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Quantum Theory 2

Quantum Theory 2. Sections 4-1,2. Topics. Atomic Spectra The Rutherford Model Summary. Extra Credit Problem. Using Planck’s model, derive Planck’s formula for the entropy per oscillator Then use 1/T = ∂ S/ ∂ E to show that the average energy per oscillator is given by

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Quantum Theory 2

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  1. Quantum Theory 2 Sections 4-1,2

  2. Topics • Atomic Spectra • The Rutherford Model • Summary

  3. Extra Credit Problem • Using Planck’s model, derive Planck’s formula for the entropy per oscillator • Then use 1/T = ∂S/∂E to show that the average energy per oscillator is given by E = e /(ee/kT - 1) Hint: for large N, ln N! ~ N ln N – N and E = e M / N due Feb. 25

  4. Extra Credit Problem Planck’s Model The number of ways of distributing M indistinguishable quanta amongst N oscillators is

  5. Self Study • Please study • Photoelectric Effect • Compton Effect

  6. Atomic Spectra The Balmer Series of Hydrogen (1885) Johann Balmer’s empirical formula 10 Ångström = 1 nano-meter

  7. Atomic Spectra The Balmer Series of Hydrogen (1885) Rydberg-Ritz formula R is the Rydberg constant

  8. The Thomson Model J.J. Thomson proposed that atoms are spheres of positive charge with embedded electrons This, however, was not a stable configuration. The charges most move and, indeed, accelerate. But then they would radiate away energy!

  9. The Rutherford Model Rutherford showed that the radiation from uranium consisted of two types: a and b Hans Geiger and his advisor Ernest Rutherford University of Manchester, England

  10. The Rutherford Model Gold foil (2000 atoms thick) Microscope Radioactive source 214Bi Scintillation screen H. Geiger and E. Marsden, Philosophical Review, 25, 507 (1913)

  11. The Rutherford Model It was quite the most incredible event that ever happened to me in my life. It was as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you Ernest Rutherford

  12. The Rutherford Model b is the impact parameter q is the scattering angle Note: The smaller the impact parameter, the greater the scattering angle

  13. The Rutherford Model For a beam of particles of area A, incident on a foil of thickness t, let’s try to calculate the fraction f of the beam particles that are scattered by more than the angle q.

  14. b Area pb2 The Rutherford Model Let I0 be the intensity of the beam of particles, that is, the number of particles per second that cross unit area Then the number N of particles scattered by more than the angle q is The quantity s = N / I0 = p b2 is called the cross section

  15. The Rutherford Model Number of nuclei per unit volume in the foil is NA is Avogadro’s number and M the molar mass. The number in volume V = At is n A t

  16. The Rutherford Model Number of particles scattered per second is the number of particles scattered per nucleus x number of nuclei The total number of incident particles is I0 A, so

  17. The Rutherford Model Rutherford derived a formula for the number DN of particles scattered at any given angle, which agreed with the data obtained by his students

  18. Self Study • Please study • EXAMPLE 4-2

  19. The Rutherford Model The size of the gold nucleus can be estimated by considering a head-on collision between it and an a particle from 214Bi in which all the 7.7 MeV kinetic energy of the a is converted into electric potential energy:

  20. Summary • Atomic Spectra • Empirical formulae were found, such as that by Rydberg and Ritz, that described the spectra of simple atoms • Discovery of the nucleus • By observing and successfully explaining the large-angle scatter of a particles from gold, Rutherford and his students showed that the mass of an atom resides in a very tiny nucleus

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