Solid State Chemistry

Question 0

What is a solid?

Answer 0

A group of atoms interconnected by arrays of strong, directional chemical bonds. It’s as if they’re “super molecules.”

Question 1

When this external mechanical force is applied to a solid, it will not flow.

Answer 1

stress

Question 2

What is a crystal?

Answer 2

A solid whose structure is highly ordered and symmetrical over macroscopic distances

Question 3

What is Haüy’s Law?

Answer 3

Stated by René-Just Haüy, when one cleaves a crystal, one will observe constant interfacial angles. Haüy concluded the outward symmetry implies a highly regular internal structure and the existence of a smallest crystal unit cell.

Question 4

What is a symmetry element?

Answer 4

An potential action such as reflection or rotation that, if the new image is symmetric with the original, will perfectly superimpose that new image over the original image.

Question 5

What is a crystal lattice?

Answer 5

A 3-D array of a network of all the points in a crystal in the same environment and of the same orientation

Question 6

What is the cubic system?

Answer 6

A crystal lattice comprising cubic unit cells. It is the lattice of highest possible symmetry.

Question 7

What is a unit cell?

Answer 7

The smallest space-filling body in volume that contains all structural information about its crystal

Question 8

What is the shape of a unit cell in a crystal?

Answer 8

Parallelepiped

Question 9

What are the cell constants?

Answer 9

The 3 edge lengths (a, b, c) and the 3 angles (α, β, γ) unique to each different kind of unit cell

Question 10

What are the minimum essential symmetry and conditions on unit cell edges and angles for a hexagonal crystal system?

Answer 10

1X 6-fold rotation a=b, α=β=90°, γ=120°

Question 11

What are the minimum essential symmetry and conditions on unit cell edges and angles for a cubic crystal system?

Answer 11

4X independent 3-fold rotation axes (each 70.53° from each other) a=b=c, α=β=γ=90°

Question 12

What are the minimum essential symmetry and conditions on unit cell edges and angles for a tetragonal crystal system?

Answer 12

1X 4-fold rotation a=b, α=β=γ=90°

Question 13

What are the minimum essential symmetry and conditions on edges and angles for a trigonal crystal system?

Answer 13

1X 3-fold rotation a=b=c, α=β=γ≠90°

Question 14

What are the minimum essential symmetry and conditions on unit cell edges and angles for an orthorhombic crystal system?

Answer 14

3X mutually perpendicular 2-fold rotations α=β=γ=90°

Question 15

What are the minimum essential symmetry and conditions on unit cell edges and angles for a monoclinic crystal system?

Answer 15

1X 2-fold rotation α=γ=90°

Question 16

What are the minimum essential symmetry and conditions on unit cell edges and angles for a triclinic crystal system?

Answer 16

None, None

Question 17

When would you use a nonprimitive unit cell over a primitive cell?

Answer 17

When the primitive cell lacks enough symmetry elements to be a unit cell

Question 18

What are the 3 types of nonprimitive unit cells used to describe crystals?

Answer 18

1) body-centered
2) face-centered
3) side-centered

Question 19

What is constructive interference?

Answer 19

When waves in phase encounter each other, reinforcing each other and thus amplifying the resulting wave. This occurs when the waves’ paths differ in length by a whole number of wavelengths. The amplitudes are additive.

Question 20

What is destructive interference?

Answer 20

When waves out of phase encounter each other and combine to cancel each other out, producing a wave with an overall amplitude lesser than the originals’ average. The amplitudes are subtractive.

Question 21

What is the Bragg law?

Answer 21

nλ = 2d sinθ

This law confirms if the waves in question are indeed diffracting on the crystal in question.

Question 22

What is the simple cubic lattice?

Answer 22

The simplest form of crystal lattice in which the lattice points form perfect cubes with only 1 lattice point: 1 in the center. Po is the only element known to crystallize into a simple cubic lattice. Each unit cell contains 1 Po atom, separated from its neighboring Po atom by 3.35 Å.

Question 23

What is the body-centered cubic (bcc) structure?

Answer 23

A cubic unit cell form with 2 lattice points: 1 in the center and 1/8 on each corner. All alkali metals crystallize into the bcc structure.

Question 24

What is the face-centered cubic (fcc) structure?

Answer 24

A cubic unit cell form with 4 lattice points: 1/2 on each face and 1/8 on each corner. Al, Ni, Cu, and Ag crystallize into the fcc structure, among other metals.

Question 25

How do you find the volume of a unit cell given its unit cell constants?

Answer 25

Cosine Law:

V_c = abc√[1-cos²(α) - cos²(β) - cos²(γ) + 2cos(α)cos(β)cos(γ)]

Question 26

What is the lattice parameter?

Answer 26

The distance between the 2 nearest lattice points, often the value d from the Bragg law when n is a whole number.

Question 27

Using the van der Waals equation of state for gas, how can we approximate the volume of a formula unit?

Answer 27

b/N_A,

where b is the van der Waals paramter of the volume excluded per mole of gas

Question 28

Using probability density functions of electron density, how can we approximate the volume of an atom?

Answer 28

V = 4/3 πr³,

where r is the radial distance from the nucleus at which the density function approaches a certain value.

Question 29

Using X-ray (or some other suitable form of radiation) crystallography, how can we find the approximate volume of a formula unit?

Answer 29

V = 4/3 πr³,

where r is the radius of one formula unit and is found by halving the “nearest neighbor distance” between 2 nearest lattice points (tangent).

Question 30

What is the nearest neighbor distance of a simple cubic?

Answer 30

a,

where a is the lattice parameter.

Question 31

What is the nearest neighbor distance of a bcc?

Answer 31

(a√3)/2,

where a is the lattice parameter.

Question 32

What is the nearest neighbor distance of a fcc?

Answer 32

(a√2)/2,

where a is the lattice parameter.

Question 33

What is the packing fraction of a simple cubic?

Answer 33

π/6

Question 34

What is the packing fraction of a bcc?

Answer 34

√(3π)/8

Question 35

What is the packing fraction of a fcc?

Answer 35

√(2π)/6

Question 36

What are the 2 ways of packing spheres?

Answer 36

Cubic close-pack (ccp) (AKA fcc) & Hexagonal close packing (hcp)

Question 37

What is cubic close-packing (ccp), or fcc?

Answer 37

1 of 2 ways to most efficiently pack spheres together.

In ccp, the stacking pattern is abcabc…, and from a top view, no interstitial spaces are visible, though they exist.

Question 38

What is hexagonal close-packing (hcp)?

Answer 38

1 of 2 ways to most efficiently pack spheres together.

Its stacking pattern is ababab…, and from a top view interstitial spaces are visible.

Question 39

What is an interstitial site?

Answer 39

A center of free volume in a unit cell.

Question 40

What are the 2 interstitial sites of a ccp, or interchangeably fcc?

Answer 40

Octahedral site & Tetrahedral site

Question 41

What is a Bragg angle?

Answer 41

The angle from the crystal plane at which the incident radiation enters the crystal lattice matrix.

Question 42

What is an octahedral site?

Answer 42

1 of 2 types of interstitial sites. It is surrounded at equal distances by 6 nearest-neighbor atoms. Octahedral sites lie at the midpoints of the edges of the fcc unit cell and at the center of the unit cell.

Question 43

How many octahedral sites are there in total per fcc unit cell?

Answer 43

4

3 in total contributed from the edges (1/4 each, 12 points) & 1 at the center of the unit cell

Question 44

How do you find the maximum radius of an interstitial atom at an octahedral site?

Answer 44

r₂= a/2 - a√(2)/4

= 0.146a,

where

a = 2r₂ + 2r₁

and

r₁ = a√(2)/4

Question 45

What is a tetrahedral site?

Answer 45

1 of 2 types of interstitial sites. It is surrounded by 4 touching spheres. In fcc, it is in the volume between a corner atom and its 3 neighboring face-centered atoms.

Question 46

How many tetrahedral sites are there in total per fcc unit cell?

Answer 46

8

Question 47

What is the ratio of the octahedral-site radius to the host-atom radius?

Answer 47

r₂/r₁ = 0.414

Question 48

What is the ratio of the tetrahedral-site radius to the host-atom radius?

Answer 48

r₂/r₁ = 0.225

Question 49

What is an ionic crystal?

Answer 49

A solid compound formed by atoms with significantly different electronegativities. To a first approximation, these ions can be treated as hard, charged spheres (wave functions do not overlap) that occupy positions on the crystal lattice.

Question 50

What elements typically react to form ionic crystals?

Answer 50

Groups I and II w/ Groups VI and VII (the great majority also crystallize in the cubic system).

Question 51

Which compounds crystallize in the rock-salt (“sodium chloride”) structure?

Answer 51

The alkali halides (except cesium halides), ammonium halides, and alkaline-earth oxides and sulfides

Question 52

What is the rock-salt structure?

Answer 52

An fcc lattice of anions whose octahedral sites are all occupied by cations, or vice versa. Each ion is surrounded by 6 equidistant ions of the opposite charge.

Question 53

When is the rock-salt structure a stable crystal structure?

Answer 53

When the cation-anion radius ratio lies between 0.414 and 0.732 (assuming cations and anions behave as incompressible charged spheres, implying their wave functions do not overlap).

Question 54

What is the cesium chloride structure?

Answer 54

An ionic crystal structure of 2 interpenetrating simple cubic lattices, 1 of anions and 1 of cations. This is achieved when the hard-sphere cation-anion radius ratio exceeds 0.732.

Question 55

What is the zinc blende, or sphalerite, structure?

Answer 55

(Named after ZnS)

An ionic crystal structure consisting of an fcc lattice of S^2- ions with Zn²⁺ ions occupying half of the available tetrahedral sites in alternation. This is achieved when the hard-sphere cation-anion radius ratio falls below 0.414.

Question 56

What is the fluorite (CaF₂) structure?

Answer 56

An ionic crystal structure based on an fcc lattice of Ca2+ ions. The F- ions occupy all 8 of the tetrahedral sites.

Question 57

What is the significance of 0.414?

Answer 57

It is a ratio of octahedral-site radius to host-atom radius for an fcc lattice. When this ratio is exceeded (>0.414), the ion inserted into the octahedral site comes into contact with ions of the opposite sign in the rock-salt structure.

Question 58

What is the significance of 0.732?

Answer 58

It is a ratio of tetrahedral-site radius to host-atom radius for a simple cubic structure.

Question 59

Characterize the ionic bonds within ionic crystals, and discuss their macroscopic implications.

Answer 59

The strength and range of these electrostatic attractions make the respective crystals hard, high-melting, brittle solids that are electrical insulators.

Question 60

Ionic solids are poor electrical conductors, so why are ionic liquids good electrical conductors?

Answer 60

Melting ionic solids breaks the strong electrostatic attractions, disrupting the lattice and setting the ions free to move.

Question 61

Do metals have a tendency for ionic bonding? Why or why not?

Answer 61

Low tendency to form ionic bonding

Small differences in electronegativity

Question 62

Do metals have a tendency for covalent bonding? Why or why not?

Answer 62

Low tendency for covalent bonding

Do not have nearly-filled subshells (to ease forming stable octet)

Question 63

What is the Drude model of metal solid structure?

Answer 63

(Proposed by Paul Drude)

A fixed array of positively charged metal ions, each localized at a site of the crystal lattice. These fixed ions were surrounded by a sea of delocalized, mobile electrons, 1 from each of the atoms. It accounts for malleability, ductility, and conductivity.

Question 64

Can a stable sodium molecule/cluster ever fill an antiboding MO in MO theory?

Answer 64

No

Question 65

What is an energy band?

Answer 65

A collection of MO sublevels extremely close in energy (imagine forming a “bold font” MO line by bunching so many MO’s together)

Question 66

The core AO electrons of Na retain their distinct, localized character of the ions at the lattice sites. Why?

Answer 66

The core AOs electrons of Na are only very slightly broadened at the equilibrium internuclear separation of the crystal

Question 67

Most metals have crystal structures of high symmetry and crystallize in which lattices? Which metals have more complex crystal structures?

Answer 67

bcc, fcc, or hcp

Ga, In, Sn, Sb, Bi, & Hg

Question 68

Metals can exist in crystal phase. Can metals also exist in liquid phase?

Answer 68

Yes.

In fact, the conductivity usually drops by only a small amount when the metal melts.

Question 69

What provides the very strong binding in most metals?

Answer 69

The electron sea

This is manifested by their high boiling points, though they have a very large range of melting points.

Question 70

What is a covalent (network) crystal?

Answer 70

A class of crystalline solids whose atoms are linked by covalent bonds. A classic example is diamond (cubic system). To bond, each hybrid orbital has at least 1 e^- that can spin-pair with a corresponding e^- from another, equivalent hybrid orbital.

These crystals have very high melting points and are hard and brittle.

Question 71

What is a molecular crystal?

Answer 71

A group of molecules held in their lattice sites by intermolecular forces like hydrogen bond and van der Waals forces. Examples include noble gases, diatomics, some covalent molecules, metal halides of low ionicity (ex.: Al₂Cl₆, FeCl₃, BiCl₃), and most organic compounds.

Question 72

How do you account for the complexity of attractive vs. repulsive forces in a molecular crystal from the large number of atoms?

Answer 72

1) Assume each atom is a sphere, with van der Waals radius, fused to each other.
2) Pack the spheres in such a way that no molecules overlap while minimizing empty space.

Question 73

How do molecular crystals crystallize?

Answer 73

If simple geometry (ex: noble gas): highly symmetric fcc lattice.

If complex geometry: lowly symmetric monoclinic or triclinic system

Question 74

Is it possible for an elemental ionic solid to exist?

Answer 74

Highly unlikely (pretty much “no”)

Question 75

Characterize halogen solids.

Answer 75

Each halogen atom has 7 valence e^- and can react with one more halogen atom. Once the single bond forms, the diatomic molecule can only interact with anoter diatomic molecule through van der Waals forces and form molecular solids with low melting points and boiling points.

Question 76

Characterize chalcogen solids.

Answer 76

Structures vary.

O: can form 1 double or 2 singles. Highly prefers double bond (except O₃), so forms weakly bound diatomics.

S: prefers 2 singles to 2 other S. Leads to (puckered) ring or chain structure. Ring is S₈ (stable at room temp.). Above 160℃ rings break open and relink into long, tangled chains (viscous liquid). Weak intermolecular.

Se: prefers 2 singles to 2 other Se. Leads to ring or chain structure. Ring is Se8 (unstable). Stable form is very long spiral chains w/ weak interchain interaction (gray crystal, metallic appearance).

Te: similar to Se.

Question 77

Characterize Group V elemental solids.

Answer 77

Similar trend to chalcogen solids.

N: prefers triples, forming diatomics.

P: 3 forms, each w/ 3 singles rather than 1 triple. White: tetrahedral P4 (weak van der Waals). Black and Red: bonded (in)directly with all other atoms in sample. Stronger.

As & Sb: Stable form like black P. Unstable form like white P.

Question 78

Characterize metalloid solids.

Answer 78

Intermediate electronegativity, so exist as solids between metallic and covalent.

Sb: metallic luster, but poor conductor of electricity and heat

Si & Ge: semiconductors

Sn: 2 crystalline forms. White: tetragonal crystal structure, metallic conductor, stable above 13℃. Gray: powder, diamond structure, poor conductor (conversion: “tin disease,” prevented by small addition of Bi or Sb).

C: graphite is stable, not diamond. Graphite is sheets of fused hexagonal rings w/ weak interactions between layers, sp² hybridization, remaining p is in extended π-bonding interactions over whole plane.

Question 79

What is a semiconductor?

Answer 79

Intermediate conductor of electricity

Question 80

What is an amorphous solid (glass)?

Answer 80

A crystal with so many defects that its crystalline order is destroyed. Otherwise resembles crystalline solid. Any substance that can be liquefied can almost certainly be prepared in an amorphous state by bypassing crystallization: cool molten material very fast.

Microscopically, strong tendency to form glass is associated w/ presence of ong/irregularly shaped molecules that can easily become tangled and disordered. Often used in fabrication into articles of any shape because the flow properties can be managed by controlling temp.

Question 81

What are the 2 common point defects in a pure crystalline substance?

Answer 81

Vacancies (missing atoms)

Interstitials (added atoms)

Question 82

What are Schottky defects?

Answer 82

Point defect in which a small fraction of the normal sites remain unoccupied.

Its concentration depends on temperature:

N = N_se^-ΔG/RT,

where N is the number of lattice vacancies per unit volume,

• N_s* is the number of atom sites per unit volume
• ΔG* is the molar free energy of formation of vacancies

Question 83

What are Frenkel defects?

Answer 83

Point defect in which atoms/ions are displaced (diffuse) from their regular lattice sites to interstitial sites. Prime examples are silver halides, where Ag⁺ occupies almost random sites.

Diffusion frequency depends on temperature:

D = D₀e^-ΔG/RT,

where D is self-diffusion coefficient.

Rates of diffusive motion vary enormously from one substance to another. In low-melting metals, very frequent. In high-melting metals, very rare.

Question 84

What is a F-center (Farbenzentrum, or color center)?

Answer 84

Simplest of a family of electronic crystal defects, and imparts a color to ionic crystals.

In this defect, an uncharged atom attempting to diffuse to the surface of an ionic solid to escape encounters an anion vacancy and is trapped in the Coulomb field of the surrounding cations.

Question 85

Explain the composition of nonstoichiometric compounds.

Answer 85

The solid compounds want to remain neutral, but the ion in question stably exists in multiple oxidation states, resulting in a varying composition of the ion and its ionic counterpart(s).

Examples include: FeO (wüstite) (2⁺/3⁺), NiO (2⁺/3⁺), Cu2S, TiO.

When NiO is prepared in 1:1 composition: pale green, electrical insulator. When prepared in excess of O₂: black, conducts electricity well. Iin black, small fraction of Ni²⁺ replaced by Ni³⁺, so compensating vacancies occur at some Ni atom sites in crystal.

Question 86

What is an alloy?

Answer 86

A mixture of elements that displays metallic properties. Related disorder to nonstoichiometric compounds (ionic).

2 types: substitutional & interstitial

Question 87

What is a substitutional alloy?

Answer 87

Defect in which some of the metal atoms in a crystal lattice are replaced by other atoms (usually of comparable size).

Ex.: brass (Cu/Zn 33%) & pewter (Sn/Cu 7% / Bi 6% / Sb 2%)

Question 88

What is an interstitial alloy?

Answer 88

Defect in which atoms of 1+ additional elements enter the interstitial sites of the host metal lattice.

Ex.: steel (mild: Fe/C 0.2%, high-carbon: Fe/C 1.5%)

Question 89

What is an alloy steel?

Answer 89

Both substitutional and interstitial alloy: Fe/Cr (sub) / V (sub) / C (int)

Question 90

What is lattice energy of a crystal?

Answer 90

The energy required to separate the crystal into its component atoms/molecules/ions at 0K.

Question 91

How do you estimate the lattice energy of a molecular crystal?

Answer 91

Lennard-Jones potential

V_LJ(R) = 4ϵ[(σ/R)¹²-(σ/R)⁶]

V_tot = 1/2 ∑_i=1^N∑_j=1^NV_LJ(R_ij),

where R_ij is distance between atom i & atom j. When everything is calculated:

lattice energy = V_tot = 8.61ϵN_A

This overestimates the true lattice energy because of zero-point energy (quantum).

Question 92

Outline the Born-Haber cycle.

Answer 92

Applies Hess’s law to determine lattice energy of ionic crystal.

1) Convert ionic solid to elements in standard states (-ΔH°_f).
2) Transform elements into gas-phase atoms (ΔH°_f).
3) Transfer e- from potential cation to potential anion (IE & EA).

Assumes ΔU≈ΔH. Actually,

ΔH = ΔU + RTΔn_g,

where Δn_g is change in number of moles of gas molecules in each step of the reaction. Estimate overapproximates, presumably because short-range repulsive interactions & zero-point energy were not taken into account.

Question 93

How do you estimate the lattice energy of an ionic crystal?

Answer 93

Coulomb’s Law

V = (q₁q₂) / (4πϵ₀R₀)

V_attraction = -e²/ 4πϵ₀R₀[2(1)+2(¹/₃)+2(¹/₅)+…]

V_repulsion = e² / 4πϵ₀R₀[2(¹/₂)+2(¹/₄)+2(¹/₆)+…]

V_net = -N_Ae² / 4πϵ₀R₀[2-(²/₂)+(²/₃)-(²/₄)+…]

lattice energy = M(N_Ae² / 4πϵ₀R₀),

where M is Madelung constant.

Question 94

What is the Madelung constant? Give its value for different lattices.

Answer 94

The series in a 3-D version of V_net approximation of lattice energy in an ionic crystal:

V_net = -N_Ae² / 4πϵ₀R₀[2-(²/₂)+(²/₃)-(²/₄)+…]

Rock salt: M=1.7476

CsCl: M=1.7627

Zinc blende: 1.6381

Fluorite: 2.5194