Characteristics of Solids The Classification of Solids Preferred for characterization of structure and properties Polycrystalline Powder (Highly crystalline) Electrons in Solids Used for characterization when single crystal ean not be easily obtained, preferred for industrial production and nd Theory Polycrystalline Powder Large Surface Area) Desirable for further reactivity and certain applications such catalysis and electrode materials Semiconducto Amorphous(Glass) No long range translationalorder Thin Film widespread use in microelectronics, telecommunications, 18, coatings, etc. Molecular solid Covalent Types of Crystals According to the ionic bonding in solids (a)ion crystalNaCD olid state Ar crystal(H, BO (mixed bond crystals(graphite) Bo Bonding onding aPrimary bonds-strong attractions between atoms ovalent donic- Metal ion(+)& Nonmetallic ion H) Covalent- local sharing of electrons between atoms MEtallic- global sharing of electrons by all atoms Secondary bonds-attraction forces between
1 Characteristics of Solids Bonding Electrons in Solids Band Theory Defects Semiconductor Classification of Solids There are several forms solid state materials can adapt ßSingle Crystal -Preferred for characterization of structure and properties. ßPolycrystalline Powder (Highly crystalline) -Used for characterization when single crystal can not be easily obtained, preferred for industrial production and certain applications. ßPolycrystalline Powder (Large Surface Area) -Desirable for further reactivity and certain applications such as catalysis and electrode materials ßAmorphous (Glass) -No long range translationalorder. ßThin Film -Widespread use in microelectronics, telecommunications, optical applications, coatings, etc. Molecular solid Ionic Covalent Metallic Types of Crystals (a)ion crystal(NaCl) (b)metallic crystal (c)covalent crystal(InSb) (d)molecule crystal (solid state Ar) (e)hydrogen bond crystal(H3BO3 ) (f)mixed bond crystals (graphite) According to the ionic bonding in solids Bonding Electrons transfer between atoms Between many atoms Electrons sharing two atoms Bonding Primary bonds ¾ strong attractions between atoms Ionic ¾ Metal ion(+) & Nonmetallic ion (-) Covalent ¾ local sharing of electrons between atoms Metallic ¾ global sharing of electrons by all atoms Secondary bonds ¾ attraction forces between molecules
Primary Bonding Solid State- Strength TThe strength of the solid depends on the molecular forces that hold the solid together monic interactions are strong because the opposite charges resist breaking of the intermolecular bonds. For I example, the melting temperature of salts is very high Covalent typically400to800°C a Some molecular interactions are strong because of the 3 ngement of the atoms. For example, diamonds ar atoms. The bonds are molecular not ion Metallic Types of Non Bonded (intermolecular, van der waals)Interactions Ion-dipole interactions Dipole-dipole lon-dipole Induced dipole Dispersion Hydroger When polar molecules e er ions he positive end of the dipole is attracted to negative ions and vice versa. hen two polar molecules with net dipoles 甲 es to be present, one to provide ions &HH closer, one end of the dipole in one lin the second molecule than ion ion interactions in io Q o (b). These forces decrease with distance as Dispersion or London forces In non-polar molecules, electrons are distributed Noble gases are atomic gases and do not have dipole metrically. This symmetry can be distorted by an liquefied suggests that forces of attraction exist between These forces are weak and are of short range. atoms of a noble gas. These forces are called Dispersio cThe distribution of electrons in an atom/ molecule uctuate over time. These fluctuations set up temporary forces. These forces exist between all atoms and molecules electrons in atoms/molecules. "Dispersion force ncre group. Thus, forr down a g ases, dispersion forces ase as Xe> Kr> Ar> Ne He
2 Primary Bonding e + Na+ e e e e e e e e e e + e e e e e e e e e F- Ionic + + e e Covalent e + + + + + + + + + e e e e e e e e Metallic Solid State ¾ Strength The strength of the solid depends on the molecular forces that hold the solid together. Ionic interactions are strong because the opposite charges resist breaking of the intermolecular bonds. For example, the melting temperature of salts is very high, typically 400 to 800°C. Some molecular interactions are strong because of the 3- D arrangement of the atoms. For example, diamonds are exceptionally hard because the solid state forms, a 3-D network such that each carbon is held close to 4 other atoms. The bonds are molecular, not ionic. Types of Non-Bonded (intermolecular, van der Waals) Interactions Dipole-dipole Ion-dipole Induced dipole Dispersion or London Hydrogen bonding H Cl + - H + Cl - Dipole-dipole interactions: when two polar molecules with net dipoles come closer, one end of the dipole in one molecule will be attracted to the opposite end in the second molecule. (a). These forces are strong, but are weaker than ion-ion interactions in ionic compounds. (b). These forces decrease with distance as 3 r 1 Ion-dipole interactions: When polar molecules encounter ions, the positive end of the dipole is attracted to negative ions and vice versa. The ion-dipole interactions require two species to be present, one to provide ions and another to provide dipoles. O H H d+ d+ d- Na+ ClInduced dipole interactions: In non-polar molecules, electrons are distributed symmetrically. This symmetry can be distorted by an ion/dipole, by inducing dipole in non-polar molecule. These forces are weak and are of short range. e e e e e e + e e e e e e + e e Dispersion or London forces: Noble gases are atomic gases and do not have dipole moments or net charges. The fact that they can be liquefied suggests that forces of attraction exist between atoms of a noble gas. These forces are called Dispersion or London forces. The distribution of electrons in an atom/molecule fluctuate over time. These fluctuations set up temporary dipoles which induce dipoles in others. The attraction between temporary dipoles is responsible for dispersion forces. These forces exist between all atoms and molecules. Strength of dispersion forces increases with number of electrons in atoms/molecules. “Dispersion force increases down a group”. Thus, for rare gases, dispersion forces increase as Xe > Kr > Ar > Ne > He
Types of Non- Bonded Interactions Interaction "ion Interaction A hydrogen atom covalently bonded to N, O, or F is attracted to the lone pair of a different atom nearby H-bonding rming a hydrogen bond. 2k/mol Dispersion/London ydrogen bonding is stronger than any other non- bonds/interaction, yet weaker than covalent Induced dipole Hydrogen bonding Hydrogen bond o covalent HF(g) covalent Hydrogen bonding Variation of ionic radii with Some general trends for Ionic radii Coordination Number 1. lonic radii increase on going down a group (Lanthanide contraction restricts the increase of heavy 2. Radii of equal charge ions decrease across a period 3. lonic radii increase with increasing coordination lumber (the higher its CN the bigger the ions seems to The radius of one ion has to be 4. The ionic radius of a given atom decreases with fixed to a reasonable value increasing charge(r(Fe2+)>rFe3+)) (O2)=1.40A→ Linus 5. Cations are usually the smaller ions in a cation/anion That value is then combination(exceptions: r(Cs*)>r(F).1 compile a set of self 6. Frequently used for rationalization of structures nt values for all other oolA rcation)/r(anion)(< 1) 81012 Ion bond and Ion Crystal ome Properties of lonic Crystals Some Properties of lon Crystal rElative stable and hard crystals Lattice energy of ion crystal Poor electrical conductors (lack of free electrons) Ionic radii in crystals High melting and vaporization temperatures ← Pauling' s Rule Transparent to visible light but absorb strongly on bonds with part covalent bond infrared light n water and polar liquids
3 Hydrogen bonding: A hydrogen atom covalently bonded to N, O, or F is attracted to the lone pair of a different atom nearby forming a hydrogen bond. Hydrogen bonding is stronger than any other nonbonded interaction, yet weaker than covalent bonds/ionic bonds O H H .. H O H Hydrogen bond Types of Non-Bonded Interactions Interaction Energy Ion-ion interaction ~250 kJ/mol H-bonding ~20 kJ/mol Ion-dipole Dipole-dipole ~2 kJ/mol Dispersion/London r(Fe3+)) 5. Cations are usually the smaller ions in a cation/anion combination (exceptions: r(Cs+) > r(F- ) ...!!!) 6. Frequently used for rationalization of structures: “radius ratio” r(cation)/r(anion) (< 1) Some General Trends for Ionic Radii Ion Bond and Ion Crystal Some Properties of Ion Crystal Lattice energy of ion crystals Ionic radii in crystals Pauling’s Rules Ion bonds with part covalent bond Some Properties of Ionic Crystals Relative stable and hard crystals Poor electrical conductors (lack of free electrons) High melting and vaporization temperatures Transparent to visible light but absorb strongly infrared light Soluble in water and polar liquids!
Lattice Enthalpy Equilibrium Distance Cohesive Energy AH is the standa molar enthalpy change for the following proces Mtas+Xgas→Ma △H<0 - otal derations are neglected the most stable crystal structure of a given compound is the Lattice Energy(U)of Ionic Compounds emble one mole of a crystalline ionic compound nto free ZTe 1 ZTe Bohr-Madelung equation LAttice Energy of lonic Compounds: Bohr-Madelung Z+z-e- equation ANZ'Z'e Find B and n at equilibrium. dE N=6.02x10=moh cThe Madelung constant endent of the ionic charges 0→B=2ze2 elf know the crystal structure, you can choose a suitable ladelung constant, and the distance between the ions r you can estimate the lattice energy of ion compoun Total energy at ro Azel To What is n? Compressibility= zns (wurtzite) 2ns (ine blende) igg Applications of lattice Enthalpy The Kapustinskii Equation Calculations Kapustinskii noticed that A/v, is almost athermal stabilities of ionic solids constant for all structures v is the number of ions in the formula unit stabilities of oxidation states of cations r=r+r, unit.pm Solubility of salts in water Variation in A /v with structure is partially calculations of electron affinity data canceled by change in ionic radii with coordination number stabilities of "non existent 125200uZZ,34.5
4 Two definitions: The lattice enthalpy change is the standard molar enthalpy change for the following process: M+ (gas) + X- (gas) ® MX(solid) If entropy considerations are neglected the most stable crystal structure of a given compound is the one with the highest lattice enthalpy. Lattice Energy (U) of Ionic Compounds: disassemble one mole of a crystalline ionic compound at 0K into free components o DHL H 0 o D L < Lattice Enthalpy o U = -DHL Equilibrium Distance & Cohesive Energy ) n 1 (1 r Z Z e E E E 0 2 attractive repulsive = - - = + + - Ep r n r B r Z Z e + - 2 - total At equilibrium: (Erepulsive) (Eattractive) 0 2 attractive r Z Z e E + - = - Find B and n at equilibrium: repulsive n r B E = n 2 Total attractive repulsive r B r Z Z e E = E + E = - + + - 0 dr dE Total = 0 r = r n 1 0 2 n 1 0 2 0 2 r n Z Z e 0 B r nB r Z Z e - + - + + - Þ - + = Þ = Total energy at r0 : ) n 1 (1 r Z Z e E 0 2 r r0 = - - + - = What is n? Compressibility Þ n » 9 Bohr-Madelung equation Lattice Energy of Ionic Compounds: Bohr-Madelung equation： N = 6.02x1023 mol-1 The Madelung constantis independent of the ionic charges and the lattice dimensions, but is only valid for one specific structure type If know the crystal structure, you can choose a suitable Madelung constant, and the distance between the ions ro , you can estimate the lattice energy of ion compound. ) n 1 (1 r ANZ Z e U 0 2 = - + - Structure Type Madelung Constant CsCl 1.763 NaCl 1.748 ZnS (Wurtzite) 1.641 ZnS (Zinc Blende) 1.638 thermal stabilities of ionic solids stabilities of oxidation states of cations solubility of salts in water calculations of electron affinity data stabilities of “non existent”compounds Applications of Lattice Enthalpy Calculations The Kapustinskii Equation Kapustinskii noticed that A /n, is almost constant for all structures νis the number of ions in the formula unit ro = r++r-, unit: pm Variation in A /n with structure is partially canceled by change in ionic radii with coordination number ) r 34.5 (1 r 125200 Z Z U 0 0 - u = + -
BONDS Coord. No Length(A Limiting and Optimal radius Ratios for Specific Coordinations 464666 ot"in touch in touch Notes: alon radii for given element increase with coordination Ion radii for given element decrease with increasing oxidation state/positive charge Anions often bigger thancations Radius ratio rules fr Rationalization for octahedral "rattle inside the octahedral site coordination: R= radius of large alf r/R>0.414, the anions are pushed apart ion. Gradius of small ion Coordination Minimum r/R Linear. 2 R+r-c0s45= Trigonal 3 Tetrahedral, 4 0.225 √ER=R+r Octahedral. 6 0.414 Cubic. 8 0.732 Close packed, 12 1.000 A simple prediction tool, but beware- it doesn't always work us Rati ZnS 4:4 miting Radius Ratios anions in the coordination polyhedron of cation are in contact with the cation d with each other Radius ratio Coordination lary (AB) agenda 2 body diagonal a rA/=1 1>x+h。>0732 073>r/>044 0414>r/E>025 篱 七/·3-1.1 0.414 0225
5 BONDS Coord. No. Length (Å) C-O 3 1.32 Si-O 4 1.66 Si-O 6 1.80 G e-O 4 1.79 G e-O 6 1.94 SnI V -O 6 2.09 PbI V -O 6 2.18 PbII -O 6 2.59 Notes: Ion radii for given element increase with coordination number (CN) Ion radii for given element decrease with increasing oxidation state/positive charge Radii increase going down a group Anions often bigger thancations not “in touch” in touch Limiting and Optimal Radius Ratios for Specific Coordinations Radius Ratio Rules Rationalization for octahedral coordination: R= radius of large ion, r=radius of small ion 0.414 R r ( 2 1)R r 2R R r 2 1 cos45 R r R Þ = Þ - = Þ = + = = + o If r/R 0.414, the anions are pushed apart If r/R £ or ³ 0.414, coordination changes: Coordination Minimum r/R Linear, 2 - Trigonal, 3 0.155 Tetrahedral, 4 0.225 Octahedral, 6 0.414 Cubic, 8 0.732 Close packed, 12 1.000 A simple prediction tool, but beware ¾ it doesn’t always work! Limiting Radius Ratios - anions in the coordination polyhedron of cation are in contact with the cation and with each other Radius Ratio Coordination no. Binary (AB) Structure-type r+/r- = 1 12 none known 1 > r+/r- > 0.732 8 CsCl 0.732 > r+/r- > 0.414 6 NaCl 0.414 > r+/r- > 0.225 4 ZnS
C.N. rra Geometry Coordination Number The critical ratio 2 Geometrical Shapes determined by geometrical 0.155-0225 Cubie hole C.N.=3 0.225-0.414 60.414-0.732 C.N.=4 80.732-1.0 ①O Cuboctahedral and Anti-cuboctahedral Common Coordination Polyhedra Structure 份 Cuboctahedron Cubic close packing Hexagonal close packing CN aB CNaB Ceramic Crystal Structures of ionic radii Greation/ranion )dictates the coordination number of anions around each cation Crystal Structure of AB I As the ratio gets larger ( i.e. as rcation/ranion 1, the coordination number gets larger and on Ratio (r+/r)
6 The critical ratio determined by geometrical analysis 2 <0.155 3 0.155-0225 4 0.225 - 0.414 6 0.414 - 0.732 8 0.732 - 1.0 C.N. rC/rA Geometry Coordination Number vs Geometrical Shapes C.N. = 3 C.N. = 4 C.N. = 6 Cubic hole Cuboctahedral hole Cuboctahedral and Anti-cuboctahedral Structure Cuboctahedron ¾¾ Cubic close packing Anti-cuboctahedron ¾¾ Hexagonal close packing Common Coordination Polyhedra Ceramic Crystal Structures The ratio of ionic radii (rcation/ranion ) dictates the coordination number of anions around each cation. As the ratio gets larger (i.e. as rcation/ranion®1), the coordination number gets larger and larger. Crystal Structure of AB vs Ion Ratio (r+/r-)
Structure Maps Plots of r versus Structure Maps-Plots of r, versu rn with structure type indicated with structure type indicated Separation of sEparation of structure types Is structure types is A, BO achieved on structure achieved on structure diagrams-but, the diagrams -but, the bo col Conclusion-Size i Conclusion-Size does matter but not does matter. but not necessarily in any AB necessarily in any 200pa 0如0174op Pauling s rules Pauling s rules for lonic Crystals a Cation environment in a polyhedron(cation anion distance and Coordination Number) Deal with the energy state of the crystal structure a Relationship between bond valence and oxidation number a Corner, edge and face sharing polyhedra .The cation anion distance =2 a Large valence and small Coordination Number cations tend not to share poly hedra elements a Rule of parsimony coordination number of the ca Paulings rules Paulings Rules for lonic Crystals for lonic Crystals 2nd Rule 2nd Rule- the electrostatic valence principle aFirst note that the strength of an electrostatic a An ionic structure will be stable to bond valence/CN strengths of electrostatic bonds that reach an anion from adjacent cations Nat in NaCl is in VI 60+1/6)=+1(sum from Na's) divided by 6 These charges are equal in magnitude so the structure is stable =+1/6
7 Structure Maps ¾¾ Plots of rA versus rB with structure-type indicated Separation of structure types is achieved on structure diagrams ¾ but, the boundaries are complex Conclusion ¾ Size does matter, but not necessarily in any simple way! Structure Maps ¾¾ Plots of rA versus rB with structure-type indicated Separation of structure types is achieved on structure diagrams ¾ but, the boundaries are complex Conclusion ¾ Size does matter, but not necessarily in any simple way! Pauling’s Rules Cation environment in a polyhedron (cationanion distance and Coordination Number) Relationship between bond valence and oxidation number Corner, edge and face sharing polyhedra Large valence and small Coordination Number cations tend not to share polyhedra elements Rule of parsimony Pauling’s Rules for Ionic Crystals Deal with the energy state of the crystal structure 1st Rule The cation-anion distance = S radii Can use r+ /rto determine the coordination number of the cation Pauling’s Rules for Ionic Crystals 2nd Rule First note that the strength of an electrostatic bond = valence / CN Cl Cl Cl Cl Na Na+ in NaCl is in VI coordination For Na+ the strength = +1 divided by 6 = + 1/6 Pauling’s Rules for Ionic Crystals 2nd Rule ¾ the electrostatic valence principle + 1 /6 + 1 /6 + 1 /6 + 1 /6 Na Na Na Na ClAn ionic structure will be stable to the extent that the sum of the strengths of electrostatic bonds that reach an anion from adjacent cations = the charge of that anion 6( + 1/6 ) = +1 (sum from Na ’s) charge of Cl = -1 These charges are equal in magnitude so the structure is stable
awnings Rules Pauling s Rules for lonic Crystals for lonic Crystals 2nd Rule- the electrostatic valence principle 3rd rule The sharing of edges, and particularly of faces, of adjacent poly hedra tend to decrease the stability of an ionic structure dn ISio l, strengths of Si-O=1, Strength=2, stable df l Al replace 1 Si, sTrength= 1+3/4=1.75, unstable af2 Al replace 2 Si, sTrength =3/4+3/4=1.5 very unstable 争争 Polyhedral Linking Paulings Rules 4th Rule: 器樂郎 An a crystal with different cations, those of high valence and small CN tend not to share polyhedral elements 是 n extension of rule3 The stability of structures with different types of lyhedral linking is vertex sharing edgesharing> ice.sharin Eeffect is largest for cations with high charge and low coordination number Especially large when r, /r approaches the lower Sit in Iv coordination is very unlikely to share limit of the polyhedral stabilit edges or faces Paulings rules Polarization of lon for lonic crystals Polarization of an ion is the distortion of the ctron cloud of the anion due to the influence of 5th Rule- Rule of Parsimony the nearby cation. .The number of different kinds of constituents in Perfect model of ionic compound a crystal tends to be small Not a purely ionic compound wards the cation and result in higher stronger bond is resulted
8 In [SiO4 ], strengths of Si-O=1, åStrength=2,stable If 1 Al replace 1 Si, åStrength = 1+3/4=1.75, unstable If 2 Al replace 2 Si, åStrength =3/4+3/4=1.5 very unstable Pauling’s Rules for Ionic Crystals •2nd Rule ¾ the electrostatic valence principle 3rd Rule: The sharing of edges, and particularly of faces, of adjacent polyhedra tend to decrease the stability of an ionic structure Pauling’s Rules for Ionic Crystals Polyhedral Linking The stability of structures with different types of polyhedral linking is vertex-sharing > edge-sharing > face-sharing effect is largest for cations with high charge and low coordination number especially large when r+ /r- approaches the lower limit of the polyhedral stability 4th Rule: In a crystal with different cations, those of high valence and small CN tend not to share polyhedral elements An extension of Rule 3 Si4+ in IV coordination is very unlikely to share edges or faces Pauling’s Rules for Ionic Crystals 5th Rule ¾ Rule of Parsimony The number of different kinds of constituents in a crystal tends to be small Pauling’s Rules for Ionic Crystals Polarization of Ion Polarization of an ion is the distortion of the electron cloud of the anion due to the influence of the nearby cation. + – + – Perfect model of ionic compound Electron cloud of anion is attracted towards the cation and result in higher electron density between the ion, stronger bond is resulted. Ionic compound with polarization of ion ¾¾ Not a purely ionic compound
Polarizing power of cation Polarizability of Anion B The ability to distort the electron a The ease of the electron cloud being distorted distribution of adjacent ions or atom. by the influence of adjacent ion or atom. bThe polarizing power of cation is favored by w The more polarizable anion would Id have higher charge and smaller size to have a higher charge and larger size higher charge density PAF+>Mg2+>Na+ Lit> Nat Polarization Effect on AB, Structures(A= Transition Metal Polarization aSame r/, larger size of anion induced more increasing Polarization part of electrons are active in the whole crystal, property of semiconductor and in bonding Hela layer , pe metal, such as Fes, smaller cation has more polarizing lavers/chains crystals to molecule crystals along vertical axi Sisa chain Covalent Crystals一 Covalent Bonding Held Together by Covalent Bond Review some important features of covalent Share electrons lead to strongest bond a Some Properties Basic Concepts of Molecular Orbital Theory Very hard half· filled orbital High melting points hybrid orbital Insulators/semiconductors Atomic orbital molecular orbital bonding(symmetric) molecular orbital tric) molecular orbital
9 Polarizing Power of Cation The ability to distort the electron distribution of adjacent ions or atom. The polarizing power of cation is favored by higher charge and smaller size Þ to have a higher charge density Al3+ > Mg2+ > Na+ Li+ > Na+ Polarizability of Anion The ease of the electron cloud being distorted by the influence of adjacent ion or atom. The more polarizable anion would have higher charge and larger size S2- > O2- S2- > ClPolarization Effect on AB2 Structures (A = Transition Metal) Same r+ /r- , larger size of anion induced more polarizable anion ® part of electrons are active in the whole crystal ® property of semiconductor and metal, such as FeS2 smaller cation has more polarizing power ® ion crystals to molecule crystals along vertical axis. Polarization Increasing Polarization in bonding low-dimensionality ¾¾ layers/chains Covalent Crystals ¾ Held Together by Covalent Bonds Share electrons lead to strongest bonds Some Properties: - Very hard. - High melting points. - Insulators/semiconductors. Covalent Bonding Review some important features of covalent bonding: ßBasic Concepts of Molecular Orbital Theory half-filled orbital hybrid orbital atomic orbital & molecular orbital bonding (symmetric) molecular orbital antibonding (antisymmetric) molecular orbital
What is the Largest Molecule in the world? Molecular Crystals D Not polymer aggregates -held together by van der waals bond Diamonds. The largest of all was the so-called Cullinan diamond, found on Jan 25, 1905, in South Africa and sporting a weight of 621.6g(3106 carats). electrie dipole moment diamond of 38.41 carats, the growth of whine an artificial davs The hardest element with the Polar molecules attract each other thermal conductor is Ag (429W/mK) Van der Waals Attraction between Non polar molecule Metallic Bond and Metallic Crystal On the average, non"polar fRee Electrons Gas Model in Metals ③ distribut. but at a The Nearly. Free- Electron Model in Metals moment the distributions are 9 Distribution of electrons in metals asymmetric. The fluctuations in the charge distributions of ● Band theory earby molecules lead to an attractive force, given also by Like copper and gold, most of the Metals are solids which require extra known elements are metals discussion to explain their special properties High electrical conductivity ..+e high heat conductivity caba umawaos=m npa am
10 What is the Largest Molecule in the world? Not polymer aggregates The largest molecules that have ever been found are Diamonds. The largest of all was the so-called Cullinan diamond, found on Jan. 25, 1905, in South Africa and sporting a weight of 621.6 g (3106 carats). The largest man-made synthetic molecule is an artificial diamond of 38.4 carats, the growth of which required 25 days. The hardest element with the highest thermal conductivity of >2000 W/mK (the best metallic thermal conductor is Ag (429 W/mK). Molecular Crystals ¾held together by van der Waals bonds weak … but everywhere. Polar molecules ¾ electric dipole moment Polar molecules attract each other: Van der Waals Attraction between Non-polar molecules: On the average, non-polar molecules are symmetric distributions, but at any moment the distributions are asymmetric. The fluctuations in the charge distributions of nearby molecules lead to an attractive force, given also by: attractive 6 r 1 U ~ - Metallic Bond and Metallic Crystal Free Electrons Gas Model in Metals The Nearly-Free-Electron Model in Metals Distribution of Electrons in Metals Band Theory Like copper and gold, most of the known elements are metals. Metals are solids which require extra discussion to explain their special properties: Ductility Shiny surface High electrical conductivity high heat conductivity
