Atomic Orbitals - Hydrogen Atom and Atomic Orbitals General...

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Hydrogen Atom and Hydrogen Atom and Atomic Orbitals General Chemistry Dr. Michael W.-Y. Yu ABCT PolyU ABCT, PolyU Outline z Stationary wavefunctions for Hydrogen Atom z Orbitals z Degeneracy z Three quantum numbers for electron in three-dimensional motion z Principal quantum number ( n ) z Angular momentum quantum number ( l ) z Magnetic quantum number ( m ) z Spin quantum number ( s ) z Shape of the orbitals z Probability density z Radial probability distribution z Radial nodes, nodal planes and angular nodes Petrucci, General Chemistry Principles & Modern Applications , 9th edition, Pearson International Edition, 2007.
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Schrödinger Equation for One-Electron Atom z A single electron moving in a spherical electric field under the influence of a nuclear charge (+Ze ) z Electron motion around the nucleus as wave and Electron motion around the nucleus as wave, and mathematically expressed by a wavefunction z Wavefunction being solved by the Schrödinger equation Kinetic energy Coulombic potential energy 2 = 2 / x 2 + 2 / y 2 + 2 / z 2 = Laplacian Operator r = distance of the electron from the nucleus Spherical Polar Coordinates ψ = R ) Y θφ = ( r ) ( θ , φ ) R(r) = radial wavefunction Y ( , ) = angular wavefunction / spherical harmonics z The potential energy depends only on the distance between the nucleus and the electron spherically symmetric the nucleus and the electron ( ) z Transformation from Cartesian coordinates ( x, y, z ) to spherical polar coordinates ( r , θ, φ ) z r = distance from the center of the atom z = angle from the positive x -axis (the “ north pole ”) z = angle about the z -axis (the “ equator ”)
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Energy of the Electron z Coulombic interaction between an electron and the nuclear charge +Z nuclear charge +Ze z V(r) = (+Ze)(e) / 4 πε o r , for hydrogen atom, Z = 1 z Solve the Schrödinger equation with z V(r) = e 2 / 4 o r for hydrogen atom z Energy = E =- h R H / n 2 n / z ( R H = m e e 4 / 8h 3 ε o 2 ; n = 1, 2, 3 …) z R=329 × 10 15 Hz = Rydberg constan R = 3.29 Hz = Rydberg constant z For other hydrogen-like atoms: E n = -Z 2 h R H / n 2 z (Z = atomic number (Z atomic number) Principal Quantum Number ( n ) z The integer that labels the energy levels of the electron z For n = 1, the lowest energy state (electronic ground state ground state) z For n >1, electronic excited states z Transition between electronic states leads to absorption / emission of a photon ( h ν = Δ E ) z E = 0, when n = , ionization state z Thus, ionization energy = E E 1
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Atomic Orbitals z Wavefunction for an electron in an atom is called an atomic orbital z Heisenberg’s Uncertainty Principle – velocity and position cannot be simultaneously specified z An orbital
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Atomic Orbitals - Hydrogen Atom and Atomic Orbitals General...

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