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 Structure of the Atom CHAPTER 6 Structure of the Atom Homework due Friday Oct. 16 th Chapter 6: 2,15, 20, 24, 28 Niels Bohr ( )

Experimental results were not consistent with Thomson’s atomic model. Rutherford proposed that an atom has a small positively charged core (nucleus) surrounded by the negative electrons. Geiger and Marsden confirmed the idea in Rutherford’s Atomic Model Ernest Rutherford ( ) Experimental observation of many large angle scattering events!

The Classical Atomic Model Consider an atom as a planetary system. Like gravity, the force on the electron an inverse-square-law force. This is good. Now, by Newton’s 2 nd Law: where v is the tangential velocity of the electron: The total energy is then: This is negative, so the system is bound, which is good. The potential energy is: So: Kinetic energy

The planetary atom will emit light of a particular frequency. An electron in an orbit will emit a light wave at the orbital frequency. Like planets in the solar system, the electron’s orbital frequency will vary according to the electron’s distance from the nucleus. So this model could, in principle, explain atoms’ discrete spectra. But all frequencies seem possible…

The Planetary Model is Doomed! Because an accelerated electric charge continually radiates energy (electromagnetic radiation), the total energy must continually decrease. So the electron radius must continually decrease! Physics had reached a turning point in 1900 with Planck’s hypothesis of the quantum behavior of radiation, so a radical solution would be considered possible. The electron crashes into the nucleus!

2. Classical laws of physics do not apply to transitions between stationary states, but they do apply elsewhere. 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, E n. Transitions can occur between them, yielding light of energy: E = E n − E n ’ = h 3. The angular momentum of the n th state is: nħ where n is called the Principal Quantum Number. n = 1 n = 3 n = 2 Angular momentum is quantized! E n > E n ’ : Emission E n < E n ’ : Absorption

Consequences of the Bohr Model The angular momentum is: But:So: Solving for r n : So the velocity is: where: a 0 is called the Bohr radius. It’s ½ the diameter of the Hydrogen atom (in its lowest-energy, or ground, state). a0a0

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

The Hydrogen Atom Energies So the energies of the stationary states are: where E 0 = 13.6 eV. Use the classical result for the energy: E n =  E 0 /n 2 or: and:

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

Transitions in the Hydrogen Atom The atom will remain in the excited state for a short time before emitting a photon of energy h 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 h and make a transition to an excited state.

Shells have letter names: K shell for n = 1 L shell for n = 2 The atom is most stable in its ground state. When it occurs in a heavy atom, the radiation emitted is an x-ray. It has the energy E (x-ray) = E u − E ℓ. Characteristic X-Ray Spectra and Atomic Number An electron from higher shells will fill the inner-shell vacancy at lower energy.

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

For large n : Substituting for E 0 : The Correspondence Principle The frequency of the radiation emitted classical is equal to the orbital frequency orb 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): E n = h n  =  E 0 /n 2 where:

Fine Structure Constant The electron’s velocity in the Bohr model: In the ground state, v 1 = 2.2 × 10 6 m/s ~ 1% of the speed of light. The ratio of v 1 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: Success: This modification improved the theory’s accuracy! The Rydberg constant for infinite nuclear mass, R ∞, is replaced by R.

Limitations of the Bohr Model 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. Failures: The Bohr model was a great step in the new quantum theory, but it had its limitations.