bohr model

• Photoelectric effect


  • Quantisation of the energy of light explained it
• Compton effect


  • Apparent momentum of photons explained it
• Rutherford's experiments


  • Changed understanding of the structure of the atom
• You will learn about the Bohr approach to explaining absorption and emission spectra
• Energy states


  Radiation is absorbed or emitted when an electron changes energy state
• Rutherford's model assumed that electrons orbit the nucleus and the Coulomb force between electron and nucleus provides centripetal force. This produces several problems

  
• Classically an accelerating charge produces radiation, which is an energy loss. This causes the electron to orbit closer to nucleus. From conservation of angular momentum the electron will speed up

  
• As the electron's speed changes the emitted radiation will change colour leading to a continuous spectrum. Other problems too
• In fact Balmer saw discrete spectral lines
• Bohr's assumptions about the hydrogen atom


  • Electrons move only in certain circular orbits, called STATIONARY STATES
  • Radiation is only emitted when an electron moves from one allowed state to another of lower energy
  • The angular momentum of the electron is restricted to integer multiples of ħ

  
• Bohr's assumptions about the hydrogen atom

  1. Analyse the motion from a classical viewpoint

  2. Use similar analysis to planetary orbits

  
• For atomic hydrogen the Bohr model predicts that the energy of an orbit is quantised

  
• Photon emission and absorption in the Bohr model

  1. Photon emitted when electron transitions from high to low energy level

  2. Photon absorbed to make electron transition from low to high energy level

  
• Transitions described by 1/λ = R(1/nlow^2 - 1/nhigh^2) result in atomic spectral series

  
• Hydrogen-like atoms


  • Atom with charge Z and one electron

  Radius of orbit r = a0Z/n^2

  Potential energy Un = -Z^2e^2/4πε0a0n^2

  Kinetic energy increases by Z^2

  Energy of nth orbit En = -mZ^2e^4/8h^2ε0^2n^2

  
• Electron orbits closer to the nucleus


  Potential energy increases

  
• Kinetic energy


  Increases by a factor of Z^2
• Change in energy ΔE

  Ehigh - Elow
• R = me^4/8h^3ε0^2c is the Rydberg constant 1.097 x 10^7 m^-1
• Experimental observations support the idea that electromagnetic disturbances can be treated as waves
• Explanation of photoelectric effect and Compton scattering relied on the idea that electromagnetic waves can be particle-like
• Light can be thought of as both a wave and a particle, depending on the experimental system
• Probability of finding a photon per unit volume

  Proportional to N/V, where N is the number of photons and V is the volume

  
• Probability per unit volume

  Proportional to intensity I, which depends on the square of the electric field E^2
• De Broglie wavelength


  λ = h/p = h/mu
• Matter waves


  Probability waves
• Wave


  A disturbance from a normal or equilibrium condition that propagates without the transport of matter
• Wave function


  Depends on the position of the particle at a given time, can be separated into spatial and temporal parts
• Probability density is given by |ψ(r)|^2

  
• Probability of finding particle between a and b


  ∫_a^b |ψ(x)|^2dx
• The wave function must be normalized such that ∫_-∞^∞ |ψ(x)|^2dx = 1
• Wave-particle duality of matter


  The concept that all particles, including electrons and photons, exhibit both wave-like and particle-like behavior
• Compton scattering formula


  A formula used to calculate the change in wavelength of a photon due to its collision with an electron: λ' - λ = (h/m\_e \* c) \* (1 - cos(θ))
• Compton shift


  The change in wavelength of a photon due to its collision with an electron
• Compton effect


  A phenomenon in physics that occurs when a photon collides with an electron, resulting in a decrease in energy and frequency of the photon and a recoil of the electron