Degrees of freedom, vibrational and rotational

Both heavy (atoms, ions, molecules) and light (electrons, photons) particles can be involved in collisions. Polyatomic molecules have internal degrees of freedom (vibrational and rotational motion of atoms) and, in this sense, they have an internal structure. 

Different kinds of molecules have different degrees of translational, vibrational, and rotational freedom and, hence, different average degrees of molecular disorder or randomness. Now, if for a chemical reaction the degree of molecular disorder is different for the products than for the reactants, there will be a change in entropy and AS0 A 0. 

Also, rotational state resolution of cross-sections can be obtained by employing a coherent state analysis [51] for the situation of weak coupling between rotational and vibrational degrees of freedom. A suitable rotational coherent state can be expressed as... 

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

Table 3.5. The number of degrees of freedom in translation, rotation and vibrations of the reacting molecules and the transition state in the gas phase reaction of CO and O2 and the temperature dependence these modes contribute to the partition function. Note that one of the modes of the transition state complex is the reaction coordinate, so that only six vibrational modes are listed.

Figure 1. Translation, rotation, and vibration of a diatomic molecule. Every molecule has three translational degrees of freedom corresponding to motion of the center of mass of the molecule in the three Cartesian directions (left side). Diatomic and linear molecules also have two rotational degrees of freedom, about rotational axes perpendicular to the bond (center). Non-linear molecules have three rotational degrees of freedom. Vibrations involve no net momentum or angular momentum, instead corresponding to distortions of the internal structure of the molecule (right side). Diatomic molecules have one vibration, polyatomic linear molecules have 3V-5 vibrations, and nonlinear molecules have 3V-6 vibrations. Equilibrium stable isotope fractionations are driven mainly by the effects of isotopic substitution on vibrational frequencies.

In the calculation of the thermodynamic properties of the ideal gas, the approximation is made that the energies can be separated into independent contributions from the various degrees of freedom. Translational and electronic energy levels are present in the ideal monatomic gas.ww For the molecular gas, rotational and vibrational energy levels are added. For some molecules, internal rotational energy levels are also present. The equations that relate these energy levels to the mass, moments of inertia, and vibrational frequencies are summarized in Appendix 6. 

For excess energy distributed in the other degrees of freedom, i.e., rotation, translation, and vibration, only collisional degradation is possible. However, it is known that degradation of vibrational energy is a slow process and so a considerable fraction of vibrationally excited species may be expected to persist for some time. On the other hand, rotational and translational energy appear to be exchanged very readily in collisions, so that species thus excited may be expected to be thermalized very rapidly. 

The remarks in the previous paragraph apply, of course, only to the case of electronically adiabatic molecular collisions for which all degrees of freedom refer to the motion of nuclei (i.e. translation, rotation and vibration) if transitions between different electronic states are also involved, then there is no way to avoid dealing with an explicit mixture of a quantum description of some degrees of freedom (electronic) and a classical description of the others.9 The description of such non-adiabatic electronic transitions within the framework of classical S-matrix theory has been discussed at length in the earlier review9 and is not included here. 

If such a monomolecular mechanism is accepted a possible explanation for the low pre-exponential factors observed is, that in the initial state (the formate ion) there are internal degrees of freedom—e.g., rotation with the axis perpendicular to the surface, and bending vibrations— which are lost in the transition to the activated state. This can be schematically represented as follows ... 

Next, let us consider the eigenstates of molecular motions in the Schrodinger equation. The molecular motions are classified into four types the translational, rotational, and vibrational motions of atomic nuclei (Fig. 1.1) and the motions of electrons. The translational motions are the uniform motions of all nuclei with three degrees of freedom (DOFs), the rotational motions are those with respect to the centroids of molecules, with three DOFs (two DOFs for linear molecules), and... 

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