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**Zumdahl’s Chapter 10 and Crystal Symmetries**

Liquids Solids

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**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

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**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?

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**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.

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**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.

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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

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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!

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**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.

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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.

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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.

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**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.

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**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 not tetragonal but rather face-centered cubic.

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**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.

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**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

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**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.

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**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.

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**Rhombohedral _ 3 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.

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**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

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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.

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**Common Cubic Classes Simple cubic Body-centered cubic**

“Primitive” P Body-centered cubic “Interior” I Face-centered cubic “Faces” F “Capped” C if only on 2 opposing faces. BCC FCC

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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.

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**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.

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**Gold’s Density from Unit Cell**

4 Å 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

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**Bravais Lattices 7 lattice systems + P, I, F, C options**

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.

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**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). –

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**Triclinic Lattice Designation**

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. – –

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**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

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**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.

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**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

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**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

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**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. m 2

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(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

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**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) (invert only) 6 / m mm (6 m + ||+|| ) 3 m ( 3 + ||+||+|| ) _

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**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!

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**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

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**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).

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Inverse Distances b/2 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) c a/3 a b

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**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

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**Bragg Formula 2 sin / = 1 / d (conveniently inverted)**

b a 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

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**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°

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**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 32 screw cell 1 a glide

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**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.

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**Nature’s Choice Symmetries**

36.0% P 21 / c monoclinic 13.7% P 1 triclinic 11.6% P 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 _

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**A B A : hexagonal close pack**

Packing in Metals A B A : hexagonal close pack A B C : cubic close pack

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**Relationship to Unit Cells**

Is FCC A B C A B C : cubic close pack

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**ABA (hcp) Hexagonal A B 90° The white lines indicate an**

elongated hexagonal unit cell with atoms at its equator and an offset pair at ¼ & ¾. 120° If we expand the cell to see it’s shape, we get a diamond at both ends…3 make a hexagon whose planes are 90° to the sides of the (expanded) cell.

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**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)

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**Phase Changes Solid-to-Gas Liquid-to-Gas Solid-to-Solid**

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

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**Heating Curve (1 mol H2O to scale)**

Csteam T 60 steam heats water becomes steam heat (kJ) Hvaporization ice warms water warms Hfusion Cice T Cwater T ice becomes water 0°C T 100°C

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**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!

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**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.

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**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.

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**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.

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**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.

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**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

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**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 1 TF TB PT TT

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**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 fusion gelation freezing vaporization condensation deposition sublimation

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**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! P T

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**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

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**Clausius–Clapeyron Parameters**

H T H ~ a + bT b = (HBP–H°) / (BP–298) a = H° – 298 b 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 H° 298K BP

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End of Presentation Last modified 30 June 2001

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