Bohr Model and Franck-Hertz Experiment Announcements

Bohr model and Franck-Hertz experiment Announcements:

• Will finish up material in Chapter 5. • There will be no class on Friday, Oct. 18. Will announce again! • Still have a few midterms–see me if you haven’t picked it up • Today will be analyzing the Bohr model with clicker questions.

Gustav Hertz: 1887-1985

James Franck: 1882-1964 http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 1 Clicker question 1 Set frequency to AD What atomic energy

levels for electrons are 3ev 5ev

2ev Photon energy consistent with this spectrum? 0 100 200 300 400 500 600 700 800 nm Electron Energy levels: A B C D E 0 eV 0 eV 0 eV 10 eV −2 eV −3 eV 7 eV −5 eV 5 eV 5 eV −5 eV −5 eV −7 eV −7 eV 3 eV −8 eV 2 eV −10 eV −10 eV 0 eV 0 eV http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 2 Clicker question 1 Set frequency to AD What atomic energy

levels for electrons are 3ev 5ev

Bohr postulated that angular momentum was quantized so: where Quantizing angular momentum leads to a quantization of radius: where

Quantizing radius leads to a quantization of energy: is the Bohr radius

where ER is the Rydberg energy: http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 4 Last lecture we found v - If the electron orbits the proton at a constant speed, the magnitude of the net force on the electron is mv2/r r +

Setting the net force (from Coulomb) equal to the mass times acceleration (mv2/r) for circular motion gives us:

which we can also write as: http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 5 What does this say about total energy? - v We now put together three pieces: F 2 2 r 1. mv = ke /r (just derived) 2. electrostatic potential energy is U = -ke2/r + 3. nonrelativistic kinetic energy is K = ½mv2 This means K = -½U and the total energy is

distance from proton 0 Note E = -K and 2K = -U potential energy The total energy, radius, and velocity are all related. Knowing just one of the three determines the other two!

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 6 A little algebra

mvr =nħ = nh/2π , n = 1, 2, 3, …..

So r = n2ħ2/m ke2 , n = 1, 2, 3, …. r = .053 nm for n =1 And v = ke2/nħ , n = 1, 2, 3, …. v = 2.2 × 106 m/s Justifies using classical mechanics, but for large Z need to be careful

E = -K = mv2/2 = -mk2e4/2n2ħ2, n = 1, 2, 3, …. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 7 Clicker question 2 Set frequency to AD Which of the following principles of classical physics is violated in deriving the Bohr model of the atom? A. Opposite charges attract with a force inversely proportional to the square of the distance between them. B. The force on an object is equal to its mass times its acceleration. C. Accelerating charges radiate energy. D. Particles have a well-defined position and momentum. E. All of the above.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 8 Clicker question 2 Set frequency to AD Which of the following principles of classical physics is violated in deriving the Bohr model of the atom? A. Opposite charges attract with a force inversely proportional to the square of the distance between them. B. The force on an object is equal to its mass times its acceleration. C. Accelerating charges radiate energy. D. Particles have a well-defined position and momentum. E. All of the above.

Note that A & B were used in the derivation of the Bohr model.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 9 Bohr’s calculation of hydrogen energy levels Using the formula for energy Hydrogen energy levels

Bohr could calculate the various transitions and they agreed with the generalized Balmer formula.

Photon energy is given by: ⎛ 1 1 ⎞ ⎛ 1 1 ⎞ E = E − E = −E ⎜ − ⎟ = −13.6 eV ⎜ − ⎟ γ n n' R ⎝ n'2 n 2 ⎠ ⎝ n'2 n 2 ⎠

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 10

€ Clicker question 3 Set frequency to AD An atom with the energy levels shown is initially n=5 −1 eV n=4 −2 eV in the ground state. A free electron with an n=3 −5 eV energy of 16.0 eV hits the atom. What possible states could the atom be in after the interaction? n=2 −10 eV A. n=1 n=1 −20 eV B. n=1, n=2, or n=3 C. n=3 D. n=2 or n=3 E. Any of the states

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 11 Clicker question 3 Set frequency to AD An atom with the energy levels shown is initially n=5 −1 eV n=4 −2 eV in the ground state. A free electron with an n=3 −5 eV energy of 16.0 eV hits the atom. What possible states could the atom be in after the interaction? n=2 −10 eV A. n=1 n=1 −20 eV B. n=1, n=2, or n=3 C. n=3 If the atom goes to n=1, 2, or 3, D. n=2 or n=3 the free electron will lose 0 eV, 10 E. Any of the states eV, or 15 eV of kinetic energy.

Free electron hits atom e Less e e KE

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 12 Clicker question 4 Set frequency to AD An atom with the energy levels shown is initially n=5 −1 eV n=4 −2 eV in the ground state. A photon with an energy of n=3 −5 eV 16.0 eV hits the atom. What possible states could the atom be in after the interaction? n=2 −10 eV A. n=1 n=1 −20 eV B. n=1, n=2, or n=3 C. n=3 D. n=2 or n=3 E. Any of the states

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 13 Clicker question 4 Set frequency to AD An atom with the energy levels shown is initially n=5 −1 eV n=4 −2 eV in the ground state. A photon with an energy of n=3 −5 eV 16.0 eV hits the atom. What possible states could the atom be in after the interaction? n=2 −10 eV A. n=1 n=1 −20 eV B. n=1, n=2, or n=3 C. n=3 For the free electron, whatever energy D. n=2 or n=3 is absorbed by the atom is deducted E. Any of the states from the free electron’s kinetic energy. For photons, the photon is absorbed and so it must transfer all of its energy to the atom. Otherwise energy would not be conserved. Therefore, atoms can only absorb photons with an energy that will exactly move the atom to another energy level.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 14 Clicker question 5 Set frequency to AD n=5 −1 eV An atom with the energy levels shown is initially n=4 −2 eV in the ground state. Which of the following is n=3 −5 eV the most complete list of photon energies that n=2 −10 eV can be absorbed by the atom?

A. 1 eV n=1 −20 eV B. 1 eV, 2 eV, 5 eV, 10 eV C. 10 eV D. 10 eV, 15 eV, 18 eV E. 10 eV, 15 eV, 18 eV, 25 eV

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 15 Clicker question 5 Set frequency to AD n=5 −1 eV An atom with the energy levels shown is initially n=4 −2 eV in the ground state. Which of the following is n=3 −5 eV the most complete list of photon energies that n=2 −10 eV can be absorbed by the atom?

A. 1 eV n=1 −20 eV B. 1 eV, 2 eV, 5 eV, 10 eV Photons with energy of 10 eV, C. 10 eV 15 eV, 18 eV, 19 eV will cause D. 10 eV, 15 eV, 18 eV the electron to jump to the n=2, E. 10 eV, 15 eV, 18 eV, 25 eV n=3, n=4, n=5 energy level. Any photon with an energy ≥ 20 eV will ionize the atom. The electron will escape. Since a free electron can have any energy, any photon with energy ≥ 20 eV can be absorbed by the atom. 20 eV will go into ejecting the electron and the rest will go into the free electron’s kinetic energy. Similar to the photoelectric effect. http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 16 Hydrogen like ions Atoms which have only one electron can be analyzed much like the hydrogen atom. An atom with atomic number Z with Z-1 electrons removed is a hydrogen like ion

The (Coulomb) force on the electron is

The increase in the force results in tighter orbits and a deeper potential well, reducing the energy (more negative).

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 17 Clicker question 6 Set frequency to AD A single electron is in the n=2 energy level around a helium nucleus (He+). What is the minimum energy photon that can remove this electron? A. 3.4 eV B. 13.6 eV C. 27.2 eV D. 54.4 eV E. None of the above

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 18 Clicker question 6 Set frequency to AD A single electron is in the n=2 energy level around a helium nucleus (He+). What is the minimum energy photon that can remove this electron? A. 3.4 eV B. 13.6 eV C. 27.2 eV The atomic number of helium is Z=2 D. 54.4 eV and the electron is in the n=2 energy E. None of the above level. So the energy of the state is

The value |En| is also referred to as the binding energy since it is a measure of how bound the electron is. It takes that amount of energy to free the electron (break its bond to the proton).

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 19 Reduced Mass • Have shown that energy for one electron atoms levels scale as E = -K = mv2/2 = -mk2Z2e4/2n2ħ2, n = 1, 2, 3, ….

• But we have one more correction – it was found that the energy levels in Hydrogen were all lower by 1 part out of 1800. This was accounted for by the “reduced mass”, because we had assumed the nucleus was infinitely heavy Reduced Mass

L = meMN/(me+MN) x vr = nħ

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 20 Reduced Mass 1    2 r = r − r E = mev + V(r) 21 2 1 2   { m r + M r = 0 1 e 1 2 E = m r˙ 2 + V(r) 2 e  M  r = − r 1 2 1 2 € 1 m M 21 E = m r˙ + M r˙ + V(r − r ) e + 2 e 1 2 2 2 1 €  m  r e r € Plugging back into energy equation 2 = 21 me + M We find: € 2 2 € 1 M 2 1 me 2 E = me 2 r˙21 + M 2 r˙21 + V(r2 − r1) 2 (me + M) 2 (me€+ M)

1 me M 2 E = r˙21 + V(r2 − r1) 2 (me + M) € http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 21

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