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 = mr2 • = reduced mass = m1m2 / (m1 + m2) r = internuclear distance m2 m1 r
Rigid Rotor Model In wavenumbers (cm-1): Separation between adjacent levels: F(J) – F(J-1) = 2BJ
Rotational Energy Levels Selection Rules: Molecule must have apermanent dipole. DJ = 1 J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.
Rotational Spectra J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.
Intensity of Transitions %T cm-1 J. Michael Hollas, Modern Spectroscopy, John Wiley & Sons, New York, 1992.
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 108 m/s Jmax≈ [(1.38 x 10-23 J/K*298 K)/(2*6.626 x 10-34 Js*3.00 x 1010 cm/s*1.91 cm-1)]1/2 -1/2 Jmax = 7 at room temperature Jmax ≈ [(1.38 x 10-23 J/K*573 K)/(2*6.626 x 10-34 Js*3.00 x 1010 cm/s*1.91 cm-1)]1/2 -1/2 Jmax = 10 at 300 °C
The Non-Rigid Rotor Account for the dynamic nature of the chemical bond: DJ = 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
More Complicated Molecules Still must have a permanent dipole DJ = 0, 1 K is a second rotational quantum number accounting for rotation around a secondary axis A.
Practical Issues Small DE 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 (1012 – 1013 s-1) than the period of rotation (10-10 s). Result:Rotational spectroscopy is only used for analytical purposes when studying low pressure gases.