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
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