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Madelung Constant, A SolidSolid StateState TheoryTheory PhysicsPhysics 545545 Structure of Matter • Properties of materials are a function of their: Atomic Structure Bonding Structure Crystal Structure Imperfections • If these various structures are known then the properties of the material can be determined. • We can tailor these properties to achieve the material needed for a particular product. ¾¾ PropertiesProperties ofof aa materialmaterial z DependsDepends onon bondingbonding ofof atomsatoms thatthat formform thethe materialmaterial z DictatesDictates howhow itit interactsinteracts andand respondsresponds toto thethe worldworld aroundaround it.it. TheseThese include:include: • Physical Properties z Form (Gas, Liquid, Solid) ,Hardness (Rigid, Ductile, Strength), Reactivity • Thermal Properties z Heat capacity, Thermal conductivity, • Electrical Properties z Conductivity, Dielectric Strength, Polarizability • Magnetic Properties z Magnetic permittivity, • Optical Properties z Spectral absorption, Refractive index, Birefringence, Polarization, Atomic Theory • Nucleus of the atom is made up of protons (+) and neutrons (-) • Number of electrons surrounding the nucleus must equal the number of protons (free state-electrically neutral). • Atomic Number - the number of protons in the nucleus • Atomic Number determines the properties and characteristics of materials Atomic Structures And Atomic Bonding • gives chemical identification Nucleus • consists of protons and neutrons • # of protons = atomic number • # of neutrons gives isotope number Electrons • participate in chemical bounding • described by orbitals ElectronElectron ConfigurationsConfigurations ValenceValence electrons:electrons: occupyoccupy thethe outermostoutermost filledfilled shell.shell. + Example:Example:Na SodiumSodium atom:atom:Only Na:Na: 1 1s1selectron22s2s 2in2p2p the63s 3s3rd1 shell, it is readily released. Once this electron is +11e released, it becomes Sodium ion (Na+). Cation: positive charge Anion: negative charge Electron Behavior • Elements cannot be fully explained by nucleus alone - requires understanding electron behavior as well. • When shells are filled, the atom is stable. • Electrons in unfilled shells are know as valence electrons. • Valence electrons are largely responsible for element behavior. • Partially filled shells mean that electrons may be given up, accepted from other atoms, or may share them with other atoms. • Manner in which this stabilization occurs determines the type of bonding. Bonding Forces and Energies FA FR r = + FN FA FR Where FN: Net force between the two atoms FA: Attractive force FR: Repulsive force Bonding Energy Energy and force relation: E = ∫ Fdr r E = F dr N ∫∞ N r r = F dr + F dr ∫∞ A ∫∞ R = + → When E A E R Bonding energy dE N = 0, E = E dr N 0 Bonding Forces And Energies + = FA FR 0 r0 r E = (F + F )dr = E + E N ∫ A R A R ∞ Primary bondings: Ionic, Covalent, Metallic bonds Secondary bondings: Van der Waals bond, Hydrogen bond Classification of Bonds Primary bonds Ionic Bonding Covalent Bonding Metallic Bonding Secondary bonds van der Waals bonds Hydrogen bonds • Van de Waals Bonds – Low temperature(~0K), Noble gases •Ionic (ie Ar, He, Kr, Ne) – Unequal sharing of outer electrons – Electrostatic attraction due to electron –egNaCl + - orbit variations Na loses electron to Cl -> ions Na , Cl – Bond due to net electrostatic attraction between – results in charge polarization of atoms ions – Weak bond – Electrons tightly bound to atoms • Covalent – Outer valance electrons shared equally between atoms – Strong bond, electrons tightly bound to atoms – No net charge - No Electrostatic forces – Eg C, Ge, Si • Metallic – Variation of Covalent – Valance electrons stripped from nucleus – Shared equally between all atoms as sea of community e’s – Bonding due to electrostatic attraction between sea of -ve electrons and sea of +ve nuclei – High thermal and electrical conductivity due to free electron gas – Eg: Na, Cu, Al, Mg, Fe Van Der Waals Bond • Formed when an atom or Hydrogen bonds: permanent dipole bonds molecule is asymmetric, creating a net polar moment in the charges. • The bond is weak and is found in neutral atoms such as inert gases. •No electron transfer or sharing Based upon the Van der Waals bonds: attraction of dipoles fluctuating dipole bonds, Ar •Bonding energy: Ar ~0.01 eV (weak) •Compared to thermal + Ar - + Ar - vibration energy kBT ~ 0.026 eV at T = 300 K Dipole-dipole interaction •Examples: inert gases Secondary Atomic & Molecular Bonds [Van der Waals Bonds] Permanent Dipole Bonds • Weak intermolecular bonds are formed between molecules which possess permanent dipoles. • A dipole exists in a molecule if there is asymmetry in its electron density distribution. Fluctuating Dipole Bonds • Weak electric dipole bonding can take place among atoms due to an instantaneous asymmetrical distribution of electron densities around their nuclei. • This type of bonding is termed fluctuation since the electron density is continuously changing. Van der Waals Bonding Energy of the van der Waals bond A B U = r 6 + r n (n ≅ 12) 3 ways to lead to a rather weak bonding - Fluctuating induced dipole bonds - Polar molecule-induced dipole bonds - Permanent dipole bonds Pauli exclusion principle: Ionic Bond • When elements donate or receive an electrons in its outer shell a charged particle or an ION is formed. • If the element gives up an electron, it is then left with at net + 1 charge, and is called a POSITIVE ION. • Charged particles are attracted to each other. Ionic Bonding NaCl Formation of ionic bond 1. Mutual ionization occurs by electron transfer 2. Ions are attracted by coulombic forces = − EA A/ r = n ER B / r , n ~ 8 o ~ 640 KJ/mole or 3.3eV/atom, Tm~ 801 C Bonding energy: 1-10 eV (strong) An ionic bond is non-directional (Ions can be attracted to one another in any direction) Ionic Bonding Some aspects to remember: 1. Electronegative atoms will generally gain enough electrons to fill their valence shell and more electropositive atoms will lose enough electrons to empty their valence shell. e.g. Na: [Ne]3s1 → Na+: [Ne] Ca: [Ar]4s2 → Ca+2: [Ar] Cl: [Ne]3s2 3p5 → Cl-: [Ar] O: [He]2s2 2p4 → O-2: [Ne] 2. Ions are considered to be spherical and their size is given by the ionic radii that have been defined for most elements (there is a table in the notes on Atomic Structure). The structures of the salts formed from ions is based on the close packing of spheres. 3. The cations and anions are held together by electrostatic attraction. Ionic Bonding Because electrostatic attraction is not directional in the same way as is covalent bonding, there are many more possible structural types. However, in the solid state, all ionic structures are based on infinite lattices of cations and anions. There are some important classes that are common and that you should be able to identify, including: CsCl NaCl Zinc Blende Fluorite Wurtzite And others…Fortunately, we can use the size of the ions to find out what kind of structure an ionic solid should adopt and we will use the structural arrangement to determine the energy that holds the solid together - the crystal lattice energy, U0. Ionic Bonding The “cation” has a + charge & the “anion” has the - charge. The cation is much smaller than the anion. Ionic Bonding Most ionic (and metal) structures are based on the “close packing” of spheres - meaning that the spheres are packed together so as to leave as little empty space as possible - this is because nature tries to avoid empty space. The two most common close packed arrangements are hexagonal close-packed (hcp) and cubic close packed (ccp). Both of these arrangements are composed of layers of close packed spheres however hcp differs from ccp in how the layers repeat (ABA vs. ABC). In both cases, the spheres occupy 74% of the available space. Because anions are usually bigger than cations, it is generally the anions that dominate the packing arrangement. hcp ccp Usually, the smaller cations will be found in the holes in the anionic lattice, which are named after the local symmetry of the hole (i.e. six equivalent anions around the hole makes it octahedral, four equivalent anions makes the hole tetrahedral). Ionic Bonding Some common arrangements for simple ionic salts: Cesium chloride structure Rock Salt structure 8:8 coordination 6:6 coordination Primitive Cubic (52% filled) Face-centered cubic (fcc) e.g. CsCl, CsBr, CsI, CaS e.g. NaCl, LiCl, MgO, AgCl Zinc Blende structure Wurtzite structure 4:4 coordination 4:4 coordination fcc hcp e.g. ZnS, CuCl, GaP, InAs e.g. ZnS, AlN, SiC, BeO Ionic Bonding Fluorite structure Anti-fluorite structure 4:8 coordination 8:4 coordination fcc e.g. Li2O, Na2Se, K2S, Na2S e.g. CaF2, BaCl2, UO2, SrF2 You can determine empirical formula for a structure by counting the atoms and partial atoms within the boundary of the unit cell (the Rutile structure box). E.g. in the rutile structure, 6:3 coordination two of the O ions (green) are fully Body-centered cubic (bcc) within the box and there are four (68% filled) half atoms on the faces for a total of 4 O ions. Ti (orange) one ion is e.g. TiO2, GeO2, SnO2, NiF2 completely in the box and there are 8 eighth ions at the corners; Nickel arsenide structure this gives a total of 2 Ti ions in the cell. This means the empirical 6:6 coordination formula is TiO2; the 6:3 ratio is hcp determined by looking at the e.g. NiAs, NiS, FeS, PtSn number of closest neighbours around each cation and anion. There are many other common forms of ionic structures but it is more important to be able to understand the reason that a salt adopts the particular structure that it does and to be able to predict the type structure a salt might have. Ionic Bonding The ratio of the radii of the ions in a salt can allow us to predict the type of arrangement that will be adopted. The underlying theory can be attributed to the problem of trying to pack spheres of different sizes together while leaving the least amount of empty space.
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