Some Important Ionic Structures •  West: p.35, Section 1.17 •  Rock Salt: NaCl •  Interpenetrated ccp lattices

ccp/fcc of Cl– Na+ in all octahedral

sites

Or: ccp/fcc of Na+,

Cl– in all octahedral

sites

Two Forms of Zinc Sulfide

•  Zinc Blende •  fcc/ccp S2–

•  Zn2+ in half Td sites (all T+) •  Zn4S4 = ZnS •  AxBy: C.N.(A) : C.N.(B) = y : x •  (C.N. of Zn) = 4 ⇒ (C.N. of S) = 4

•  Wurtzite •  hcp S2–

•  Zn2+ in half Td sites •  ZnS

Fluorite

•  , CaF2

•  fcc/ccp cations (Ca2+) •  Anions (F–) in all Td sites •  Or: primitive cubic (PC) F–,

Ca2+ in alternate cubes •  AxBy: C.N.(A) : C.N.(B) = y : x •  ⇒ (C.N. of Ca) = 2 × 4 = 8

Antifluorite and

Fluorite

•  , Na2O •  fcc/ccp anions (O2–) •  Cations (Na+) in all Td sites •  Or: primitive cubic Na+,

O2– in alternate cubes

Antifluorite

Polyhedral Views of Antifluorite

Edge-sharing NaO4

Tetrahedra

Edge-sharing ONa8 Cubes

Cube View of Antifluorite, Mexico City

CsCl

•  PC Cl–, Cs+ at body center of every cube

•  Or: PC Cs+, Cl– at body center

•  C.N. = 8 for both anions & cations

•  C.N. ≠ 12 ⇒ not close-packed •  Collectively, CsCl is bcc

Covalent Structures

•  e.g. Diamond •  fcc/ccp C, with C in half

Td sties •  cf. zinc blende •  All C’s equivalent,

tetrahedral •  Maximum C.N. of C,

versus 12 for metals •  Carborundum: SiC, with

alternating Si and C

Radius Ratios for Ionic Compounds

•  West, Section 3.2.3, ionic model •  Minimize distance and maximize CN

between unlike ions •  At the minimum radius ratio, r +/r – :

•  Anions just touching •  Cation just fills cavity •  If cation smaller, goes in lower CN

(smaller) cavity •  e.g. LiI, r +/r – = 0.436, Li+ goes in 6 CN site

Intersticial CN Geometry r +/r –

4 Tetrahedral ≥ 0.23

6 Octahedral ≥ 0.41

8 Cubic ≥ 0.73

4 atoms of an hcp or ccp lattice

2r−( )2+ 2r−( )

2= 2r− + 2r+( )

2

4 r−( )2+ 2 r−( )

2= 2r− + 2r+( )

2

6r− = 2r− + 2r+

r+

r−=

6 − 22

= 0.23

Radius Ratio Rules •  Octahedral intersticial site: West, p.133-4 •  Only need to think of square plane

•  Tetrahedral (CN = 4) site: West, p.134 •  Cubic (CN = 8) site: West, p.134 •  CN = 2, 3, 4, 6, 8, 12 West, Table 3.3

•  If r +/r – > 1, use r –/r + e.g. CsF

r +/r – < 0.41: Cation too small, → lower CN site

r +/r – = 0.41: CN = 6 site just right, w/ anions touching

0.41 < r +/r – < 0.73: Still CN = 6, cation pushes anions apart

r +/r – = 0.73 = cubic site (CN = 8)

General Principles of Ionic Solids

•  Ions are charged, polarizable spheres

•  MZ+ and XZ – held together by electrostatic forces

•  Coulomb’s law: F = [(Z+e)(Z–e)] / r 2

•  Lattice energy = Σ Fi

•  Ions prefer highest CN with neighbors of opposite charge (and are in contact)

•  Ions of like charge prefer to be far apart

•  Local electroneutrality prevails

•  p.130-1

Lattice Energy of Ionic Crystals •  3D array of point charges •  Lattice energy, U

Na+(g) + Cl–(g) → NaCl(s) ΔH = U

•  U ≡ internal enthalpy change when one mole of the solid is formed from the gaseous ions @ 1 atm, 0K (⇒ negative U values)

•  U depends on structure, ions, charge, CN, distance from opposite ion •  Two principal forces involved:

(a) Electrostatic attraction & repulsion: F = [(Z+e)(Z–e)] / r 2

V = ∫∞ F dr = – (Z+Z–e 2) / r (b) Short-range repulsion if electron clouds too close:

V = B / r n B is cst, n = 5 to 12 •  Combine V’s to maximize |U| and find re, internuclear separation (over) •  Section 3.2.5; Smart & Moore, p.61-70

r

Electrostatic Term for NaCl

•  V = – (Z+Z–e 2) / r •  Na+ in center of unit cell:

Nearest neighbor: Cl– on 6 face centers 2nd nearest neighbor: Na+ on 12 edge centers 3rd nearest neighbor: Cl– on 8 corners

V = – [(Z+Z–e 2) / r ] [6 – 12/(√2) + 8/(√3) – 6/(√4) + …] VTot = – [(Z+Z–e 2) / r ] N A

A = Madelung constant, depends on lattice structure Table 3.5: Rock salt A = 1.748 Wurtzite A = 1.641 Fluorite A = 2.520

Calculating Lattice Energy (U) •  V = B / r n

•  Repulsive term, regardless of ion charge

•  Internuclear separation, r

•  U = [–(Z+Z–e 2 NA) / r] + [BN / r n]

•  r = re = equ’m internuclear separation, where:

dU / dr = 0 = [(Z+Z–e 2 NA) / r 2] – [nBN / r n+1]

B = (Z+Z–e 2A r n–1) / n

•  Umin = – [(Z+Z–e 2 NA) / re ][1 – (1 / n)]

•  Born-Landé Equation

•  NaCl: U = – (859.4 + 98.6 - 12.1 + 7.1) kJ/mol = – 765.8 kJ/mol (van der Waals and vibrational terms, p. 140)

Some Lattice Energies

•  Umin = – [(Z+Z–e 2 NA) / re][1 – (1 / n)] •  e, N, A, n: constants for same structure

•  ⇒ U depends on Z+ , Z–

, re

•  Charge very important, since Z+ × Z–

•  e.g. SrO & LiCl, similar re’s, but 3369 kJ/mol vs. 861 kJ/mol

•  LiX: decreasing U as re increases •  U proportional to MP (and sublimation energy)

MP: MgO(2800°C), CaO(2572°C), BaO(1923°C) and NaCl(800ºC)

Kapustinskii’s Equation •  Madelung’s Constant, A ∝ CN (Table 3.5)

wurtzite (A = 1.641), rock salt (A = 1.748), CsCl (A = 1.763)

•  For given anion & cation, re ∝ CN (Fig. 3.3)

•  v = # of ions per formula unit, A / v almost constant (Smart & Moore, Table 1.14)

•  Use rock salt (A = 1.748), re for octahedral, plug in N, e, n & re (= rc + ra):

Umin = – [(107900 v Z+Z–) / (rc + ra)] kJ/mol

•  Can calculate U (or rc, ra) for any known or hypothetical solid

•  Ion can be non-spherical; e.g.: SO42–, PO4

3–, OH–, NH4+, etc.

•  Effective (thermochemical) radius, Table 3.7

Born-Haber Cycle •  Ay–

(g) + Bx+(g) → AxBy (s)

•  Impossible to measure directly

•  ΔHf o, heat of formation from elements in standard states, can be

measured

•  Hess’s Law allows U to be solved, compared to calculation

•  If U is known, allows other energies to be confirmed or determined

•  Can calculate ΔHf o

of unknown compounds

•  p.141-2: NaCl

•  Discrepancies ⇒ covalency or CFSE

U

ΔHfo[MCl2(s)] = ΔHatm

o[M(s)] + IE1(M) + IE2(M) + D(Cl–Cl) + [– 2EA(Cl)] + U[MCl2(s)]

ΔHfo[MgCl2(s)] = – 641 kJ/mol

NaCl2(s)? Assume U[NaCl2(s)] = U[MgCl2(s)] = – 2523 kJ/mol

⇒ ΔHfo[NaCl2(s)] = 2190 kJ/mol

NaCl2 doesn’t exist, due to IE2(Na) (Table 3.9, p.144)

U

U