Rotational Contribution

Rotational Contribution. For linear molecules, the rotational energy levels are given, to a good first approximation, by the expression 

The summation for unsymmetrical molecules is over all /, and for symmetrical molecules is over even or odd values of J for para- or ortho-species. Since qxot cannot be expressed in closed form, its evaluation as a sum with the thermodynamic functions calculated from equations (3)—(5), (7), and (9)—(11) is necessary, for example for hydrogen at moderate temperatures. However, if is small, then the summation can be replaced by an integral and evaluated as (07) where a is the symmetry number taking the value 2 or 1 for symmetrical or unsymmetrical molecules, respectively. The same result appears as the leading term in a power series for rot which may be derived  

For non-linear, asymmetric top molecules the rotational partition function involves all three moments of inertia and the resulting expression is complex. For most practical purposes the result reduces to the classical high-temperature approximation 

Since only the product IJyh of the three principal moments of inertia is required, a simplified procedure can be used. The symmetry number a is the number of indistinguishable positions into which the molecule can be turned by simple rigid rotations it is given directly by the point group of the molecule. Equation (14) is valid for polyatomic molecules at or above room temperature. The rotational contributions to the thermodynamic functions foUow from equation (14) using equations (3), (4), (5), and (8), and the practical equations are given in Table 1. 

Herzberg Infrared and Raman spectra of Polyatomic Molecules , Van Nostrand, Princeton, 1945, p. 508. 

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Calculate the rotational contribution to the entropy of adsorption of benzene on carbon at 35°C, assuming that the adsorbed benzene has one degree of rotational freedom. 

Calculate the rotational contribution to the entropy of adsorption of ammonia on silica at -30°C, assuming (n) that the adsorbed ammonia retains one degree of rotational freedom and (b) that it retains none. In case (n) assume that the nitrogen is bonded to the surface. 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

N-protonation the absolute magnitude of the Ad values is larger than for Af-methylation <770MR(9)53>. Nuclear relaxation rates of and have been measured as a function of temperature for neat liquid pyridazine, and nuclear Overhauser enhancement has been used to separate the dipolar and spin rotational contributions to relaxation. Dipolar relaxation rates have been combined with quadrupole relaxation rates to determine rotational correlation times for motion about each principal molecular axis (78MI21200). NMR analysis has been used to determine the structure of phenyllithium-pyridazine adducts and of the corresponding dihydropyridazines obtained by hydrolysis of the adducts <78RTC116>. 

This argument can be generalized to any number of subsystems and energy levels. For the case of a molecular system in a given electronic state, the factorization into translational, vibrational, and rotational contributions gives... 

If we deal with N isolated non-interacting entities such as the molecules in a gas at low density, we can further divide up molecular energies with reasonable accuracy into their electronic, vibrational and rotational contributions... 

Summarizing, in order to calculate rate and equilibrium constants, we need to calculate and AGq. This can be done if the geometry, energy and force constants are known for the reactant, TS and product. The translational and rotational contributions are trivial to calculate, while the vibrational frequencies require the ftill force constant matrix (i.e. all energy second derivatives), which may involve a significant computational effort. 

Translational and Rotational Contributions to Enthalpy for Molecule For translation... 

Calculation of Thermodynamic Properties We note that the translational contributions to the thermodynamic properties depend on the mass or molecular weight of the molecule, the rotational contributions on the moments of inertia, the vibrational contributions on the fundamental vibrational frequencies, and the electronic contributions on the energies and statistical weight factors for the electronic states. With the aid of this information, as summarized in Tables 10.1 to 10.3 for a number of molecules, and the thermodynamic relationships summarized in Table 10.4, we can calculate a... 

Figure 10.10 Internal rotation contribution to the heat capacity of CH3-CCI3 as a function of temperature. Reprinted from K. S. Pitzer. Thermodynamics, McGraw-Hill, Inc., New York, 1995, p. 374. Reproduced with permission of the McGraw-Hill Companies.

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 A4.6 gives the internal rotation contributions to the heat capacity, enthalpy and Gibbs free energy as a function of the rotational barrier V. It is convenient to tabulate the contributions in terms of VjRTagainst 1/rf, where f is the partition function for free rotation [see equation (10.141)]. For details of the calculation, see Section 10.7c.

Fig. 3.8. The Q-branch Raman width alteration with condensation of nitrogen. The theoretical results for the strong (A) and weak (B) collision limits are shown together with experimental data for gaseous [89] ( ) and liquid nitrogen [145] ( ) (point a is taken from the CARS experiment of [136]). The broken curves in the inset are A and B limits whereas the intermediate solid curve presents the rotational contribution to line width at y = 0.3. The straight line estimates the contribution of vibrational dephasing [143], and the circles around it are the same liquid data but without rotational contribution.

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Fig. 3.13. Density-dependence of the Qo, branch line width y of methane (the dashed line is for pure vibrational dephasing, supposed to be Unear in density), (o) experimental data (with error bars) [162] Top part rotational contribution yR and its theoretical estimation in motional narrowing limit [162] (solid line) the points were obtained by subtraction of dephasing contribution y

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

For nonlinear molecules, the rotational contribution is RT, for a total contribution of 3 RT. Therefore, we have the following expressions ... 

Fig. 4 Rotational contribution to the molar heat capacity C for ortho, para and nominal hydrogen. Note that 1 cal/deg-mol = 4.18 J K Lmor1.

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

TABLE 1. Rotational contributions to the Lb Cotton effects for a chiral benzene compound with a single chiral substituent giving a negative vibronic contribution to the Lb Cotton effects... 

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