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Canonical Ensemble Lecture: Rotational and Vibrational Energy Levels in Molecules

Explore rotational and vibrational energy levels in diatomic molecules, equipartition theorem, and classical uncertainty. Homework due next Friday, assigned problems listed. Specific heat for H2, classical uncertainty scenarios, and examples of degrees of freedom.

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Canonical Ensemble Lecture: Rotational and Vibrational Energy Levels in Molecules

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  1. Lecture 18 — The Canonical Ensemble Chapter 6, Wednesday February 20th • Rotational energy levels in diatomic molecules • Vibrational energy levels in diatomic molecules • More on the equipartition theorem Reading: All of chapter 5 (pages 91 - 123) Homework 5 due next Friday (22nd) Homework assignments available on web page Assigned problems, Ch. 5: 8, 14, 16, 18, 22

  2. Rotational energy levels for diatomic molecules l = 0, 1, 2... is angular momentum quantum number I = moment of inertia CO2 I2 HI HCl H2 qR(K) 0.56 0.053 9.4 15.3 88

  3. Vibrational energy levels for diatomic molecules n = 0, 1, 2... (harmonic quantum number) w w = natural frequency of vibration I2 F2 HCl H2 qV(K) 309 1280 4300 6330

  4. Specific heat at constant pressure for H2 CP = CV + nR H2 boils w CP (J.mol-1.K-1) Translation

  5. More on the equipartition theorem Classical uncertainty: Where is the particle? V(x) V = ∞ V = ∞ V = 0 W = 9 x x = L

  6. More on the equipartition theorem Classical uncertainty: Where is the particle? V(x) V = ∞ V = ∞ V = 0 W = 18 x x = L

  7. More on the equipartition theorem Classical uncertainty: Where is the particle? V(x) V = ∞ V = ∞ V = 0 W = 36 x x = L

  8. More on the equipartition theorem Classical uncertainty: Where is the particle? V(x) V = ∞ V = ∞ V = 0 W = ∞ S = ∞ x x = L

  9. More on the equipartition theorem: phase space Area h Cell: (x,px) dpx px dx x

  10. More on the equipartition theorem: phase space Area h Cell: (x,px) dpx px dx x

  11. More on the equipartition theorem: phase space Area h Cell: (x,px) dpx px dx x

  12. More on the equipartition theorem: phase space In 3D: Uncertainty relation: dxdpx = h dpx dx

  13. Examples of degrees of freedom:

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