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Physics 102: Lecture 25 At i Spectroscopy Atomic S t & Quantum Q t Atoms At Physics 102: Lecture 25, Slide 1 From last lecture – Bohr model Angular momentum is quantized Ln = nh/2π n = 1, 2, 3 ... Energy is quantized mk 2 e 4 Z 2 13.6 ⋅ Z 2 En = − ≈− eV ( where h ≡ h / 2π ) 2 2 2 2h n n Radius is quantized 2 2 n2 ⎛ h ⎞ 1 n = ( 0.0529 nm ) rn = ⎜ ⎟ 2 Z ⎝ 2π ⎠ mke Z Velocity too! En= -13.6 Z2/n2 = ½ mvn2 Physics 102: Lecture 25, Slide 2 +Ze Transitions + Energy Conservation • Each orbit has a specific energy: energ : En= -13.6 Z2/n2 • Photon absorbed when electron j jumps from f low l energy to t high hi h energy. Photon emitted when electron l t jumps j from f high hi h energy to t low energy orbit: E2 – E1 = h f = h c / λ Physics 102: Lecture 25, Slide 3 E2 E1 Demo: Line Spectra p In addition to the continuous blackbody spectrum, elements emit a discrete set of wavelengths which show up as lines in a diffraction grating. En= -13.6 Z2/n2 656 nm H n=3 n=2 This is how neon signs & Na lamps work! Spectra give us information on atomic t i structure t t Physics 102: Lecture 25, Slide 4 n=1 Checkpoint 1.1 Electron A falls from energy level n=2 to energy level n=1 ( (ground d state), t t ) causing i a photon h t to t be b emitted. itt d Electron B falls from energy level nn=33 to energy level nn=11 (ground state), causing a photon to be emitted. n=3 n=2 Which photon has more energy? 1) Photon A 2) Photon B A B n=1 Physics 102: Lecture 25, Slide 5 Spectral Line Wavelengths Calculate the wavelength of photon emitted when an electron in the hydrogen yd oge atom ato ddrops ops from o tthee n=2 state to the t e ground g ou d state ((n=1). ). E2= -3.4 eV n=3 n=2 Z2 E n = −13.6eV 2 n hf = E2 − E1 = −3.4eV − (−13.6eV) = 10.2eV E1= -13.6 eV Ephoton = Physics 102: Lecture 25, Slide 6 n=1 hc λ hc 1240 λ= = ≈ 124nm 10.2eV 10.2 ACT: Spectral Line Wavelengths Compare the wavelength of a photon produced from a transition from o n=3 3 to n=2 with w t that t at of o a photon p oto produced p oduced from o a transition ta sto n=2 to n=1. (Α) λ32 < λ21 (Β) λ32 = λ21 (C) λ32 > λ21 E32 < E21 Physics 102: Lecture 25, Slide 7 so n=3 3 n=2 λ32 > λ21 n=1 ACT/Checkpoint 1.2 The electrons in a large group of hydrogen atoms are eexcited c ted to the t e n=33 level. eve . How ow many a y spectral spect a lines es will w be produced? A. 1 B. 2 n=3 n=2 C 3 C. D. 4 E 5 E. n=1 Physics 102: Lecture 25, Slide 8 The Bohr Model is incorrect! To be consistent with the Heisenberg Uncertainty Principle Principle, which of these properties cannot be quantized (have the exact value known)? Electron Radius Would know location Electron Energy Electron Velocity Would know momentum Electron Angular Momentum B in But, i the h Bohr B h model: d l 2 2 2 h 1 n n ⎛ ⎞ rn = ⎜ = ( 0.0529 0 0529 nm ) ⎟ 2 Z ⎝ 2π ⎠ mke Z Physics 102: Lecture 25, Slide 9 Quantized radii and velocities for electron orbitals Checkpoint 2 +Ze Bohr Model Quantum Atom So what keeps the electron from “sticking” to the nucleus? Centripetal Acceleration Pauli Exclusion Principle Heisenberg Uncertainty Principle Physics 102: Lecture 25, Slide 10 Quantum Mechanics Q Theory used to predict probability distributions QM Physics 102: Lecture 25, Slide 11 Quantum Mechanical Atom • Predicts available energy states agreeing with Bohr. Bohr • Don’t have definite electron position, only a probability b bilit function. f ti • Each orbital can have 0 angular momentum! • Each electron state labeled by 4 numbers: n = pprincipal p quantum q number (1, ( 2, 3, …)) l = angular momentum (0, 1, 2, … n-1) p of l ((-l < ml < l)) ml = component ms = spin (-½ , +½) Physics 102: Lecture 25, Slide 12 Quantum Mechanics (vs. Bohr) Electrons are described by a probability function, not a definite radius! Physics 102: Lecture 25, Slide 13 Quantum Numbers Each electron in an atom is labeled by 4 #’s n = Principal p Q Quantum Number ((1,, 2,, 3,, …)) • Determines the Bohr energy l = Orbital Quantum Number (0, (0 1, 1 2, 2 … n-1) n 1) • Determines angular momentum • l <n always true! h L = l(l + 1) 2π ml = Magnetic Quantum Number (-l , … 0, … l ) • zz-component component of l • | ml | <= l always true! ms = S Spin i Q Quantum t N Number b ((-½ ½ , +½) • “Up Spin” or “Down Spin” Physics 102: Lecture 25, Slide 14 h Lz = ml 2π ACT: Quantum numbers For which state of hydrogen is the orbital angular momentum required to be zero? 1. n=1 2. n=2 3. n=3 Physics 102: Lecture 25, Slide 15 The allowed values of l are 0, 1, 2, …, n-1. When n=1, l must be zero. Spectroscopic p p Nomenclature “Shells” “Subshells” l =0 0 is “s s state state” l =1 is “p state” l =2 is “d state” l =3 is “f state” l =4 is “g g state” n=1 is “K K shell shell” n=2 is “L shell” n=3 n 3 is “M M shell shell” n=4 is “N shell” n=5 n 5 is “O O shell shell” 1 electron in ground state of Hydrogen: n=1, l =0 is denoted as: 1s1 n=1 Physics 102: Lecture 25, Slide 16 l =0 1 electron Electron orbitals IIn correct quantum mechanical h i l description d i i off atoms, positions ii off electrons not quantized, orbitals represent probabilities Carbon orbitals imaged in 2009 using electron microscopy! Physics 102: Lecture 25, Slide 17 Quantum Numbers How many unique electron states exist with n n=2? 2? l = 0 : 2s2 ml = 0 : ms = ½ , -½ 2 states l = 1 : 2p p6 ml = +1: ms = ½ , -½ ml = 0: ms = ½ , -½ ml = -1: 1 ms = ½ , -½ ½ 2 states 2 states 2 states t t There are a total of 8 states with n=2 Physics 102: Lecture 25, Slide 18 ACT: Quantum Numbers How many unique electron states exist with n=5 and ml = +3? A) 0 B) 4 C) 8 D) 16 E) 50 l l l l = 0 : ml = 0 = 1 : ml = -1, 0, +1 = 2 : ml = -2,, -1,, 0,, +1,, +2 Only y l = 3 and l = 4 have ml = +3 = 3 : ml = -3, -2, -1, 0, +1, +2, +3 ms = ½ , -½ 2 states l = 4 : ml = -4, -3, -2, -1, 0, +1, +2, +3, +4 ms = ½ , -½ 2 states There are a total of 4 states with n=5, ml = +3 Physics 102: Lecture 25, Slide 19 Pauli Exclusion Principle In an atom with many electrons only one electron quantum state ((n,, l,, ml, ms)). is allowed in each q This explains the periodic table! Physics 102: Lecture 25, Slide 20 Electron Configurations # electrons Atom Configuration 1 2 H He 1s1 1s2 3 Li 1s22s1 4 Be 1s22s2 5 B etc 1s22s22p1 10 Ne (n=1 shell filled noble gas) 2s shell filled 1s22s22p6 s shells hold up to 2 electrons Physics 102: Lecture 25, Slide 21 1s shell filled 2p shell filled (n=2 shell filled noble gas) p shells hold up to 6 electrons The Periodic Table s (l =0) n = 1, 2 2, 3, ... p (l =1) Also s d (l =2) f (l =3) What determines the sequence? Pauli exclusion & energies Physics 102: Lecture 25, Slide 22 Summary • Each electron state labeled by 4 numbers: n = principal i i l quantum t number b (1, (1 2, 2 3, 3 …)) l = angular momentum (0, 1, 2, … n-1) ml = componentt off l (-l ( < ml < l)) ms = spin (-½ , +½) • Pauli Exclusion Principle explains periodic table • Shells fill in order of lowest energy. gy Physics 102: Lecture 25, Slide 23