Degrees of freedom, vibrational

Calculate A52 at = 0.1 for argon at 77 K that forms a weak adsorption bond with the adsorbent, having three vibrational degrees of freedom. 

In an ideal molecular gas, each molecule typically has translational, rotational and vibrational degrees of freedom. The example of one free particle in a box is appropriate for the translational motion. The next example of oscillators can be used for the vibrational motion of molecules. 

In contrast to the bimoleciilar recombination of polyatomic radicals ( equation (A3.4.34)1 there is no long-lived intennediate AB smce there are no extra intramolecular vibrational degrees of freedom to accommodate the excess energy. Therefore, the fonnation of the bond and the deactivation tlirough collision with the inert collision partner M have to occur simultaneously (within 10-100 fs). The rate law for trimoleciilar recombination reactions of the type in equation (A3.4.47) is given by... 

For chemically bound molecules, it is usual to analyse tlie vibrational energy levels in teniis of normal modes, a non-linear (or linear) molecule witli V atoms has 3 V - 6 (or 3 V - 5) vibrational degrees of freedom. There is a... 

The presence of nonlinearity in an Arrhenius plot may indicate the presence of quantum mechanical tunnelling at low temperatures, a compound reaction mechanism (i.e., the reaction is not actually elementary) or the unfreezing of vibrational degrees of freedom at high temperatures, to mention some possible sources. 

The Franck-Condon principle reflected in tire connection between optical and tliennal ET also relates to tire participation of high-frequency vibrational degrees of freedom. Charge transfer and resonance Raman intensity bandshape analysis has been used to detennine effective vibrational and solvation parameters [42,43]. 

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... 

The difference between the energy of a molecule at 0 K and its enthalpy at 298 depends on the thermal contr ibution due to vibration at the two temperatures. If the molecule in question is rigid, with few vibrational degrees of freedom, this contribution will be small, as it is for propene and cyclopropane. For larger molecules with a good deal of vibrational freedom, the difference will be conespondingly larger. 

The integrations over the eleetronie eoordinates eontained in I)f p , as well as the integrations over vibrational degrees of freedom yield "expeetation values" of the eleetrie dipole moment operator beeause the eleetronie and vibrational eomponents of i and f are identieal ... 

However, we know that a non-linear moleeule has three rotational and three translational degrees of freedom, all of whieh ean be assigned to symmetry speeies (Seetion 4.3.1). These are indieated in Table 6.5 and subtraeted from the total number of degrees of freedom to give the total number of vibrational degrees of freedom. 

When experimental data are not available, methods of estimation based on statistical mechanics are employed (7,19). Classical kinetic theory suggests a contribution to CP of S R for each translational degree of freedom in the molecule, a contribution of S R for each axis of rotation, and of R for each vibrational degree of freedom. A cmde estimate of CP for small molecules can be obtained which neglects vibrational degrees of freedom ... 

Since only the vibrational degrees of freedom take part in a solid-state reaction, the sole reason for this change may be the increase in their frequencies in the transition state... 

In general, the number of phonon branches for a carbon nanotube is very large, since every nanotube has 6N vibrational degrees of freedom. The symmetry types of the phonon branches for a general chiral nanotube are obtained using a standard group theoretical analysis [194]... 

Estimate the molar heat capacity (at constant volume) of sulfur dioxide gas. In addition to translational and rotational motion, there is vibrational motion. Each vibrational degree of freedom contributes R to the molar heat capacity. The temperature needed for the vibrational modes to be accessible can be approximated by 6 = />vvih/, where k is Boltzmann s constant. The vibrational modes have frequencies 3.5 X... 

There are 78 vibrational degrees of freedom for TgHg and it has been shown that the molecule has 33 different fundamental modes under Oh symmetry, 6 are IR active, 13 are Raman active, and 14 vibrations are inactive. The experimental fundamental IR active vibrational frequencies have been assigned as follows 2277 (v Si-H), 1141 (vas Si-O-Si), 881 5 O-Si-H), 566 ( s O-Si-O), 465 (v O-Si-O), and 399 cm ( s O-Si-O). These generally agree well with calculated values The IR spectrum recorded in the solid state shows bands at 2300 and 2293 cm ... 

The electronic, rotational and translational properties of the H, D and T atoms are identical. However, by virtue of the larger mass of T compared with D and H, the vibrational energy of C-H> C-D > C-T. In the transition state, one vibrational degree of freedom is lost, which leads to differences between isotopes in activation energy. This leads in turn to an isotope-dependent difference in rate - the lower the mass of the isotope, the lower the activation energy and thus the faster the rate. The kinetic isotope effects therefore have different values depending on the isotopes being compared - (rate of H-transfer) (rate of D-transfer) = 7 1 (rate of H-transfer) (rate of T-transfer) 15 1 at 25 °C. 

In general a nonlinear molecule with N atoms has three translational, three rotational, and 3N-6 vibrational degrees of freedom in the gas phase, which reduce to three frustrated vibrational modes, three frustrated rotational modes, and 3N-6 vibrational modes, minus the mode which is the reaction coordinate. For a linear molecule with N atoms there are three translational, two rotational, and 3N-5 vibrational degrees of freedom in the gas phase, and three frustrated vibrational modes, two frustrated rotational modes, and 3N-5 vibrational modes, minus the reaction coordinate, on the surface. Thus, the transition state for direct adsorption of a CO molecule consists of two frustrated translational modes, two frustrated rotational modes, and one vibrational mode. In this case the third frustrated translational mode vanishes since it is the reaction coordinate. More complex molecules may also have internal rotational levels, which further complicate the picture. It is beyond the scope of this book to treat such systems. 

The characters Xj for the examples in the previous section were calculated following the method described in Section 8.9, that is, on the basis of Cartesian displacement coordinates. Alternatively, it is often desirable to employ a set of internal coordinates as the basis. However, they must be well chosen so that they are sufficient to describe the vibrational degrees of freedom of the molecule and that they are linearly independent The latter condition is necessary to avoid the problem of redundancy. Even when properly chosen, the internal coordinates still do not usually transform following the symmetry of the molecule. Once again, the water molecule provides a very simple example of this problem. 

Hinshelwood (51) used reasoning based on statistical mechanics to show that the energy probability factor in the kinetic theory expressions (e E,RT) is strictly applicable only to processes for which the energy may be represented in two square terms. Each translational and rotational degree of freedom of a molecule corresponds to one squared term, and each vibrational degree of freedom corresponds to two squared terms. If one takes into account the energy that may be stored in 5 squared terms, the correct probability factor is... 

These activated complexes differ from ordinary molecules in that in addition to the three normal translational degrees of freedom, they have a fourth degree of translational freedom corresponding to movement along the reaction coordinate. This degree of freedom replaces one vibrational degree of freedom that would otherwise be observed. 

Figure 6P.2 contains the Raman spectra of pyridine adsorbed on silica gel at three temperatures as reported by Schrader and Hill [Rev. Sci. Instrum, 46 (1335), 1975]. Four bands are apparent at 991, 1006, 1032, and 1069 cm F If the band intensities are proportional to surface concentrations and if each band is associated with one vibrational degree of freedom of an adsorbed species, what is your interpretation of these data ... 

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