1. Rotational Spectra Simplest Case: Diatomic or Linear Polyatomic molecule Rigid Rotor Model: Two nuclei joined by a weightless rod J = Rotational quantum number (J = 0, 1, 2, …) I = Moment of inertia =  r 2  = reduced mass = m 1 m 2 / (m 1 + m 2 ) r = internuclear distance m1m1 m2m2 r

2. Rigid Rotor Model In wavenumbers (cm -1 ): Separation between adjacent levels: F(J) – F(J-1) = 2BJ

3. Rotational Energy Levels Selection Rules: Molecule must have a permanent dipole.  J =  1 J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.

4. Rotational Spectra J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992. J” → J’F(J’)-F(J”) 3 → 42(1.91)(4)15.3 cm -1 4 → 52(1.91)(5)19.1 cm -1 5 → 62(1.91)(6)22.9 cm -1 6 → 72(1.91)(7)26.7 cm -1 7 → 82(1.91)(8)30.6 cm -1 8 → 92(1.91)(9)34.4 cm -1 9 → 102(1.91)(10)38.2 cm -1

5. Intensity of Transitions J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992. %T cm -1

6. Are you getting the concept? Calculate the most intense line in the CO rotational spectrum at room temperature and at 300 ° C. The rigid rotor rotational constant is 1.91 cm -1. Recall: k = 1.38 x 10 -23 J/K h = 6.626 x 10 -34 Js c = 3.00 x 10 8 m/s J max ≈ [(1.38 x 10 -23 J/K*298 K)/(2*6.626 x 10 -34 Js*3.00 x 10 10 cm/s*1.91 cm -1 )] 1/2 -1/2 J max = 7 at room temperature J max ≈ [(1.38 x 10 -23 J/K*573 K)/(2*6.626 x 10 -34 Js*3.00 x 10 10 cm/s*1.91 cm -1 )] 1/2 -1/2 J max = 10 at 300 ° C

7. The Non-Rigid Rotor Account for the dynamic nature of the chemical bond:  J = 0,  1 D is the centrifugal distortion constant (D is large when a bond is easily stretched) Typically, D < 10 -4 *B and B = 0.1 – 10 cm -1

8. More Complicated Molecules Still must have a permanent dipole  J = 0,  1 K is a second rotational quantum number accounting for rotation around a secondary axis A.

9. Practical Issues Small  E of rotational transitions make lines difficult to resolve. Collisional broadening blurs spectra unless in the gas phase at low pressure. In the solution phase collisions occur more frequently (10 12 – 10 13 s -1 ) than the period of rotation (10 -10 s). Result: Rotational spectroscopy is only used for analytical purposes when studying low pressure gases.