Degrees of freedom for rotation

As was shown for translational and rotational motions, there are three degrees of freedom for vibrational motion for every center of mass in the molecule. The number six on the right hand side term of equation (2.9) arises from the total number of degrees of freedom for translational and rotational motion, which do not belong to vibrational motion. It should be known that for linear molecules, there are only two degrees of freedom for rotational motion. This is why for this case there is a special equation for the calculation of the degrees of freedom for vibrational motion (2.10). 

As described in detail on page 770 and in Table 28-1, nonlinear molecules consume 3 degrees of freedom for rotation, whereas linear molecules exhibit only 2 degrees of rotational freedom. There are several examples where molecules that contain three atoms (i.e., CO2, CS2, N2O) are linear because the bond angle is 180°. Acetylene (i.e., HCsCH) is a four-atom linear molecule that exhibits only 2 degrees of freedom for rotation. Molecules exhibit fewer rotational degrees of fi eedom if rotation is hindered. 

Obviously, this approximation is excellent when N corresponds to 1 mol of molecules. Molar properties are obtained via division by N, and multiplication by Navo. where the gas constant R = kN vo- For diatomic molecules with 2 degrees of freedom for rotation and 1 for vibration. 

Resorcinarenes have some degrees of freedom for rotation around the C—C bonds of the connecting CHa-groups, which results in five conformations, namely crown, boat, chair, diamond and saddle. The hydroxy groups at the wider rim must be linked in order to fix the receptor in the specific cavitand conformation and to improve binding properties. 

The translational motion of a molecule (i.e. movement through space) can be described in terms of three degrees of freedom relating to the three Cartesian axes. If there are 3n degrees of freedom in total and three degrees of freedom for translational motion, it follows that there must be (3n — 3) degrees of freedom for rotational and vibrational motion. For a non-linear molecule there are three degrees of rotational freedom, but for a linear molecule, there are... 

For discussion of dynamics of lamellar smectic phases it is important to include another variable, the layer displacement u (r) [3] or, more generally, the phase of the density wave [4]. This variable is also hydrodynamic for a weak compression or dilatation of a very thick stack of smectic layers (L oo) the relaxation would require infinite time. On the other hand, the director in the smectic A phase is no longer independent variable because it must always be perpendicular to the smectic layers. Therefore, total number of hydrodynamic variables for a SmA is six. For the smectic C phase, the director acquires a degree of freedom for rotation about the normal to the layers and the number of variables again becomes seven. 

A molecule may have two or three degrees of freedom for rotation. We distinguish three families of molecules. 

Figure 10.2 Translational, rotational, and vibrational degrees of freedom for a simple diatomic molecule.

The transition state theory of reaction rates [21] provides the link between macroscopic reaction rates and molecular properties of the reactants, such as translational, vibrational, and rotational degrees of freedom. For an extensive discussion of transition state theory applied to surface reactions we refer to books by Zhdanov [25] and by Van Santen and Niemantsverdriet [27]. The desorption of a molecule M proceeds as follows ... 

For a molecule with N atoms, its 3iV degrees of freedom would be split into three translational degrees of freedom (corresponding to x-, y-, and z-directions), and three rotational degrees of freedom for nonlinear molecules and two for linear ones. Therefore, 3N—6 and 3N— 5 vibrational degrees of freedom exist for nonlinear and linear molecules, respectively. Vibrational frequencies can be obtained from convenient tabulations (see, for example, Shimanouchi, 1972 Chase et al., 1985). 

The number of angles required to specify a molecule s orientation depends on whether it is linear or nonlinear. It takes only two angles, 0 and , to specify the orientation for a linear molecule, as illustrated in Fig. 8.2. Thus there are two rotational degrees of freedom for a linear molecule. It takes three angles, 6, 0, and nonlinear molecule in space, so a nonlinear molecule has three rotational degrees of freedom. 

The 3N degrees of freedom for nuclear motion are divided into 3 translational, 3 (or 2) rotational, and 3N-6 (or 3N-5) vibrational (degrees of freedom. (The translations and rotations are often called nongenuine vibrations.) The 9 irreducible representations in (9.104) include the 3 translations and the 3 rotations. To find the symmetry species of the 3 vibrations, we must find the symmetry species of the translations and rotations. 

In this expression, N is the number of times a particular irreducible representation appears in the representation being reduced, h is the total number of operations in the group, is the character for a particular class of operation, jc, in the reducible representation, is the character of x in the irreducible representation, m is the number of operations in the class, and the summation is taken over all classes. The derivation of reducible representations will be covered in the next section. For now, we can illustrate use of the reduction formula by applying it to the following reducible representation, I-, for the motional degrees of freedom (translation, rotation, and vibration) in the water molecule ... 

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