PHYS1008: The Bohr Atom

Orion: the red colour is (mostly) hydrogen.

Objectives: by the end of this you will understand Why we need photons how Bohr managed to understand the hydrogen atom How atoms radiate What energy levels are Why the Bohr model didn't really work

Planck's Radiation Law

• Wien noticed that experimental curve looked like Maxwell-Boltzmann distribution. for molecules

• Consider box of Z different atoms: velocities of each will be random, but will "average out" to give smooth curve. Energy is equally divided between different molecules

  E = 1/2 m v²
  

• on the average: also average energy increases with temperature.
• But suppose light is a particle...

• Planck 1900 suggested that E.M. radiation is emitted in lumps (quanta) which became known as "photons"

  f λ = c 
  

• (frequency x wavelength = speed)

  E = h f
  

• (energy of photon = Planck's constant x frequency)
• Energy contained in quantum increases with frequency so costs more energy to emit short wavelength light.
• Planck's constant

  h = 6.6 x 10-34 Js
   = 4.1 x 10-15 eV s
  

• Hence at room temperature average photon has
• E ~1/40 eV

  E = hf = hc/λ
  

• λ ~ 50000 nm which is very long wavelength I.R.
• As temperature increases,
  1. total energy increases
  2. wavelength for maximum decreases
• The photo-electric effect also involves light:maybe that can have a photon explanation.

Einstein and the Photo-electric Effect

Since particles have energy, photons must carry energy (obviously light has energy: that's how you get a sun tan!)

• Einstein (1903): (note this was what he actually got the Nobel prize for) adapted Planck's hypothesis to say that light is absorbed in quanta (photons). Most energetic electrons in metal need extra boost to escape:

• K.E. = 1/2mv²= hf-φ where φ is work function: differs for different metals. Slope is universal ⇒ Planck's constant

  >

• So we have to take photons seriously!

• fλ = c 
  

  (frequency x wavelength = speed)

• E = h f
  

• Electrons are ejected as soon as light strikes (no need for energy to accumulate), since photons arrive randomly

• "Classical" corks bob up and down

• "Quantum" corks are either stationary or ejected.

• what is frequency of orange light λ= 600 nm,
  1. 4x10¹⁴
  2. 4x10^-14
  3. 6x10^-7
  4. 1.7x10⁶
• what is energy of photon of orange light (f = 4x10¹⁴)
  1. 200eV
  2. 10^-19eV
  3. 2eV
  4. 4^-30eV
• Roughly how many photons does 60W bulb produce each second (energy of each photon is 2eV = 3.31x10^-19 J)?
  1. 2x10^-20
  2. 2x10¹⁹
  3. 3x10¹⁴
  4. 2x10²⁰

What is light?

• Particle? Newton, Descartes
• Wave? Young, Huyghens
• Yes? Einstein
• Light travels as wave, but arrives and departs as particle
• Why are we not aware of discrete arrival of photons?

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  • 	λ	f	E
    γ-rays	10-13	3 x 1021	12 MeV
    UV	10-7	3 x 1015	12 eV
    Visible	6 x 10-7 m	5 x 1014	2 eV
    Radio	300m	106	4 x 10-9 eV

  • Gives the "energy" part of the spectrum.

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  Compton Effect

  Further evidence that Planck and Einstein were right came later with the Compton Effect Compton 1923.

  • A photon can scatter off an electron like a billiard ball, in which case, instead of Ee = h f we would get Ee = h f - h f'

  e.g if a photon with frequency f = 10¹⁸Hz scatters of a stationary electron, and the resultant photon has a frequency f' = 10¹⁷Hz, what is the energy of the electron?
  • Problems:
    • How can something be a wave and a particle at the same time?
  e.g if a photon with frequency f = 10¹⁸Hz scatters of a stationary electron, and the resultant photon has a frequency f' = 10¹⁷Hz, what is the energy of the electron?

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  The Bohr atom

  (Bohr 1917)

  Model for H. atom must explain

  • Spectrum
    \color{red}{ \frac{1}{\lambda } = R\left( {\frac{1}{{n^2 }} - \frac{1}{{m^2 }}} \right)}
  • m > n, n = 1, 2, 3...

    n = 1 m = 2, 3, 4...
    n = 2 m = 3, 4, 5...
    n = 3 m = 4, 5, 6...
    

  • This implies only photons of certain definite energies are emitted . . .
  • Rutherford's observation of massive nucleus
  • Stability

  ---

  • Let's build an electric solar system!

  • electrostatic force = centripetal force FE = QeQp = mv² 4πε₀r² r

  ---

  However,

  Qe= - Qp = e
  

  • From this, total energy
    \color{red}{ E_{tot} = \frac{1}{2}mv^2 - \frac{{e^2 }}{{4\pi \varepsilon _0 r}} = - \frac{1}{2}\frac{{e^2 }}{{4\pi \varepsilon _0 r}}}
  • This energy must somehow be quantized. How? For a photon, energy is related to frequency
  • E = n h f
  • (n is the number of photons of a given energy: n = 0,1,2,3,...). For electron,
  • this must be frequency of orbital motion
    \color{red}{ f = \frac{v}{{2\pi r}} \Rightarrow E = \frac{{nhv}}{{2\pi r}}}

  ---

  • However, v and r are not easy to find. Bohr took the angular momentum
  • \color{red}{ L = mvr = \frac{{\frac{1}{2}mv^2 r}}{{\frac{1}{2}v}}}
  • \color{red}{ L = mvr = \frac{{\frac{1}{2}mv^2 r}}{{\frac{1}{2}v}} = \frac{{2Er}}{v}}
  • and assumed that angular momentum was quantised
    \color{red}{ L = mvr = \frac{{\frac{1}{2}mv^2 r}}{{\frac{1}{2}v}} = \frac{{2Er}}{v} = \frac{{2nh}}{{2\pi }}}

  ---

  • This factor 2 vanishes in more sophisticated models, as we will see later
    \color{red}{ L = n\hbar }
  • Where
    \color{red}{ \hbar = \frac{h}{{2\pi }} = 1.05 \times 10^{ - 34} Js = 6.56 \times 10^{ - 16} eVs}
    (say h-bar).
  • This is Bohr's quantization condition.
  • This means only certain orbits allowed:
  • why?

  • because!

  ---

  To find radius of orbits
  • \color{red}{ {\rm{F}}_{\rm{E}} = \frac{{mv^2 }}{r} = \frac{{e^2 }}{{4\pi \varepsilon _0 r^2 }},mvr = n\hbar }
  • \color{red}{ R_n = \frac{{4\pi \varepsilon _0 n^2 \hbar ^2 }}{{me^2 }}}
  • R₁ ~ 5 x 10^-11 m ~ 1/2 Å ~ .05 nm in agreement with prejudice about size of atoms

  ---

  Bohr Energies

  • These levels have energies

    \color{red}{ \begin{array}{l} E_n = - \frac{1}{2}\frac{{e^2 }}{{4\pi \varepsilon _0 R_n }} = - \frac{1}{2}\frac{{e^2 }}{{4\pi \varepsilon _0 }}\frac{{me^2 }}{{4\pi \varepsilon _0 n^2 \hbar ^2 }} \\ = - \frac{1}{2}\left( {\frac{{e^2 }}{{4\pi \varepsilon _0 }}} \right)^2 \frac{m}{{n^2 \hbar ^2 }} = - \frac{{13.6}}{{n^2 }}eV \\ \end{array}}

  • So how does this explain the spectra?

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  Radiation and the Bohr atom

  • Energies of these levels:

    En = = - 13.6   eV
             n²
    

  • Since these are the only allowed levels, energy can only be emitted when electron jumps from one to the other

  • e.g. n = 3 ⇒ n = 2 δE = E3 - E₂ = 1.9 eV

  • what wavelength light does this correspond to?
  • This transition gives 1.9 eV photon which is red line in H.

  ---

  • In general
    \color{red}{ E_\gamma = E_n - E_m = 13.6\left( {\frac{1}{{m^2 }} - \frac{1}{{n^2 }}} \right)}
  • and
    \color{red}{ E_\gamma = hf = \frac{{hc}}{\lambda }}
  • so
    \color{red}{ \frac{1}{\lambda } = \frac{{13.6}}{{hc}}\left( {\frac{1}{{n^2 }} - \frac{1}{{m^2 }}} \right)}
  • (units are somewhat confused: need h = 4.1x10^-15 eV)

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  n = 1 Lyman series n = 2 Balmer Series n = 3 Paschen Series

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  Are these energy levels so weird?

  • e.g. energy levels for block of wood. "Transitions" from one to another ⇒ fixed amount of energy as sound.
  • That's why only get certain energies as photons in atoms

  • The Old Picture

  • Electrons in orbit round nucleus (i.e. the Bohr model) lets us explain spectrum of H and ionized He⁺ but not other atoms
  • Problems:
  • • Does not explain He atom
    • Does not explain detailed spectrum
    • Does not explain periodic Table

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  So after Planck, Einstein, Bohr, we understand quite a lot
  • Spectra Problems
    • What gives the formulae?
    • Why do atoms only emit certain wavelengths?
    • How are absorption and emission related?
  • P-E effect Problems
    • The energy of electrons does NOT depend on intensity of light,
    • DOES depend on frequency.
    • No electrons emitted below critical frequency f₀, which depends on metal
  • Thomson-Millikan Problems:
    • Why is the electron so much lighter than an atom?
    • What is the positive charge that must be in the atom?
  • Radioactivity Problem:
    • Where does the radioactivity come from?
  • Black Body Problems:
    • Where does the B-B curve come from?
    • Why does it depend on temperature?

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  • X-ray Problems:
    • What are X-rays?
    • Where does Moseley's formula come from?
  • Rutherford's Problems:
    • Why don't the electrons fall into nucleus?
    • Why are all atoms the same?
  • Planck-Einstein Problem:
    • How can something be a wave and a particle at the same time?
  • Bohr-atom Problems:
    • How does the He atom work?
    • Where does the Periodic Table come from?
  • Now for the real explanation......