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CHEM 107, Fall 2016 Electrons & Quantum Mechanics Class #16 Electrons, Quantum Numbers, and Energy Levels CHEM 107 L.S. Brown Texas A&M University Uncertainty Principle • Curious result of wave interpretation • “Position and energy (or momentum) cannot be specified simultaneously” • Mathematically: • Experiments prove that energy levels exist • How can we explain this? • “Quantum Mechanics” Application of wave concepts to atoms & electrons Electrons • Electrons in atoms can only have certain quantized energies. • If we measure electron energy, the position CAN’T be specified (Δx)(Δp) > h/4π Electron Diffraction Electrons • Electrons show properties of waves AND particles. (Like light does!) • Electrons in atoms best described as “delocalized waves” Aluminum Graphite (single crystal) © 2016, L.S. Brown 1 CHEM 107, Fall 2016 Standing Waves n=3 Ψ n=2 0 L n=1 Wavefunctions & Intensity Ψ2 n=3 n=2 n=1 Equation • We can write a general equation for the allowed standing waves: Ψ = sin nπx L Where n = 1, 2, … n is called a quantum number Schrödinger Equation HΨ=EΨ • This is a differential equation in disguise • H is an “operator,” Ψ is a “wave function,” and EΨ is an energy. • H includes kinetic and potential energies Intensity at any point is proportional to the square of the amplitude of the wave. Schrödinger Equation For a hydrogen atom: ∂2 Ψ ∂ 2 Ψ ∂ 2 Ψ e 2 + + Ψ = EΨ ∂x 2 ∂y 2 ∂z 2 4πε0 r Schrödinger Equation • Solving equation gives Ψ and E for allowed states. • Use spherical coordinates: Ψ = Ψ(r, θ, φ ) • Solution contains 3 different quantum numbers © 2016, L.S. Brown 2 CHEM 107, Fall 2016 Orbitals & Quantum Numbers • Q. numbers: n, ℓ, and mℓ • Also called “principal,” “azimuthal,” “magnetic” • A set of the 3 numbers defines an orbital • Orbital = the wave representation of an electron in an atom Quantum Numbers • n - principal quantum number influences energy and size of the orbital ● n = 1, 2, 3, ... ● • ℓ - azimuthal quantum number ● ● shape of orbital (mainly) ℓ = 0, 1, 2, ..., (n – 1) Allowed combinations Quantum Numbers • n = 1, 2, 3, … • ℓ = 0, 1, 2, ..., (n – 1) • mℓ - magnetic q. number orientation of orbital (mainly) ● mℓ = –ℓ, ..., 0, ... +ℓ n l ml # of orbitals type of orbitals 1 0 0 1 1s 2 0 1 0 -1,0,+1 1 3 2s 2p 3 0 1 2 0 -1,0,+1 -2,-1, 0,+1,+2 1 3 5 3s 3p 3d ● Combining these rules lets us find the allowed orbitals Some Orbital Wavefunctions (DON’T try to memorize these!) r 1 Ψ2s = (2a)-3 / 2 (2 - ) exp(-r / 2a) a 4π 1 r 3 Ψ2 pz = (2a)-3 / 2 exp(-r / 2a) cos θ 3 a 4π 1 r 3 Ψ2 px = (2a)-3 / 2 exp(-r / 2a) sin θ cos φ 3 a 4π 1 r 3 Ψ2 py = (2a)-3 / 2 exp(-r / 2a) sin θ sin φ 3 a 4π The meaning of Ψ • Orbitals are wavefunctions, defined mathematically • Physical interpretation? • Ψ 2 is the probability of finding the electron at some point in space • “Pictures” of orbital shapes are graphs of Ψ 2 © 2016, L.S. Brown 3 CHEM 107, Fall 2016 Representing Orbitals Electron probability • We depict orbitals in different ways • Graph of Ψ2 (or “electron density”) • Pictorial representations 1s orbital Most probable distance: Bohr radius = 53 pm Distance from nucleus 1s orbital • s orbital wavefunctions contain no angular terms ☛ Spherical orbitals p orbitals • Set of 3 p orbitals for each n-value (2p, 3p, ...) • All same shape “dumbbell” or “hourglass” • All mutually perpendicular © 2016, L.S. Brown 4