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Important Equations c = λν c = 3 x 108 m per second λ = wavelength ν = frequency hc Ephoton = h ν = λ E = hν h = Planck’s constant = 6.62 x 10-34 J s or h = 6.62 x 10-34 kg m2 s-1 Week 5 CHEM 1310 - Sections L and M 1 “Quantum” Light is quantized [Max Planck, 1901] – Quite a surprise as light was thought to be continuous Energy can be gained or lost only in integer multiples of h ν. ΔE = n(h ν) n is an integer (1,2,3,…) Each unit of size hν is called a packet or quantum Week 5 CHEM 1310 - Sections L and M 2 Energy Has Mass! E = mc2 m= E c2 Êcˆ E = hn = hÁ ˜ Ë l¯ Week 5 m= hc λ c2 λ= λ= CHEM 1310 - Sections L and M = h λc h mc h mv 3 1 Hydrogen’s Atomic Spectrum Week 5 CHEM 1310 - Sections L and M 4 Light & Energy Are Quantized DE n = hc ln n is excited state orbitals n = 1, 2, 3, … Week 5 CHEM 1310 - Sections L and M 5 Bohr Model Discrete energy levels available to electrons in which to move Week 5 CHEM 1310 - Sections L and M Niels Bohr (1885-1962 ) Nobel Prize Physics, 1922 6 2 Bohr Model Another View Week 5 Niels Bohr (1885-1962 ) Nobel Prize Physics, 1922 CHEM 1310 - Sections L and M 7 What Bohr’s Calculations Led To Angular momentum, radius, and energy of the electrons traveling in discrete orbits Angular Momentum = mevr h = n n = 1,2,3,..... 2p Week 5 Z2 E n = - 2 (2.18x10-18 J) n n2 a 0 = radius of each orbital Z a 0 called the bohr radius, a constant rn = n = orbitals, excited states n = 1,2,3,... n = 1 called ground state Z is the postive charge on the nucleus (1 of H, 2 for He, etc.) CHEM 1310 - Sections L and M 8 Heisenberg Uncertainty Principle Dx ! mDv " n 4p imprecision of position imprecision of momentum Werner Heisenberg 1927 You cannot measure/observe something without changing that which you are observing/measuring. Week 5 CHEM 1310 - Sections L and M 9 3 Schrödinger Equation Ĥψ= Eψ Ĥ is a Hamiltonian operator in linear algebra E = total energy of the atom ψ= a wave function which defines an electrons position in 3D space (x, y, z), called an orbital ψ2 = the probability that an electron is in a certain region of space; this defines the shape of the orbital (s, p, d, f) Week 5 CHEM 1310 - Sections L and M 10 Movement of an Electron The H e- can be visualized as a standing wave around the nucleus. Not the planetary orbits assumed by Bohr. Week 5 CHEM 1310 - Sections L and M 11 Solutions to the Schrödinger Eqn 1. n = principal quantum number ψ(n, l, ml) n = 1, 2, 3, … n is related to the size and energy of the orbital 2. l = angular (azimuthal) quantum number l = 0, 1, …. (n-1) l is related to the shape of the orbital l = 0 is called an s orbital l = 1 is called a p orbital ψ(n, l = 2 is called a d orbital l = 3 is called an f orbital l = 4 is called a g orbital Week 5 CHEM 1310 - Sections L and M l, ml) 12 4 Solutions to the Schrödinger Eqn 3. ml = magnetic quantum number ml = -l, … , 0, ….+l ψ(n, l, ml) ml relates to the orientation of the orbital 4. Although not a solution to the Schrödinger Equation, a 4th quantum number is ms = electron spin quantum number ms = +1/2, -1/2 denoted by ↑↓ Week 5 CHEM 1310 - Sections L and M 13 Hydrogen Atom – n is related to the size and energy of the orbital – l is related to the shape of the orbital – ml relates to the orientation of the orbital n l orbital designation ml # of orbitals 1 0 0 1 0 1 2 0 1 2 3 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 0 0 -1, 0, +1 0 -1, 0, +1 -2, -1, 0, +1, +2 0 -1, 0, +1 -2, -1, 0, +1, +2 -3, -2, -1, 0, +1, +2, +3 1 1 3 1 3 5 1 3 5 7 2 3 4 Week 5 CHEM 1310 - Sections L and M 14 Representations of Orbitals ψ(1, 0, 0) Week 5 ψ(2, 0, 0) ψ(3, 0, 0) CHEM 1310 - Sections L and M 15 5 Representations of p-Orbitals Week 5 CHEM 1310 - Sections L and M 16 Representations of d-Orbitals Week 5 CHEM 1310 - Sections L and M 17 6