Internal rotation, partition function for

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For light rotors it may be possible to observe sufficient torsional transitions for the internal rotation partition function to be determined by direction summation (p. 271). If only a few low-lying torsional levels are observed, they may be fitted to a suitable potential from which higher torsional energy levels can be computed. Thermodynamic functions of hydrogen peroxide calculated by use of this procedure give a value of 5 (298.15 K) in excellent agreement with the calorimetric value. ... 

D Anna et al. (2003) showed that the NO3 reaction with CH2O proceeds by H-atom abstraction so that the sole products of this reaction are HNO3 HCO. Mora-Diez and Boyd (2002) examined the mechanism of the reactions between NO3 and formaldehyde and acetaldehyde theoretically comparisons with experiment are consistent with a direct abstraction mechanism. Alv ez-Idaboy et al. (2001a) reached a similar conclusion for NO3 reaction with formaldehyde, acetaldehyde, propanal, n-butanal, and 2-methylpropanal. Their calculations showed that all reactions proceed via abstraction of the a-carbonyl H-atom the dependence of the rate constant on molecular size was shown to be attributable to the increase in the internal rotational partition function with the size of the aldehyde. 

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

With equations (10.138) and (10.139) the partition function for free rotation can be written. However, when the internal rotation can be described by a... 

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.

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

Show that, for the bimolecular reaction A + B - P, where A and B are hard spheres, kTsr is given by the same result as jfcSCT, equation 6.4-17. A and B contain no internal modes, and the transition state is the configuration in which A and B are touching (at distance dAR between centers). The partition functions for the reactants contain only translational modes (one factor in Qr for each reactant), while the transition state has one translation mode and two rotational modes. The moment of inertia (/ in Table 6.2) of the transition state (the two spheres touching) is where p, is reduced mass (equation 6.4-6). 

Here the partition functions refer to internal degrees of freedom (subscript int for internal), QAB = 8i 2(ii,d2)kBT/ h2, that is, a rotational partition function where A and B are considered as point masses separated by the distance d, and Z = itd2 v) is related to the (hard-sphere) collision frequency Zab defined in Eq. (4.16), that is, Zab = Z[A][B. ... 

Partition functions for internal rotation will be included in the Qvlb with this expression. 

With respect to the third assumption, for a number of years, Tanaka has shown that the internal particle partition function of the guest molecules differ significantly in the cages, with restricted rotation and vibration, particularly for those molecules larger than methane. Indeed, some restriction on such motions is the basis, for example, in Raman spectroscopic determination of differing environments of the methane molecule, in the gas, in solution, and in the hydrate cages. ... 

Evaluation of the entropy change in adsorption by the statistical mechanics approach can be found in numerous sources. Making use of Refs. [22,23], we will outline what is required for the present purpose. The partition function for a gaseous molecule qm is the product of the translational component qlr and the internal components rotational rot vibrational qviu and electronic qt. ... 

In all of the aforementioned discussions, we left unspecified the internal partition function of a single molecule. This, in general, includes contributions from the rotational, vibrational, and electronic states of the molecule. Assuming that these degrees of freedom are independent, the corresponding internal partition function may be factored into a product of the partition functions for each degree of freedom, namely,... 

The partition function for a molecule is formed of the partition functions for individual types of energy increments (motions), i.e. from the translational, rotational, internal rotational (free rotation, hindered rotation), vibrational, electronic and nuclear spin partition functions... 

All the linear (i.e. noncyclic) alkanes have internal rotations about the C—C bonds. For each internal rotation, if there is no energy barrier to rotation, the partition function for free internal (one-dimensional) rotation is... 

However, a complete set of molecular energy levels needed for calculation of the partition function (Eq. (1.16)) is not available in most cases. The arising problem can be simplified through the approximation that the different types of motion such as vibration, rotation, and electronic excitations are on a different timescale and therefore are unaffected by each other and can be treated as decoupled motions. This leads to a separation of Q into factors that correspond to separate partition functions for electronic excitations, translation, vibration, external molecular rotation, and hindered and free internal rotation ... 

In molecules or radicals, such as ethyl, internal rotations around bonds such as CH3—O—CH2 occur. Accordingly, the partition function for a free rotor is defined as... 

Troe has described how one can estimate the value of the partition function basic expression for the density of internal states at the dissociation limit, which treats the vibrations in RadiRad ) as harmonic. Multiplicative factors are then estimated to allow, in turn for (i) the anhar-monicity of the vibrations (ii) the energy dependence of the density of vibrational states (iii) an overall rotation factor, which allows for the existence of centrifugal barriers and (iv) an internal rotation factor allowing for the barriers associated with internal rotors. 

Internal Partition Functions for Polyatomic Molecules.— The internal partition function for a polyatomic molecule comprises contributions from nuclear spin and electronic levels, and from rotational and vibrational degrees of freedom. On the assumption that the corresponding energies are additive and independent, these contributions can be factored, and the corresponding contributions to the thermodynamic functions are additive. 

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