Five Chapter 8

THE BOHR MODEL OF THE ATOM

• The observed spectra for the various elements cannot be explained by the Rutherford model of the atom
• This model has a central nucleus containing most of the mass, and is positively charged. The electrons were placed around the nucleus.
• An explanation was needed as to why the electrons didn't collide with the nucleus. The electrons were imagined to be in orbits like the planets around a star.
• The Rutherford model of the atom is a planetary model. In this model, electrons orbit the nucleus which has a positive charge. The centripetal force for the orbit is caused by Coulomb forces.
• Maxwell's theories of electrodynamics state that accelerating charges cause the emission of electromagnetic waves.
• Electrons in orbit are continually accelerating (centripetally) and therefore would radiate electromagnetic waves with energy. This means that the energy of the electron would decrease, and it would spiral in towards the nucleus. As it spirals, the radiation emitted would gradually increase in frequency.
• Maxwell's theory predicts an outcome for the Rutherford model of the atom, which does not match observations.
• Experiment showed the existence of discrete energy levels in the atom. Atoms could only absorb or emit discrete amounts of energy. These observations are contrary to Maxwell's predictions, and the Rutherford model of the atom.
• Niels Bohr tried to solve these problems by proposing a quantum approach to the motion of electrons within the atom.
• Bohr assumed that classical mechanics and electrodynamics did not apply at the subatomic level.
• He made three postulates regarding the motion of electrons within atoms.
  1. Only a very special few orbits are "allowed": each allowed orbit is characterized by a different specific amount of electron energy.
  2. In an allowed orbit, an electron is exempt from the classical laws of electromagnetism and does not radiate energy.
  3. Electrons may "jump" from energy orbit to energy orbit with the energy difference between these two states being given off or absorbed in the form of a single photon of electromagnetic radiation.
• STATIONARY STATE - Each of the allowed orbits of the electron.
• Essentially, Bohr states that only certain orbits are possible, electrons could move between these orbits, and when they do so, they absorb or emit energy in the form of a photon whose energy equals the energy difference between stationary states.
• There remains a question. What makes the allowed orbits possible but not other orbits?

Analysing Atomic Spectra

• With the Bohr model of the atom, we can now explain the spectra of the elements.
• Normally, in nature, everything tends to lowest possible energy.
• Ground State - The state of an atom which is not "excited."
  • Electrons are in a stationary state with the lowest possible energies.
• First excitation state - The state of an atom which has absorbed the smallest amount of energy it can absorb.
  • Electrons are no longer in the stationary state with the lowest possible energy.
• When atoms absorb energy in collisions with electrons, or by other processes, they quickly emit this excess energy in the form of light.
• If we measure the wavelength for the emitted light, the energy of the photon can be calculated.

E = hf

C = f l

Therefore

E_p = (hc)/l

• If an atom absorbs 4.89 eV of energy to lift it to its first excitation state, and then emits light with a wavelength of 2.54 X 10^-7 m, the energy of the emitted photon is given by

E_p = (6.63 X 10^-34J)(3.00 X 10⁸m/s)/2.54 X 10^-7m

or

E_p = 7.83 X 10^-19 J = 4.89 eV

• The energy of the photon tells us that atoms which are raised to their first excitation level, return to their ground state by emitting a photon whose energy is equal to the difference between the energy of the first excited state and the ground state.
• If the incident electrons have energy greater than the second excitation state, then the photons emitted correspond to transitions between
  • second excited state to ground
  • second excited state to first excited state.
  • first excited state to ground
• In all cases, the photons emitted correspond to all possible transitions between the excited states with less than the energy of the incident electrons.
• In general, for the emission or absorption of photons we can say:

E_p = E_f - E_i

• E_p = energy of the emitted or absorbed photon, E_i = energy of the initial energy level, and E_f = the energy of the final energy level. If E_p < 0, then the photon was emitted from the atom, and if E_p > 0, then the photon was absorbed by the atom.
• To calculate the wavelength or frequency of the emitted or absorbed photon substitute the energy of the photon into

E_p = (hc)/l = hf

• To produce an emmision spectrum, (spectrum with bright lines in it), energy must be added to the atoms. This can be accomplished in various ways. One of the more common ways is to pass an electrical current through a gas.
• An absorption spectrum can be produced by passing white light through a cool gas to produce dark lines in the resulting spectrum.
• In this process, atoms absorb energy from the white light. This removes the photons from the white light with energies corresponding to the possible transitions between ground and excited states in the atoms
• In the absorption spectrum of sodium, dark gaps appear at wavelengths of 259.4 nm, 254.4 nm, and 251.2 nm. For the 259.4 nm line, the energy of the absorbed photon is given by:

E_p = (hc)/l

E_p = (6.6c X10^-34)(3.00 X 10⁸ m/s)/2.59 X 10^-7m

E_p = 7.68 X 10^-19 J = 4.80 eV

• Thus, there are two energy levels in sodium that are 4.80 eV apart. For the other two lines, the energy differences are 4.89 eV and 4.95 eV respectively.
• Energies for photons of visible light range from 1.6 eV to 3 eV.
• These energies are not sufficient to cause excitation to the first excited state for common gases. Therefore the gases do not absorb energy from the visible light and do not give off energy when irradiated by visible light.
• This explains why gases are invisible in appearance to us.
• Emission spectra have more lines than absorption spectra, because atoms are usually in the ground state, and transitions are then from ground state to first, ground to second, etc. The atoms do not remain in excited states long enough for transitions from second to third, second to fourth etc.
• Atoms can receive energy in two ways:
  • collisions with high speed particles - the atom absorbs the energy from the particle that corresponds to the differences in energy between excited states. The particle retains any left over energy.
  • By absorbing a photon - the photon is only absorbed if the energy it has is exactly equal to the energy difference between excited states of the atom.

Energy Levels of Hydrogen

• In the middle of the 19^th century, hydrogen was known to be the simplest atom.
• To Balmer, the hydrogen spectrum in the visible region showed regularity. The equation which describes this regularity is:

• Further investigation of hydrogen's spectrum, showed that the infra red and ultraviolet regions of the electromagnetic spectrum showed similar regularity.
• The equation describing this regularity is a generalization of Balmers equation.

• In this formula, l = wavelength, n₁ = the number of the initial energy level, n₂ = the number of the final energy level, and R = Rydberg constant = 1.097 X 10⁷/m. If l < 0, then a photon is emitted, and if > 0 then a photon is absorbed.
• If we consider absorption of a photon by an atom, we have

• In gravitational systems of the first course we chose our reference level to be infinitely distant from the object producing the gravitational field. If we stay consistant with this for the atom, then as the absorbed photons become more energetic, the final energy level of the excited electron must have an energy approaching zero. That is to say, E₂ would approach zero. This would be true regardless of the energy level (E₁) that the internal electron started on. Therefore we can replace E₁ with E_n as long as E_n < E₂.
• Under these considerations we can continue our calculations by looking at what happens in the limit that E₂ approaches zero (i.e. the atom is ionized and n₂ approaches infinity).

• This last equation now gives us the energy of the n^th energy level of the hydrogen atom.
• Substituting known values into this equation gives.

E_n = ((-1.097 X 10⁷ m^-1)(6.63 X 10^-34Js)(3.00 X 10⁸m/s))/n²

or

• This last equation allows us to calculate all the energy levels of the hydrogen atom.

Level Value of n Energy Energy above Ground

ground state 1 -13.6 eV 0.0 eV

1st excited state 2 -3.4 eV 10.2 eV

2nd excited state 3 -1.51 eV 12.1 eV

3rd excited state 4 -0.85 eV 12.8 eV

4th excited state 5 -0.54 eV 13.1 eV

• The information above, allows us to construct an energy level diagram for hydrogen

January 20, 2014