Chap16 solutions

Physical Chemistry

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16 Spectroscopy 1: rotational and vibrational spectroscopy Solutions to exercises Discussion questions E16.1(b) (1) Doppler broadening . This contribution to the linewidth is due to the Doppler effect which shifts the frequency of the radiation emitted or absorbed when the atoms or molecules involved are moving towards or away from the detecting device. Molecules have a wide range of speeds in all directions in a gas and the detected spectral line is the absorption or emission profile arising from all the resulting Doppler shifts. As shown in Justification 16.3, the profile reflects the distribution of molecular velocities parallel to the line of sight which is a bell-shaped Gaussian curve. (2) Lifetime broadening . The Doppler broadening is significant in gas phase samples, but lifetime broadening occurs in all states of matter. This kind of broadening is a quantum mechanical effect related to the uncertainty principle in the form of eqn 16.25 and is due to the finite lifetimes of the states involved in the transition. When τ is finite, the energy of the states is smeared out and hence the transition frequency is broadened as shown in eqn 16.26. (3) Pressure broadening or collisional broadening . The actual mechanism affecting the lifetime of energy states depends on various processes one of which is collisional deactivation and another is spontaneous emission. The first of these contributions can be reduced by lowering the pressure, the second cannot be changed and results in a natural linewidth. E16.2(b) (1) Rotational Raman spectroscopy . The gross selection rule is that the molecule must be anisotrop- ically polarizable, which is to say that its polarizability, α , depends upon the direction of the electric field relative to the molecule. Non-spherical rotors satisfy this condition. Therefore, linear and symmetric rotors are rotationally Raman active. (2) Vibrational Raman spectroscopy . The gross selection rule is that the polarizability of the molecule must change as the molecule vibrates. All diatomic molecules satisfy this condition as the molecules swell and contract during a vibration, the control of the nuclei over the electrons varies, and the molecular polarizability changes. Hence both homonuclear and heteronuclear diatomics are vibrationally Raman active. In polyatomic molecules it is usually quite difficult to judge by inspection whether or not the molecule is anisotropically polarizable; hence group theoretical methods are relied on for judging the Raman activity of the various normal modes of vibration. The procedure is discussed in Section 16.17(b) and demonstrated in Illustration 16.7. E16.3(b) The exclusion rule applies to the benzene molecule because it has a center of symmetry. Consequently, none of the normal modes of vibration of benzene can be both infrared and Raman active. If we wish to characterize all the normal modes we must obtain both kinds of spectra. See the solutions to Exercises 16.29(a) and 16.29(b) for specified illustrations of which modes are IR active and which are Raman active.
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