Transcript Document 7205079

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


True dipolar molecules have permanently
shifted electron distributions which attract
one another strongly  R–4 (longer range).


Non-polar molecules bind exclusively by
London potential  R–6 (short-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.
Diamonds are not
Real diamond is 10. tetragonal but rather
face-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





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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


Body-centered cubic


“Primitive” P
BCC

FCC
“Interior” I
Face-centered cubic


“Faces” F

“Capped” C if only on
2 opposing faces.
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


Volume NAv cells:



(8) + ½ (6) = 4
M = 4(197 g) = 788 g
(4.0710–10 m)3  Nav
3.9010–5 m3 = 39.0 cc
 = M / V = 20.2 g/cc
4Å
Bravais Lattices

capped
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.
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, bodycentered, 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

4
m
m
Abbreviated 4/mmm
m
Nature’s Favorite for Organics

m




Monoclinic
abc
 =  = 90° < 
Then b is a 2-fold axis
and  to m
So it is 2/m

2
b is a 2 because the
crystal is infinite.
(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.
So it is 6/m mm
6
m
m
m
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 + ||+|| )
_
3m
( 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

reinforced


d

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 a/3
makes more sense, 0.
Shown is (3,2,0)
b/2
c
a
b
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–2hkl = (h/a)–2 + (k/b)–2 + (l/c)–2
for use in Bragg
c
Bragg Formula
a



b
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

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

cell 2
cell 1
a glide
Simultaneous rotation
by 360°/n with a m/n
translation along axis.
32 screw
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%
 13.7%
 11.6%
 6.7%
 6.6%
 25.4%


P _21 / c
monoclinic
P1
triclinic
P 2 1 21 21
orthorhombic
P 21
monoclinic
C2/c
monoclinic
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
Is FCC
A
B
C
A
A B C : cubic close pack
ABA (hcp) Hexagonal
A
B
A
90°
120°
whose planes are 90° to the
sides of the (expanded) cell.
The white lines indicate an
elongated hexagonal unit
cell with atoms at its equator
and an offset pair at ¼ & ¾.
If we expand the cell to see
it’s shape, we get a diamond
at both ends…3 make a hexagon
Alloys (vary properties of metals)

Substitutional


Intersticial


Heteroatoms swap originals, e.g., Cu/Sn (bronze)
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


Hfusion significant
Vfusion small
Solid-to-Gas





Hsublimation very large
Vsublimation very large
Liquid-to-Gas

E.g., red to white P
Solid-to-Liquid


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)
Csteam T
60
steam heats
water becomes steam
heat
(kJ)
Hvaporization
ice
warms
Hfusion
Cice T
water warms
ice becomes water
0
0°C
T
Cwater T
100°C
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
Solid
Liquid
Gas
T
Phase Diagram Landmarks

P
PC
Triple Point (PT,TT)


Critical Point (PC,TC)

1
PT

beyond this exist no
liquid/vapor property
differences.
P = 1 bar

TT TF TB TC T
where SLG coexist.
Normal fusion TF and
boiling TB points.
Inducing Phase Changes

P
Below PT or above PC

fusion

Deposition of gas to solid
induced by dropping T or
raising P
Sublimation is reverse.
freezing
 Between PT and PC
vaporization
 Liquid condensation vs.
condensation
vaporization.
gelation
 Normally, pressure on
deposition
sublimation
liquid solidifies it (unless
solid < liquid)
T
Impure (solution) Phase Diagram

P
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)


Click to see the new liquid
regions and

T
Lower P wins, remember?
2 colligative properties in 1!
Clausius–Clapeyron Lab Fix

dP/dT = PHvap/RT 2


P’=Pe–[H/R][(1/T)–(1/T ’)]


But only if H  f(T)
If H ~ a + bT


from thermodynamics
P
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.
Clausius–
Clapeyron
T
Clausius–Clapeyron Parameters

H
H ~ a + bT


b = (HBP–H°) / (BP–298)
a = H° – 298 b
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
HBP
H°
298K
BP
T
End of Presentation
Last modified 30 June 2001