Average rotational energy

A linear molecule, such as any diatomic molecule, carbon dioxide, and ethyne (acetylene, HC=CH), can rotate about two axes perpendicular to the line of atoms, and so it has two rotational modes of motion. Its average rotational energy is therefore 2 X jkT = kT, and the contribution to the molar internal energy is NA times this value ... 

A nonlinear molecule, such as water, methane, or benzene, can rotate about any of three perpendicular axes, and so it has three rotational modes of motion. The average rotational energy of such a molecule is therefore 3 X jkT = ]kT. The contribution of rotation to the molar internal energy of a gas of nonlinear molecules is therefore... 

For the average rotational energy of a linear molecule, use Eq. 8.65 to find... 

When the quasi-diatomic Franck-Condon model was compared with the experimental results it was found that it could predict the observed vibrational distribution as well as the observation that the translational energy is much greater than the rotational energy. The theory could not, however, predict the observed proportionality between the average rotational energy and the available energy. A simple classical description of the impulsive dissociation of a rotating molecule does predict this observed linear proportionality. 

Waite et al., (147) have also tried to use the observed rotational distribution to derive a potential energy surface for the excited states of ICN. They have been able to obtain not only the average rotational energy at 266 nm but have also been able to reproduce the shape of the rotational distribution. It will be interesting to see if the same curves can do as well at other wavelengths. 

In the (classical) high-temperature limit, when hi/s -C kBT, the exponential can be expanded to first order and the average vibrational energy is kBT. The approximation hvs -C kis l is, however, not well satisfied for typical molecular vibrational frequencies, except at temperatures that exceed several thousand degrees. The average rotational energy of a rigid rotor is... 

Recent calculations (see Section 3.1) show that the activated complex is non-linear, that is, the average rotational energy is (3/2)ksT and Ea = Eq + (.E ib) — (E ib). %2 = 4395 cm-1 and the two vibrational frequencies associated with the activated complex are 3772 cm-1 and 296 cm-1, respectively (remember that the third vibrational degree of freedom of the non-linear triatomic molecule is the reaction coordinate which is not included in (/A). The thermal energies associated with the... 

These experiments show conclusively that the available energy is released almost entirely as internal excitation of the products. The observation that the diatomic product can subsequently excite M atoms electronically demands a degree of excitation which precludes the formation of 2P1/2 halogen atoms, which requires 21.7 kcal/mole (0.94 eV) for I and 10.5 kcal/mole (0.46 eV) for Br. Where electric deflection analysis has been performed [34-36], the averaged rotational energy yield, is about 5 kcal/mole (0.22 eV),... 

The rotational distribution of He2(d Du), and its average rotational energy and the fraction of the total excess energy, are listed in Table 6. 

The slight difference in the average rotational energy of the two forms enhances the heat capacity due to the LeChatelier shift in the equilibrium position as the temperature is changed. This effect is exhibited in the curve for ot labeled e-H2. Equilibrium-hydrogen, e-H2, is hydrogen that is kept in the presence of a catalyst to ensure that the equilibrium between 0-H2 and P-H2 is established at all temperatures. The curve for e-H2 is typical of the heat capacity of a reactive mixture maintained in equilibrium as the temperature is changed. 

The rotational energy distribution of desorbing H2 deviates from a Boltzmann distribution, but the average rotational energy (368 + 67 K, measured in temperature units) is much less than expected for a Boltzmann distribution at the TPD peak temperature of 780 K. 

The average rotational energy of H2, HD, and D2 are the same, within experimental uncertainty. 

It is interesting to compare the centrifugal barrier for the loss of H and CH3 from ethane neutrals and ethane ions. For this purpose, we treat ethane as a diatomic molecule with a reduced mass of 7.5 amu. The ethane neutral and ion can be assumed to have the same moment of inertia which is approximately / = j,r2 = 3 x 10 " kg m2. Thus at room temperature the assumed two-dimensional rotor with its average rotational energy of RT = 207 cm will have an average 7 of 15 and an angular momentum of 1.5 x lO J sec. As the molecule or ion dissociates to 2 CH3 or CjHj + H, the reduced mass associated with methyl loss remains the same at 7.5 amu, whereas it will reduce to about 1 amu for the H-atom loss. Because the reduced mass plays a role in the centrifugal barrier, the barriers for H and CH3 loss will be very different. In addition, the ionic and neutral dissociations will be different because n and a are different. 

The average energy of any of the energies can also be obtained from Eq. (9.3). For instance, the average rotational energy is expressed as... 

The average rotational energy transfer (ARET) is then given by... 

Fig. 2. Average rotational energy transfer (ARET) for different collision energies. Solid line quantum mechanical wave packet propagation using the MCTDH method (from Ref. [22]) dashed line MQCB method (equation (47)) dotted line classical dynamics.

Using j=j (y+1) S /27, derive the average rotational energy per mole for a diatomic molecule using a partition function. How fast can you do this derivation if it is a test question ... 

Using Ej 1)1 — ). calculate the average rotational energy per mole for a diatomic... 

An average rotational energy is computed by averaging over the longest vibrational period /vib of the product ... 

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