CHAPTER 6 Structure of the Atom The Atomic Models of Thomson and Rutherford Rutherford Scattering The Classic Atomic Model The Bohr Model of the Hydrogen Atom Successes & Failures of the Bohr Model Niels Bohr (1885-1962) Bohr picture: web.gc.cuny.edu/ ashp/nml/copenhagen/ Homework due Wednesday Oct. 15th Chapter 5: 12,15, 22, 23, 28 Chapter 6: 2,15, 20, 24, 28

The Bohr Model of the Hydrogen Atom Bohr’s general assumptions: 1. Electrons reside in stationary states, and do not radiate energy. They have well-defined energies, En. Transitions can occur between them, yielding light of energy: E = En − En’ = hn n = 1 n = 3 n = 2 En > En’ : Emission En < En’ : Absorption 2. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere. Angular momentum is quantized! 3. The angular momentum of the nth state is: nħ where n is called the Principal Quantum Number.

Consequences of the Bohr Model The angular momentum is: a0 So the velocity is: But: So: Solving for rn: where: a0 is called the Bohr radius. It’s ½ the diameter of the Hydrogen atom (in its lowest-energy, or ground, state).

Bohr Radius The Bohr radius, is the radius of the ground state of the Hydrogen atom: The ground state of the Hydrogen atom has a diameter:

The Hydrogen Atom Energies Use the classical result for the energy: and: So the energies of the stationary states are: or: En = - E0/n2 where E0 = 13.6 eV.

The Hydrogen Atom Emission of light occurs when the atom is in an excited state and decays to a lower energy state (nu → nℓ). where n is the frequency of a photon. R∞ is the Rydberg constant.

Transitions in the Hydrogen Atom The atom will remain in the excited state for a short time before emitting a photon of energy hn and returning to a lower stationary state. In equilibrium, all hydrogen atoms exist in the ground state, n = 1. Also, hydrogen in the ground state (n = 1) can absorb a photon of energy hn and make a transition to an excited state.

The Correspondence Principle Bohr’s correspondence principle is rather obvious: In the limit where classical and quantum theories should agree, the quantum theory must reduce the classical result.

The Correspondence Principle The frequency of the radiation emitted nclassical is equal to the orbital frequency norb of the electron around the nucleus. This should agree with the frequency of the transition from n + 1 to n (when n is very large): En = hnn = - E0 /n2 For large n: Substituting for E0: where:

Fine Structure Constant The electron’s velocity in the Bohr model: In the ground state, v1 = 2.2 × 106 m/s ~ 1% of the speed of light. The ratio of v1 to c is called the fine structure constant.

Successes and Failures of the Bohr Model The electron and hydrogen nucleus actually revolve about their mutual center of mass. The electron mass is replaced by its reduced mass: The Rydberg constant for infinite nuclear mass, R∞, is replaced by R. This modification improved the theory’s accuracy!

Limitations of the Bohr Model The Bohr model was a great step in the new quantum theory, but it had its limitations. Failures: Works only for single-electron (“hydrogenic”) atoms. Could not account for the intensities or the fine structure of the spectral lines (for example, in magnetic fields). Could not explain the binding of atoms into molecules.