Zumdahl’s Chapter 10 and Crystal Symmetries

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