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Quantum theory and models of the atom Guess now. It has been found experimentally that: (a) light behaves as a wave; (b) light behaves as a particle; (c) electrons behave as particles; (d) electrons behave as waves; (e) all of the above are true; (f) none of the above are true.
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Quantum theory and models of the atomeggnchips.bucket.s3-website-eu-west-1.amazonaws.com/.../Photoelectric2.pdf · Quantum theory and models of the atom I Radiation intensity is proportional

Jun 09, 2019

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Page 1: Quantum theory and models of the atomeggnchips.bucket.s3-website-eu-west-1.amazonaws.com/.../Photoelectric2.pdf · Quantum theory and models of the atom I Radiation intensity is proportional

Quantum theory and models of the atom

Guess now.It has been found experimentally that:

(a) light behaves as a wave;

(b) light behaves as a particle;

(c) electrons behave as particles;

(d) electrons behave as waves;

(e) all of the above are true;

(f) none of the above are true.

Page 2: Quantum theory and models of the atomeggnchips.bucket.s3-website-eu-west-1.amazonaws.com/.../Photoelectric2.pdf · Quantum theory and models of the atom I Radiation intensity is proportional

Quantum theory and models of the atom

I Radiation intensity is proportional to T 4

I Materials glow white/yellow at 2000K.

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Quantum theory and models of the atom

I The Wavelengths of the peaks follow λpT = 2.90× 10−3

I This is Wien’s law.

I Draw a graph of T against λ.

I What type of curve is it?

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Quantum theory and models of the atom

I Example: Sun temperature

I If λp =500 nm for the Sun, what is the temperature

T =2.90x10−3

500x10−9= 6000K

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Quantum theory and models of the atom

Page 6: Quantum theory and models of the atomeggnchips.bucket.s3-website-eu-west-1.amazonaws.com/.../Photoelectric2.pdf · Quantum theory and models of the atom I Radiation intensity is proportional

Quantum theory and models of the atom

I Example: Star colour

I The surface temperature of a star is 32,500K. What colourwould the star appear?

I Answer:

λp =2.x10−3

32, 500= 89.2nm

I This is the UV part of the spectrum.

I In the visible part the curve will be descending, therefore, theshortest visible wavelength will be the strongest. The star will,therefore, appear blue.

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Quantum theory and models of the atom

I Planck’s formula

I =2πhe2λ−5

ehc

λkT − 1I Planck’s assumption to justify the formula was that

I E = nhf , where n is an integer.

I n is called the quantum number.

I Planck suggested that energy of molecular vibration is only amultiple of a base energy value.

I Called Planck’s quantum hypothesis.

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Quantum theory and models of the atom

I Example: Photon energy. Calculate the energy of a photon ofblue light λ = 450 nm (in air).

I Approach

E = hf c = f λ ∴ E =hc

λI Answer

6.63x10−34 × 3.0x108

4.5× 10−7= 4.4× 10−19J

or4.4× 10−19

1.6× 10−19= 2.8eV

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Quantum theory and models of the atom

I Example: Photons from a lightbulb.Esitmate how many visible light photons a 100W light-bulbemits per second. (Assume energy efficiency is 3% i.e 97%energy is lost as heat).

I Approachλ = 500nm

E =hc

λI Answer

3W = 3Js−1 3 =nhc

λ

∴ n =3

hf=

3× 500× 10−9

6.63× 10−34 × 3.0× 108= 8× 1018

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Quantum theory and models of the atom

I Exercise: Compare a light source of 1000nm (IR) with a lightsource of 100nm (UV).

E = hf =hc

λ=

hc

1000× 10−9

For λ = 1000nm n =6.63× 10−34 × 3.0× 108

1000× 1.6× 10−38

=12.43

1000× 10−12= 12.4× 109

λ = 100nm n = 12.4× 1010 which 10 times more electrons.

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Quantum theory and models of the atom

Photoeclectric effect

I Video: A simple gold leaf electroscope.

I Video: Lonnie’s lab. UC Berkeley

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Quantum theory and models of the atom

What is the kinetic energy and the speed of an electron ejectedfrom a sodium surface who’s work function is W0 = 2.28eV when

(a) λ = 410nm

(b) λ = 550nm

I Approach: Find the energy of the photons. E = hf =hc

λIf E >W0 we get electrons. The maximum kinetic energy isE −W0

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Quantum theory and models of the atom

Answer

I m = 9.11× 10−31kg. E =1

2mv2

E =hc

λ= 3.03 eV

E −W0 = 3.03− 2.28 ∴ v =

√2E

m

=

√2× 0.75× 1.6× 10−19

9.11× 10−31= 0.513× 106 = 5.1× 105ms−1

I (b) No electrons released.Note: If v is close to 0.1c we have to look at the relativisticequation.

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Quantum theory and models of the atom

Example 2What is the lowest frequency and the longest wavelength neededto emit electrons from sodium?An exercise left for the student.

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Quantum theory and models of the atom

Energy, mass and momentum

I We know E = hf

I Photons travel at the speed of light.

I Need a relativistic formulae. (p 6= mv for a photon)

p =mv√1− v2

c2

I Since v = c for a photon, p becomes infinite which isobviously wrong.

I Relativistic equation gives E 2 = p2c2 + m2c4

I Setting m = 0 we get E 2 = p2c2 ∴ p =E

c

I Since E = hf for a photon, p =hf

c=

h

λ

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Quantum theory and models of the atom

I Example. The 1019 photons emitted per sec from the 100Wlightbulb are focused on a piece of black paper and absorbed.

(a) Calculate the momentum of one photon.(b) Estimate the force of these photons on the paper.

I Approach:

p =h

λ. The momentum changes from p to zero. Use

Newton’s second law, F =∆p

∆t.

I Answer

p =h

λ= 6.63.

If N = 1019 photons, F = 10−8 N.

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Quantum theory and models of the atom

I Sun

I If the number of photons is large, n × 10−8N can be aconsiderable force!

I Solar sail is an example.

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Quantum theory and models of the atom

I Photosynthesis

I Chlorophyll in plants captures the energy of sunlight tochange CO2 to useful carbohydrate. 9 photons are needed totransform one molecule of CO2 to carbohydrate and O2.Assuming λ = 670 nm (chlorophyll absorbs most strongly inthe range 650nm - 700nm), how efficient is the photosyntheticprocess? The reverse chemical reaction releases an energy of4.9eV/molecule of CO2.

I ApproachEfficiency is the minimum energy required (4.9eV) divided bythe actual energy absorbed ( 9× the energy of one photon).

Answer: hf =hc

λ.

9× hcλ = 2.7× 10−18 J. = 17eV. ∴4.9

17= 29% efficient.

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Quantum theory and models of the atom

I Compton effect. AH Compton (1892 - 1962)

I λ′ = λ+h

mec(1− cosφ)

I λ′ − λ is the Compton shift.

I Used to measure bone density for osteoperosis.

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Quantum theory and models of the atom

Photon interactions - Summary We get:

1. Photoelectric effect. Knocks an electron out of an atom.Photon disappears.

2. Excited state of atom (see later). Photon disappears.

3. Scattered electron and photon. Compton effect.

4. Pair production. Creates matter (e.g. e+ and e−). Photondisappears.

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Quantum theory and models of the atom

Wave nature of electron.

I Louis de Broglie (1892-1987)

I He proposed that the wavelength of a material particle wouldbe related to its momentum (in same way as for a photon).

p =h

λwhich implies λ =

h

pI Note that this is valid classically, p = mv for (v << c) and

relativistically p =mv√

1− v2

c2

.

I λ =h

pis called the de Broglie wavelength of a particle.

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Quantum theory and models of the atom

Example. Wavelength of a ball.Calculate the de Broglie wavelength of a 0.20 kg ball moving withspeed of 15ms−1.

I Approach

Use λ =h

p.

I Answerλ = 2.2× 10−34m

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Quantum theory and models of the atom

ExampleWhat is the wavelength of an electron that has been acceleratedthrough a potential of 100V.

I Approach

Kinetic energy =1

2mv2 = eV , λ =

h

mvI Answer λ = 1.2× 10−10 m = 0.12nm.

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Quantum theory and models of the atom

Electron diffraction

I Assume electrons strike perpendicular to the surface of a solidand their energy is low. K = 100eV.If the smallest angle at which diffraction maximum occurs is24◦, what is the separation between the atoms on the surface.

Page 25: Quantum theory and models of the atomeggnchips.bucket.s3-website-eu-west-1.amazonaws.com/.../Photoelectric2.pdf · Quantum theory and models of the atom I Radiation intensity is proportional

Quantum theory and models of the atomElectron diffraction

I ApproachTreat electrons as waves. Constructive interference occurswhen the difference in path = λ, that is, a sin θ = λ

K =p2

2me=

h2

2meλ2∴ λ =

√h

2meK= 0.123nm.

∴ a =λ

sin θ=

0.123

sin 24◦= 0.30 nm.

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Quantum theory and models of the atom

No-one has actually seen an electron but . . .

they are good for looking at small things. The ebola virus attacksa cell. Transmission electron microscope (x 50,000). Scanningelectron microscope (x35,000)

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Quantum theory and models of the atom

Transmission electron microscope (x 50,000).Scanning electron microscope (x35,000)

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Quantum theory and models of the atom

Spectra

I Excited gasses in a discharge tube produce discrete spectra.

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Quantum theory and models of the atom

I The emission spectra is a characteristic of the material andcan serve as a finger print.

I Also if a continuous spectrum passes through a rarefied gas,dark lines appear where the emission lines occurred. This iscalled an absorption spectrum.

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Quantum theory and models of the atom

Hydrogen is the simplest atom. One electron. Simplest spectrum.

I J.J. Balmer (1825 - 1898) showed that the 4 lines in the

visible spectrum have wavelengths that fit1

λ= R(

1

22− 1

n2)

for n = 3, 4, . . .

I R is the Rydberg constant. R = 1.0974× 107m−1

I Balmer lines extend into the UV region ending at λ = 365nm.

I Later experiments found that if1

22was replaced with

1

12,

1

32,

1

42.

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Quantum theory and models of the atom

In general

I1

λ= R(

1

n′2− 1

n2)

I n′ = 1 for Lyman series.

I n′ = 2 for Balmer series.

I n′ = 3 for Paschen series.

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Quantum theory and models of the atom

The Bohr model

I Electrons in orbits.

I Angular momentum = L = mvrn =nh

2πfor n = 1, 2, 3, . . .

I F =1

4πε0

(Ze)e

r2Coulomb’s law.

I Newton’s second law gives F = ma.

I The acceleration of the particle is a =v2

rn

I ∴1

4πε0

Ze2

rn2=

mv2

rn