zumdahl’s chapter 10 and crystal symmetries
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Zumdahl’s Chapter 10 and Crystal Symmetries. Liquids Solids. Intermolecular Forces The Liquid State Types of Solids X-Ray analysis Metal Bonding Network Atomic Solids Semiconductors. Molecular Solids Ionic Solids Change of State Vapor Pressure Heat of Vaporization Phase Diagrams - PowerPoint PPT PresentationTRANSCRIPT
Zumdahl’s Chapter 10 and Crystal Symmetries
Liquids Solids
Contents
Intermolecular Forces The Liquid State Types of Solids
X-Ray analysis Metal Bonding Network Atomic
Solids Semiconductors
Molecular Solids Ionic Solids Change of State
Vapor Pressure Heat of Vaporization
Phase Diagrams Triple Point Critical Point
Intermolecular Forces
Every gas liquifies. Long-range attractive forces overcome thermal
dispersion at low temperature. ( Tboil ) At lower T still, intermolecular potentials are
lowered further by solidification. ( Tfusion ) Since pressure influences gas density, it also
influences the T at which these condensations occur. What are the natures of the attractive forces?
London Dispersion Forces
AKA: induced-dipole-induced-dipole forces Electrons in atoms and molecules can be
polarized by electric fields to varying extents. Natural electronic motion in neighboring atoms
or molecules set up instantaneous dipole fields. Target molecule’s electrons anticorrelate with
those in neighbors, giving an opposite dipole. Those quickly-reversing dipoles still attract.
Induced Dipolar Attraction
Strengths of dipolar interaction proportional to charge and distance separated.
So weakly-held electrons are vulnerable to induced dipoles. He tight but Kr loose.
Also l o n g molecules permit charge to separate larger distances, which promotes stronger dipoles. Size matters.
+ •••••• –– •••••• +
Permanent Dipoles
Non-polar molecules bind exclusively by London potential R–6 (short-range)
True dipolar molecules have permanently shifted electron distributions which attract one another strongly R–4 (longer range). Gaseous ions have strongest, longest range
attraction (and repulsion) potentials R–2. Size being equal, boiling Tpolar > Tnon-polar
Strongest Dipoles
“Hydrogen bonding” potential occurs when H is bound to the very electronegative atoms of N, O, or F.
So H2O ought to boil at about – 50°C save for the hydrogen bonds between neighbor water molecules.
It’s normal boiling point is 150° higher!
The Liquid State (Hawaii?)
The most complex of all phases. Characterized by
Fluidity (flow, viscosity, turbulence) Only short-range ordering (solvation shells) Surface tension (beading, meniscus, bubbles)
Bulk molecules bind in all directions but unfortunate surface ones bind only hemispherically.
Missing attractions makes surface creation costly.
Type of Solids
While solids are often highly ordered structures, glass is more of a frozen fluid. Glass is an amorphous solid. “without shape”
In crystalline solids, atoms occupy regular array positions save for occasional defects.
Array composed by stacking of the smallest unit cell capable of reproducing full lattice.
Types of Lattices
While there are quite a few Point Groups and hundreds of 2D wallpaper arrangments, there are only SEVEN 3D lattice types. Isometric (cubic), Tetragonal, Orthorhombic,
Monoclinic, Triclinic, Hexagonal, and Rhombohedral.
They differ in the size and angles of the axes of the unit cell. Only these 7 will fill in 3D space.
Isometric (cubic)
Cubic unit cell axes are all THE SAME LENGTH MUTUALLY
PERPENDICULAR E.g.,“Fools Gold” is
iron pyrite, FeS2, an unusual +4 valence.
Tetragonal
Tetragonal cell axes: MUTUALLY
PERPENDICULAR 2 SAME LENGTH
E.g., Zircon, ZrSiO4. This white zircon is a Matura Diamond, but only 7.5 hardness.
Real diamond is 10.Diamonds are nottetragonal but ratherface-centered cubic.
Orthorhombic
Orthorhombic axes: MUTUALLY
PERPENDICULAR NO 2 THE SAME
LENGTH E.g., Aragonite, whose
gem form comes from the secretion of oysters; it’s CaCO3.
Monoclinic
Monoclinic cell axes: UNEQUAL LENGTH 2 SKEWED but
PERPENDICULAR TO THE THIRD
E.g., Selenite (trans. “the Moon”) a fully transparent form of gypsum, CaSO4•2H2O
Triclinic
Triclinic cell axes: ALL UNEQUAL ALL OBLIQUE
E.g., Albite, colorless, glassy component of this feldspar, has a formula NaAlSi3O8. Silicates are the most
common minerals.
Hexagonal
Hexagonal cell axes: 3 EQUAL C2
PERPENDICULAR TO A C6
E.g., Beryl, with gem form Emerald and formula Be3Al2(SiO3)6
Diamonds are cheaper than perfect emeralds.
Rhombohedral
Rhombohedral axis: CUBE stretched (or
squashed) along its diagonal. (a=b=c)
DIAGONAL is bar 3 “rotary inversion”
E.g., Quartz, SiO2, the base for amethyst with it purple color due to an Fe impurity.
_3
Identification (Point Symmetry Symbols)
Lattice Type Isometric Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Rhombohedral
Essential Symmetry Four C3
C4
Three perpendicular C2
C2
None (or rather “i” all share)
C6
C3
Classes
Although there’s only 7 crystal systems, there are 14 lattices, 32 classes which can span 3D space, and 230 crystal symmetries.
Only 12 are routinely observed. Classes within a system differ in the
symmetrical arrangement of points inside the unit cube.
Since it is the atoms that scatter X-rays, not the unit cells, classes yield different X-ray patterns.
Common Cubic Classes
Simple cubic “Primitive” P
Body-centered cubic “Interior” I
Face-centered cubic “Faces” F
“Capped” C if only on 2 opposing faces.
BCC
FCC
Materials Density
Density of materials is mass per unit volume. Unit cells have dimensions and volumes. Their contents, atoms, have mass.
So density of a lattice packing is easily obtained from just those dimensions and the masses of THE PORTIONS OF atoms actually WITHIN the unit cell.
Counting Atoms in Unit Cells
INTERIOR atoms count in their entirety.
FACE atoms count for only the ½ inside.
EDGE atoms count for only the ¼ inside.
CORNER atoms are only 1/8 inside.
Gold’s Density from Unit Cell
Gold is FCC. a = b = c = 4.07 Å # Au atoms in cell:
1/8 (8) + ½ (6) = 4 M = 4(197 g) = 788 g
Volume NAv cells: (4.0710–10 m)3 Nav
3.9010–5 m3 = 39.0 cc = M / V = 20.2 g/cc
4 Å
Bravais Lattices
7 lattice systems + P, I, F, C options P: atoms only at the corners. I: additional atom in center. C: pair of atoms “capping” opposite faces. F: atoms centered in all faces.
Totals 14 types of unit cells from which to “tile” a crystal in 3d, the Bravais Lattices.
Adding point symmetries yields 230 space groups.
capped
New Names for Symmetry Elements What we learned as Cn (rotation by 360°/n),
is now called merely n. 3’s a 3-fold axis. Reflections used to be but now they’re m
(for mirror). So mmm means 3 mirrors. In point symmetry, Sn was 360°/n and then
but now it is just n, still a 360°/n but now followed by an inversion (which is now 1).
––
Triclinic Lattice Designation
Triclinic: All 7 lattice systems
have centrosymmetry, e.g., corner, edge, face, & center inversion pts!
Designation: 1
These are inversion points only because the crystal is infinite!
While all 7 have these, triclinic hasn’t other symmetry operations.
It’s 1 means inversion.
–
–
Cubic (isometric) Designation
The principal rotation axes are “4”, but it is the four 3 axes that are identifying for cubes.
The 4–fold axes have an m to each.
Each 3–fold axis has a trio of m in which it lies. All 3 to be shown.
The cube is m 3 m All its other symmetries
are implied by these.3
m
m
The Three Cubic Lattices
Where before we called them simple, body-centered, and face-centered cubics, the are now P m3m, I m3m, and F m3m, resp.
The cubic has the highest and the triclinic the lowest symmetry. The rest of the Bravais Lattices fall in between. We will designate only their primitive cells.
It will help when we get to a real crystal.
Ortho vs. Merely Rhombic
Orthorhombic all 90° but a b c. Trivial.
It’s mmm because:
Rhombohedral all s = but 90°; a = b = c
It’s 3m because:3–
–
m
Last of the Great Rectangles
Tetragonal all 90° and a = b c
Principle axis is 4 which is m
But it is also || to mm So it is designated as
4/m mm Abbreviated 4/mmm
4
m
m
m
Nature’s Favorite for Organics
Monoclinic a b c = = 90° < Then b is a 2-fold axis
and to m So it is 2/m
b is a 2 because the crystal is infinite.2
m
(finally) Hexagonal
Hexagonal refers to the outlined rhomboid ( =120° ) of which there are six around the hexagon! So a 6
That 6 has a m and two || mm. m is a mirror because
the crystal’s infinite.
6
mm
m
So it is 6/m mm
Lattice Notation Summary
Lattice Type Isometric “Cubic” Tetragonal Orthorhombic Monoclinic Triclinic Hexagonal Rhombohedral
Crystal Symmetries m 3 m ( m4 + 3+||+||+|| ) 4 / m mm (4 m + ||+|| ) mmm (m m m) 2 / m ( 2 m) 1 (invert only) 6 / m mm (6 m + ||+|| ) 3 m ( 3 + ||+||+|| )
_
_ _
X-ray Crystal Determination
Since crystals are so regular, planes with atoms (electrons) to scatter radiation can be found at many angles and many separations.
Those separations, d, comparable to , the wavelength of incident radiation, diffract it most effectively. The patterns of diffraction are characteristic of
the crystal under investigation!
Diffraction’s Source
X-rays have d. X-rays mirror reflect
from adjacent planes in the crystal.
If the longer reflection exceeds the shorter by n, they reinforce. If by (n+½), cancel!
2d sin = n , Bragg
d
reinforced
d sin
Relating Cell Contents to
Atomic positions replicate from cell to cell. Reflection planes through them can be
drawn once symmetries are known. Directions of the planes are determined by
replication distances in (inverse) cell units. Interplane distance, d, is a function of the
direction indices (Miller indices).
Inverse Distances
The index for a full cell move along axis b is 1. Its inverse is 1.
That for ½ a cell on b is ½. Its inverse is 2.
Intersect on a parallel axis is ! Its inverse makes more sense, 0.
Shown is (3,2,0)
a
b
c
a/3
b/2
Interplane Spacings (cubic lattice)
Set of 320 planes at right (looking down c). Their normal is yellow. (h,k,l) = (3,2,0)
Shifts are a/h, b/k, c/l Inverses h/a, k/b, l/c Pythagoras in inverse! d–2
hkl = (h/a)–2 + (k/b)–2 + (l/c)–2 for use in Bragg
Bragg Formula
2 sin / = 1 / d (conveniently inverted)
Let the angles opposite a, b, and c be , , and . (All 90° if cubic, etc.)
Then Bragg for cubic, orthorhombic, monoclinic, and triclinic becomes:
2 sin / = [ (h/a)2 + (k/b)2 + (l/c)2 + 2hkcos/ab + 2hlcos/ac + 2klcos/bc ]½
a b
c
Unit Cell Parameters from X-ray
Triclinic Monoclinic Orthorhombic Tetragonal Rhombohedral Hexagonal Cubic
a b c; a b c; = = 90° < a b c; = = = 90° a = b c; = = = 90° a = b = c; = = 90° a = b = c; = = 90°; = 120° a = b = c; = = = 90°
New Space Symmetry Elements
Glide Plane Simultaneous mirror
with translation || to it. a, b, or c if glide is ½
along those axes. n if by ½ along a face. d if by ¼ along a face.
Screw axis, nm
Simultaneous rotation by 360°/n with a m/n translation along axis.
cell 2
cell 1 32 screwa glide
Systematic Extinctions
Both space symmetries and Bravais lattice types kill off some Miller Index triples!
Use missing triples to find P, F, C, I E.g., if odd sums h+k+l are missing, the unit cell is
body-centered and must be I. Use them to find glide planes and screw axes.
E.g., if all odd h is missing from (h,k,0) reflections, then there is an a glide (by ½) c.
http://tetide.geo.uniroma1.it/ipercri/crix/struct.htm
Nature’s Choice Symmetries
36.0% P 21 / c monoclinic 13.7% P 1 triclinic 11.6% P 21 21 21 orthorhombic
6.7% P 21 monoclinic 6.6% C 2 / c monoclinic 25.4% All (230 – 5 =) 225 others!
75% these 5; 90% only 16 total for organics. Stout & Jensen, Table 5.1
_
Packing in Metals
A B A : hexagonal close pack A B C : cubic close pack
Relationship to Unit Cells
A B C : cubic close pack
Is FCC
A
B CA
ABA (hcp) Hexagonal
The white lines indicate anelongated hexagonal unitcell with atoms at its equatorand an offset pair at ¼ & ¾.
If we expand the cell to seeit’s shape, we get a diamondat both ends…3 make a hexagonwhose planes are 90° to the
sides of the (expanded) cell.
120°
90°
A
A
B
Alloys (vary properties of metals)
Substitutional Heteroatoms swap originals, e.g., Cu/Sn (bronze)
Intersticial Smaller interlopers fit in interstices (voids) of
metal structure, e.g., Fe/C (steels) Mixed
Substitutional and intersticial in same metal alloy, e.g., Fe/Cr/C (chrome steels)
Phase Changes
Phase changes mean Structure reorganization Enthalpy changes, H Volume changes, V
Solid-to-Solid E.g., red to white P
Solid-to-Liquid Hfusion significant Vfusion small
Solid-to-Gas Hsublimation very large Vsublimation very large
Liquid-to-Gas Hvaporization large Vvaporization very large
All occur at sharply defined P,T, e.g., P 1 bar; Tfusion normal FP
Heating Curve (1 mol H2O to scale)
0°C 100°CT
heat(kJ)
0
60
icewarms
ice becomes water
water warms
water becomes steamsteam heats
Cice THfusion
Cwater T
Hvaporization
Csteam T
Equilibrium Vapor Pressure, Peq
At a given P,T, the partial pressure of vapor above a volatile condensed phase.
If two condensed phases present, e.g., solid and liquid, the one with the lower Peq will be the more thermodynamically stable. The more volatile phase will lose matter by gas
transfer to the less (more stable) one because such equilibrium are dynamic!
Liquid Vapor Pressures
Measure the binding potential in the liquids. Vary strongly with T since the fraction of
molecules energetic enough at T to break free is e–Hvap / RT.
Will be presumed ideal. Equal 1 bar at “normal” boiling point, Tboil. Decrease as liquid is diluted with another.
Temperature Dependence of P
The thermodynamic relationship between Gibbs Free energy, G, and gas pressure, P, can be shown to define P as a function of T.
We’ll see this in Chapter 6.
PT / P’T’ = e–Hvap / RT / e–Hvap / RT’ or Just the ratio of molecules capable of overcoming Hvap
P = P’ e –[Hvap / R] [ (1/T ) – (1/T ’) ]
The infamous Clausius-Clapeyron equation.
Raoult’s Law: PA varies with XA
Ideal solutions composed of molecules with A–A binding energy the same as A–B. Vapor pressures are consequence of the
equilibrium between evaporation and condensation. If evaporation slows, P falls.
But only XA of liquid at surface is A, then its evaporation rate varies directly with XA.
PA = P °A XA and PB = P °B XB Where P ° means P of pure (X=1) liquid.
Consequences of Ideality
Measured vapor pressures predict mole fractions (hence concentrations) of solutes. Pressure – solution equilibria predict solute –
solution equilibria. While gases are adequately ideal, solutions
almost never are ideal. Positive deviations of P from P°X imply A–B
interactions are not as strong as A–A ones.
Pure Compound Phase Diagram
Predicts the stable phase as a function of Ptotal and T.
Characteristic shape punctuated by unique points. Phase equilibrium lines Triple Point Critical Point
P
T
Solid Liquid
GasGas
Phase Diagram Landmarks
Triple Point (PT,TT) where SLG coexist.
Critical Point (PC,TC) beyond this exist no
liquid/vapor property differences.
P = 1 bar Normal fusion TF and
boiling TB points.
P
T
PC
TC
PT
TT
1
TF TB
Inducing Phase Changes
Below PT or above PC Deposition of gas to solid
induced by dropping T or raising P
Sublimation is reverse. Between PT and PC
Liquid condensation vs. vaporization.
Normally, pressure on liquid solidifies it (unless solid < liquid)
P
T
depositionsublimation
vaporizationcondensation
fusion
freezing
gelation
P
T
Impure (solution) Phase Diagram
Adding a solute to a pure liquid elevates its Tboil by lowering its vapor pressure. (Raoult’s Law) It also stabilizes liquid
against solid (lowers Tfusion) Lower P wins, remember?
Click to see the new liquid regions and 2 colligative properties in 1!
Clausius–Clapeyron Lab Fix
dP/dT = PHvap/RT 2
from thermodynamics
P’=Pe–[H/R][(1/T)–(1/T ’)]
But only if H f(T)
If H ~ a + bT where b related to CP
P=P’(T/T ’)b/R e–[a/R][(1/T)–(1/T ’)]
assumes only CP are fixed. A better approximation.
P
T
Clausius–Clapeyron
Clausius–Clapeyron Parameters
H ~ a + bT b = (HBP–H°) / (BP–298) a = H° – 298 b
H
T298K BP
H°
HBP
Molecule Tbp°C Hbp H°
C5H12 36.1 25.8 26.4
C5H11OH 138. 44.4 57.0
C7H16 98.5 31.8 36.6
End of Presentation
Last modified 30 June 2001