Anharmonic Oscillator Thermal

v" = vibrational quantum number

Interatomic distance, r v" = vibrational quantum number

Interatomic distance, r

Energy Vibrational Molecule Anharmonic

Internuclear distance, r

FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function.

Internuclear distance, r

FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function.

combination bands (i.e., those that are the sum of two or more fundamental vibrations) occur.

Because the vibrational energy level spacing is relatively large (typically of the order of lO3 cm-1) compared to their thermal energy, most molecules at room temperature are in their lowest vibrational energy level and light absorption normally occurs from v = 0.

For a purely vibrational transition, the selection rule for absorption of light requires that there be a changing dipole moment during the vibration. This oscillating dipole moment produces an electric field that can interact with the oscillating electric and magnetic fields of the electromagnetic radiation. Thus heteronuclear diatomic molecules such as NO, HC1, and CO absorb infrared radiation and undergo vibrational transitions, whereas homonuclear diatomic molecules such as 09

and H2, whose dipole moments remain constant during vibration, do not.

Since at room temperature most molecules are in the v" =0 state, and At' = ±1 is by far the strongest transition, most molecules go from v" = 0 to v' = 1. (We follow the Herzberg convention that the quantum number of the upper state is designated by a prime, and the lower state by a double prime; e.g., in this example, v" = 0 -> v' = 1.) As a result, a single vibrational absorption band (with associated rotational structure) is normally observed in the infrared, the region corresponding to the energy level differences given by Eq. (C).

(2) Rotational energy and transitions If a molecule has a permanent dipole moment, its rotation in space produces an oscillating electric field; this can also interact with electromagnetic radiation, resulting in light absorption.

In the idealized case for rotation of a diatomic molecule, one assumes the molecule is analogous to a dumbbell with the atoms held at a fixed distance r from each other; that is, it is a rigid rotor. The simultaneous vibration of the molecule is ignored, as is the increase of internuclear distance at high rotational energies arising from the centrifugal force on the two atoms.

For this idealized case, the rotational energy levels (in cm~1) are given by

where B, the rotational constant characteristic of the molecule, is given by

I is the moment of inertia of the molecule, given by I = /lir2, where ¡x is the reduced mass defined by ti~l = [(M^1) + (M^1)], Ma and Mu are the atomic masses, and r is the fixed, internuclear distance. J is the rotational quantum number; its allowed values are 0, 1, 2,... .

For a real rotating diatomic molecule, known as a nonrigid rotor, Eq. (D) becomes

The constant D is characteristic of the diatomic molecule and is much smaller than B; generally, D ~ 10 AB. The second term in Eq. (F) generally becomes important at large values of / when centrifugal force increases the separation between atoms.

Because of the requirement of a permanent dipole moment, only heteronuclear molecules can absorb radiation and change their rotational energy. For the idealized case of a rigid rotor, the selection rule is A J = ±1. For the energy levels given by Eq. (D), the energy level splitting between consecutive rotational energy levels is given by

where /' is the quantum number of the upper rotational state involved in the transition. Thus the spacing between rotational energy levels increases with increasing rotational quantum number. Splittings are small compared to those between vibrational energy levels, typically of the order of 10 cm~' in the lower levels; this corresponds to absorption in the microwave region. Indeed, these spacings are sufficiently small that the population of the rotational energy levels above J = 0 is significant at room temperature because the thermal energy available is sufficient to populate the higher rotational levels.

The Boltzmann expression can be used to calculate the relative populations of molecules in any rotational state / compared to the lowest rotational state J = 0 at temperature T (K):

In Eq. (H), k is the Boltzmann constant (1.381 X 10~23 J K"1 ) and E} is the energy of the Jth rotational level given by Eq. (D) for the ideal rigid rotor, or Eq. (F) for the nonideal case.

The exponential energy factor in Eq. (H) gives decreasing populations with increasing J, but the degeneracy factor (2/ + 1) works in the opposite direction. As a result, rotational populations increase initially with increasing /, reach a peak, and subsequently decrease.

The combination of increased spacing between energy levels as J increases and the significant population of molecules in higher rotational energy levels means that the absorption of microwave radiation occurs from a number of different initial states, resulting in a series of absorption lines, rather than a single line as seen in the pure vibrational infrared spectra of diatomic molecules. From the spacing of these lines, the rotational constant B, and hence the moment of inertia and the internuclear spacing, can be obtained using Eq. (G).

(3) Vibration-rotation Molecules, of course, vibrate and rotate simultaneously. It is a good approximation that the total energy of the molecule (excluding trans lation) is the sum of the vibrational (V), rotational

(R), and electronic (E) energy of a molecule; that is,

£tolai = Ev + Er + Ee. For the case where there is no

ci I I I I I I I I I

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