 0.1 nm
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HT Fig. 22.1 Block Diagram of X-Ray Crystallography
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Pet. Fig. 12.43 Experimental Basis of Crystallography
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Pet. Fig. 12.44 Diffraction by crystal planes
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General Comments
• crystals have 3D repeating pattern of
molecular arrangements
• “unit cell” is smallest repeating unit • whole crystal can be built
by stacking unit cells in all directions,
without gaps/voids
Illustration:
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Crystal Systems 7 systems exist
see HT Fig. 22.2
all unit cells are parallelepipeds, (six faces, opposite faces parallel)
but differ in length/angle relationships
in addition, different “lattice types” may exist
for each system,
e.g., simple,
body centered,
face centered
altogether: 14 lattice types
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HT Fig. 22.2 Crystal Systems
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Crystal systems …
In CHEM 101/3: deal mostly w/ cubic system;
(once w/ hexagonal system)
See Pet. Fig. 12.38
Distinguish
• simple cubic
• body centered cubic, bcc
• face centered cubic, fcc
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Pet. Fig. 12.38 Cubic Crystal Systems
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Counting Particles in a Unit Cell (see Pet. Fig. 12.42)
• basic assumption: atoms/molecules are hard spheres in touching contact
• counting particles in unit cell:
body center: 1 (not shared)
face center: 12 (shared by 2 cells)
edge center: 14 ( shared by 4 cells)
corner : 18 (shared by 8 cells)
Total count of particles per unit cell :
Simple cubic: 8 x 1/8 = 1
bcc : (8 x 1/8) + 1 = 2
fcc : (8 x 1/8) + (6 x 1/2) = 4
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Pet. Fig. 12.42 Counting in Unit Cells
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Coordination Number = # of nearest neighbors, in touching contact
Practice:
Analyze Pet. Fig. 12.38 &
12.42
simple cubic: 6
bcc: 8
fcc 12
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Close Packing
= max. occupation of space by spherical objects, atoms in particular.
(see Pet. Fig. 12.39)
1st layer (“a”) of spheres:
6 spheres surround a central sphere
in hexagonal fashion
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Pet. Fig. 12.39 Close Packing
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close packing …
2nd layer (b)
spheres center on dips (dimples, indentations)
of first layer
notice: only 1/2 of dips are filled
leaving 2 types of dips in the 2nd layer:
type “c”, can see through layer a (octahedral in Pet.)
type “h”, can see tops of layer a (tetrahedral in Pet.)
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close packing …
3rd layer
• if spheres center on type “c” dips:
layers repeat abcabcabc…
= cubic closest packing (ccp),
has face centered cubic (fcc) unit cell
see Pet. Fig. 12.40
coordination # = 12 , # per unit cell = 4
• if spheres center on type “h” dips:
layers repeat ababab…
= hexagonal closest packing (hcp),
has “body centered” hexagonal (“bch”) unit cell (this statement is not quite true; the “central yellow” atom is somewhat off center) see Pet. Fig. 12.41
coordination # = 12 , # per unit cell = 2
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blank, deliberately
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Pet. Fig. 12.40 Cubic Closest Packing (ccp)
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Pet. Fig. 12.41 Hexagonal Closest Packing (hcp)
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Packing Efficiency (PE) • unit cell is only partially occupied by atoms
(“hard spheres”)
• the rest is empty space (voids)
• PE = space occupiedvol. of unit cell x 100%
• can be determined by simple geometric
calculations; see HT Fig. 22.3, 22.4, 22.5
Summary:
system PE (%)
simple cubic 52.4
bcc 68.0
closest packing
ccp (= fcc) 74.0
hcp (= bch)
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HT Fig. 22.3 PE in Simple Cubic Unit Cell
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HT Fig. 22.4 PE in bcc Unit Cell
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HT Fig. 22.5 PE in fcc Unit Cell
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Applied Problems Need the following as background
• vol. of unit cell → PE vol. of particles
• crystal type # of particles per unit cell
• mass of unit cell
= (# of particles) x (mass of 1 particle) (m = MM/NA)
• density = mass of unit cell
volume of unit cell
Practice: Sample Final, #11
Pet. 12: 9, 10, 62 - 66
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Practice Germanium, Ge, crystallizes in a (complex) cubic system; Density, D, of Ge = 5.36 g/cm3 ; Length of unit cell edge = 565 pm. Q. What is the # of Ge atoms per unit cell? Approach a.) Determine volume of unit cell, V. Use cubic formula, V = a3 b.) Determine mass of unit cell, from density. c.) Determine mass of a Ge atom. d.) Relate mass of 1 Ge atom to mass of unit cell → # of atoms per unit cell
a.) V = (565 pm)3 = (565 pm x 1 x 10-12 m
1 pm )3 = 1.804 x 10-28 m3
b.) mass of unit cell: D = m/V; m = D V
m = 5.36g
cm3 x (100cm
m )3 x 1.804 x 10-28 m3= 9.67 x 10-22 g
c.) mass of Ge atom:
m (Ge) = MM NA
= 72.6 g/mol
6.02 x 1023 atoms/mol = 1.206 x 10-22 g/atom
d.) # of Ge atoms per unit cell:
mass of unit cellmass of Ge atom =
9.67 x 10-22 g 1.206 x 10-22 g = 8.02, close to 8.
∴there are 8 Ge atoms in a unit cell,
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Ionic Crystal Structures
Skip in 2008
• type of structure depends, to large extent, on
ratio of ionic radii (usually cation/anion)
• 2 important cases:
1.) if ratio between 0.4 - 0.7,
get “rock- salt” (=NaCl) structure,
e.g., NaCl, RbI, CaO, AgCl
2.) if ratio between 0.7 – 1.0,
get CsCl structure,
e.g., CsCl, CsI
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NaCl Structure • see Pet. Fig. 12.48
• anions occupy ccp (fcc; abcabc ...) positions;
• somewhat expanded (puffed up);
no longer touching each other
• “octahedral holes” become large enough
to accommodate cations for a tight fit (the center of the fcc cell is such an “octahedral hole”)
• can also be seen as 2 interpenetrating fcc systems
• unit cell has 4 Cl– & 4 Na+ ions
• coordination # (# of cations touching anions & vv.):
= 6
i.e., Na+ has 6 Cl– neighbors;
Cl– has 6 Na+ neighbors
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Pet.Fig.12.48 NaCl Unit Cell
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CsCl Structure
• see Pet. Fig. 12.49
• anions occupy simple cubic (“primitive”)
positions;
• somewhat expanded
• cations fit into body center positions
• can also be perceived as 2 inter-penetrating
simple cubic networks
• unit cell has 1 Cl– & 1 Cs+ ion (mistake in pre notes)
• coordination # = 8
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Pet.Fig. 12.49 CsCl Unit Cell