Physics 102: Lecture 25

Atom ic S pect roscopy & Q uant um At oms

Physics 102: Lecture 25, Slide 1 From last lecture – Bohr model

Angular is quantized

Ln = nh/2π n = 1, 2, 3 ... Energy is quantized mk24 e Z 213.6⋅ Z 2 Ehn =−22 ≈− 2 eV ()where h ≡ / 2π 2h nn Radius is quantized

2 22+Ze ⎛⎞hn1 n rn ==⎜⎟ 2 ()0.0529 nm ⎝⎠2π mke Z Z

2 2 2 Velocity too! En= -13.6 Z /n = ½ mvn Physics 102: Lecture 25, Slide 2 Transitions + Energy Conservation

• Each orbit has a specific energy: energ : 2 2 En= -13.6 Z /n E2 • absorbed when jfljumps from low energy t thiho high energy. Photon emitted when eltlectron j umps f rom high hih energy t o low energy orbit: E1

E2 – E1 = h f = h c / λ

Physics 102: Lecture 25, Slide 3 Demo: Line Spectra

In addition to the continuous blackbody spectrum, elements emit a discrete set of wavelengths which show up as lines in a diffraction grating. H n=3 2 2 656 nm En= -13.6 Z /n n=2

This is how neon signs & Na lamps work!

Spectra give us information on n=1 atitomic stttructure

Physics 102: Lecture 25, Slide 4 Checkpoint 1.1

Electron A falls from n=2 to energy level n=1 (ground st tt)ate), causi ng a ph htoton t o b e emitt ittded.

Electron B falls from energy level nn3=3 to energy level nn1=1 (ground state), causing a photon to be emitted.

n=3 Which photon has more energy? n=2 1) Photon A A B 2) Photon B

n=1 Physics 102: Lecture 25, Slide 5 Spectral Line Wavelengths

Calculate the wavelength of photon emitted when an electron in the hydroge n ato m dr ops fr om th e n=2 state to t he g rou nd state (n=1). n=3 Z 2 E = − . eV n=2 n 13 6 2 E2= -3.4 eV n

hf = E2 − E1 = −3.4eV − (−13.6eV) =10.2eV E = -13.6 eV 1 n=1

hc hc 1240 E = λ = = ≈124nm photon λ 10.2eV 10.2

Physics 102: Lecture 25, Slide 6 ACT: Spectral Line Wavelengths

Compare the wavelength of a photon produced from a transition from n=3 to n=2 w it h t hat o f a p hoto n p roduced fro m a t ra ns it io n n=2 to n=1.

(Α) λ32 < λ21 n=3 n=2 (Β) λ32 = λ21

(C)λ32 > λ21

E32 < E21 so λ32 > λ21 n=1

Physics 102: Lecture 25, Slide 7 ACT/Checkpoint 1.2

The in a large group of are exc ited to t he n=3 leve l. How ma ny spect ra l lines w ill be produced? A. 1 n=3 B. 2 n=2 C. 3 D. 4 E. 5 n=1

Physics 102: Lecture 25, Slide 8 The Bohr Model is incorrect!

To be consistent with the Heisenberg , 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 BihBhdlBut, in the Bohr model:

2 ⎛⎞hn1 22 nQuantized radii rn ==⎜⎟ 2 (0. 0529 nm) and velocities for 2π mke Z Z ⎝⎠ electron orbitals Physics 102: Lecture 25, Slide 9 Checkpoint 2

+Ze

Bohr Model Quantum

So what keeps the electron from “sticking” to the nucleus? Centripetal Acceleration Pauli Exclusion Principle Heisenberg Uncertainty Principle

Physics 102: Lecture 25, Slide 10 Theory used to predict probability distributions

QM

Physics 102: Lecture 25, Slide 11 Quantum Mechanical Atom • Predicts available energy states agreeing with Bohr. • Don’t have definite electron position, only a probability bilit f uncti on. • Each orbital can have 0 angular momentum! • Each electron state labeled by 4 numbers: n = pppqrincipal ( 1, 2, 3, …) l = angular momentum (0, 1, 2, … n-1)

ml = compp(onent of l (-l < ml < l)

ms = (-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 Q uantum Number (,,,(1, 2, 3, … ) • Determines the Bohr energy

l = Orbital Quantum Number (0 , 1 , 2 , … n -1) • Determines angular momentum h L =+ll(1) • l < n always true! 2π

ml = (-l , … 0, … l ) • z-component of h l Lm= z l • | ml | <= l always true! 2π

ms = SiSpin Quan tum Num ber (-½+½)½ , +½) • “Up Spin” or “Down Spin” Physics 102: Lecture 25, Slide 14 ACT: Quantum numbers

For which state of hydrogen is the orbital angular momentum required to be zero?

1. n=1 The allowed values of l are 2. n=2 0, 1, 2, …, n-1. When n=1, l must be zero. 3. n=3

Physics 102: Lecture 25, Slide 15 Spppectroscopic Nomenclature “Shells” “Subshells” n=1 is “K shell” l =0 is “s state” n=2 is “L shell” l =1 is “p state” n=3 is “M shell” l =2 is “d state” n=4 is “N shell” l =3 is “f state” n=5 is “O shell” l =4 is “g state”

1 electron in ground state of Hydrogen: n=1, l =0 is denoted as: 1s1

n=1 l =0 1 electron

Physics 102: Lecture 25, Slide 16 Electron orbitals IhildiifiifIn correct quantum mechanical description of atoms, positions of 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 nn2=2?

l = 0 : 2s2

ml = 0 : ms = ½ , -½ 2 states l = 1 : 2p6

ml = +1: ms = ½ , -½ 2 states ml = 0: ms = ½ , -½ 2 states ml = -1: ms = ½ , -½ 2 sta tes

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 = 0 : ml = 0 Only l = 3 and l = 4 l = 1 : ml = -1, 0, +1 have ml = +3 l = 2 : ml = -2, -1,,, 0, +1, +2

l = 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 is allowed in each qq(,uantum state (n, l, ml, ms). This explains the !

Physics 102: Lecture 25, Slide 20 # Electron Configurations electrons Atom Configuration 1 H 1s1 2 2 He 1s 1s shell filled (n=1 shell filled - noble gas)

3 Li 1s22s1

4 Be 1s22s2 2s shell filled 5 B 1s22s22p1 etc (n=2 shell filled - 10 Ne 1s22s22p6 2p shell filled noble gas)

s shells hold up to 2 electrons p shells hold up to 6 electrons

Physics 102: Lecture 25, Slide 21 The Periodic Table s (l =0) Also s p (l =1)

d (l =2) , 3, ... 22 n = 1,

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 = priilincipal quant um numb b(123er (1, 2, 3, …) l = angular momentum (0, 1, 2, … n-1)

ml = componentfl(t of l (-l < ml < l)

ms = spin (-½ , +½) • Pauli Exclusion Principle explains periodic table • Shells fill in order of lowest energy.

Physics 102: Lecture 25, Slide 23