CHE 349 Physical Chemistry for Life Sciences Lecture 19 Schr dinger equation 22 2 24 0 0 8 854 10 12 2 2 4 32 2 02 2 2 2 Quantized energy of the hydrogen atom equation Simplistic energy of the hydrogen atom equation This constant can be expressed in many unit systems like Rydberg constant 2 179 10 11 2 179 10 18 1312 13 60 Hydrogenic Wavefunctions 1 2 2 2 0 0 1 1 0 0 0 1 1 1 0 32 0 2 14 2 0 32 2 0 2 0 2 14 2 0 52 2 0 2 14 2 0 52 2 0 2 14 2 0 52 2 0 2 0 529 0 529 10 10 2 2 2 2 0 0 2 1 magnetic quantum number positive integers 1 2 3 for a given n zero and positive integers up to principle quantum number orbital angular momentum quantum number Hydrogen atom The permissible values of the quantum numbers are 22 12 22 2 2 4 0 1 2 2 4 0 2 2 4 0 12 1 1 2 2 12 Many Electron Atoms Hamiltonian operator for two electrons formula to The symbols s p d and f are assigned to functions with l 0 1 2 and 3 respectively for a given l integers from Pauli Exclusion Principle This principle states at most two electrons can be in each orbital Two electrons in the same orbital must have opposite spins No two electrons in an atom can have the same four quantum numbers To ensure this the wave function describing the system of electrons must be antisymmetric upon the exchange of any pair of electrons In simpler terms if you swap the positions of two electrons the overall wave function must change its sign This antisymmetry helps prevent the violation of the Pauli exclusion principle and is crucial for understanding the behavior of electrons in atoms and molecules Ionization Energy The first ionization energy from a many electron element 1 2 is the minimum energy required to remove an electron The second ionization energy is the energy required to remove the second electron The ionization energy generally increases upon going from left to right in a row of the periodic table Similarly it increases as you move across groups in the periodic table like from the bottom of Group 2 all the way to the top of Group 17 In contrast it decreases going down a column of the periodic table Electron affinity Electron affinity is defined as the energy released upon binding of an electron to a gas phase atom The electron affinity increases upon going from left to right in a row of the periodic table Similarly it increases as you move across groups in the periodic table like from the bottom of Group 2 all the way to the top of Group 17 In contrast it decreases going down a column of the periodic table Ionization energy and electron affinity directly reflects the orbital energies of many electron atoms Electronegativity is defined as the tendency of an atom participating in a covalent bond to attract the shared electrons to itself Mulliken electronegativity definition 12 1 According to Mulliken the electronegativity is one half of the sum of the first ionization energy and electron affinity of an atom NOTE X is the symbol for electronegativity Hybridization The three types of hybrid AOs are The order of increasing energy levels is 1s 2s 3d Equation for the hybrid orbital Equation for the hybrid orbital Equation for the hybrid orbital 2 3 and 2 3 2 2 2 12 2 12 2 2 13 2 23 2 3 14 2 34 2 3s 3p 4s Hybridization Example Bond Angle 180 2 3 120 109 5 Radial Wave Function vs Probability Distribution Hydrogen Atom The left hand side of this image represents the radial wave functions The right hand side represents the radial probability distribution Visualizing an electron distribution in space Three things happen to all orbital types s p d f as n increases They become larger extending farther from the nucleus They contain more nodes This is similar to a standing wave that has regions of significant amplitude separated by nodes points with zero amplitude They become higher in energy as n increases An atom or ion with the electron s in the lowest energy orbital s is said to be in its ground state whereas an atom or ion in which one or more electrons occupy higher energy orbitals is said to be in an excited state Type of orbital of angular nodes s p d f 0 1 2 3 0 1 2 3 0 1 0 1 2 1 0 1 2 3 2 1 0 1 2 3 0 1 2 3 4 4 4 4