Rotation, internal partition function

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

If we use the approximation of factoring these internal partition functions into electronic, vibrational, and rotational partition functions, Q(interual) = QeiecQvibQrot and wc Call write (Sec. IX) ... 

The integration on the rhs of (1.67) extends over all possible locations and orientations of the N particles. We shall refer to the vector XN=Xt,..., XN as the configuration of the system of the N particles. The factor q, referred to as the internal partition function, includes the rotational, vibrational, electronic, and nuclear partition functions of a single molecule. We shall always assume in this book that the internal partition functions are separable from the configurational partition function. Such an assumption cannot always be granted, especially when strong interactions between the particles can perturb the internal degrees of freedom of the particles involved. 

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,... 

It is instructive to recognize the three different sources that contribute to the liberation free energy. First, the particle at a fixed position is devoid of momentum partition function (though it still has all other internal partition functions such as rotational and vibrational). Upon liberation, the particle... 

Note that the rotational partition function of the entire molecule, as well as the internal partition functions of s, are included in the pseudo-chemical potential. In classical systems, the momentum partition function is independent of the environment, whether it is a gas or a liquid phase. 

The internal partition function is a sum of Boltzmann factors, multiplied by their degeneracies, Pj, over the vibrational and rotational states of the molecule. In the summation over the rotational states it is necessary to consider symmetry selection rules. This leads to a factor S in and + RlnS... 

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... 

Here, q includes the rotational as well as the internal partition function of a single molecule. The quantity (X 7X, I) is the local density of particles at X", given a particle at X coupled to the extent. Clearly, the whole integral on the rhs of (3.86) does not depend on the choice of X (for instance, we can take R = 0 and 2 = 0 and measure X relative to this choice). 

In this equation, (qmt)i is the partition function associated with the internal degrees of freedom (rotations and vibrations) of species i. In addition, the ( inOi can be written as products of vibrational ( vib) and rotational ( roOi partition functions. It is also instructive to write (qvib)R Rf as the product of terms associated with the conserved modes, which correlate with vibrational modes in the reactants, and the transitional modes, which correlate with relative translational and rotational motions in the reactants. Then ... 

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. 

Free Internal Rotation of a Single Symmetric Top. Although there are few examples of free internal rotation, it is instructive to consider this case first since restricted internal rotation has been generally considered as a perturbation of free internal rotation. The partition function for a single free internal rotation has been obtained by classical mechanics, and may be conveniently expressed by ... 

The internal partition function for molecules having inversion may be factored, to a good approximation, into overall rotational and vibrational partition functions. Although inversion tunnelling results in a splitting of rotational energy levels, the statistical weights are such that the classical formulae for rotational contributions to thermodynamic functions may be used. The appropriate symmetry number depends on the procedure used to calculate the vibrational partition function. 

Now you need to include the internal partition functions, qA, qB, and qAB because the internal degrees of freedom, such as the rotational symmetry, can change upon dimerization (see Equation (15.26)). 

Internal partition functions are relative to the vibrational, rotational, electronic and nuclear movements. We usually find that these movements are independent, although we know that this is not always so between the vibrations and rotations. We also find that the forces exerted on the molecnle by the exterior have no inflnence on these internal degrees of freedom. 

Statistical mechanics is that branch of physical chemistry dealing with partition functions. We appeal to statistical mechanics for two results, discussed in Appendix 6.A.. First, a partition function of a molecule can be expressed as a product of the partition function for the motion of the center of mass and an internal partition function. Second, the internal partition function can be approximately represented as a product of contributions from each bound mode. To apply the theory we need to compute partition fimctions (or to look them up in the JANAF tables ). In what follows, we are extremely cavalier and take all rotational partition functions to be equal, and larger by an order of magnitude than all vibrational partition functions that we also take to be equal to one another. This provides a reasonable order of magnitude but is otherwise not a recommended procedure. 

Qs(T) is the partition function of the reaction coordinate Q (T) is the internal partition function associated with the electronic degrees of freedom of AB and the (3Aab - 4) vibrational-rotational degrees of freedom after removing the reaction coordinate and three coordinates for overall translation is the partition function per unit volume for... 

As usual it will be assumed that the canonical partition function Q of the system may be written as the product of an internal partition function which does not depend on the configurational coordinates (but includes the contributions of the internal degrees of freedom and the kinetic energy of rotation and translation) and a configurational partition function... 

The main assumption involved here is that the internal partition function is independent of density. It is valid for monatomic molecules, but can lead to significant errors for polyatomic molecules, where their rotation and vibration is effected by density, especially at high values of it. 

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

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

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... 

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

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). 

Evaluation of die rotational components of the internal energy and entropy using the partition function of Eq. (10.19) gives... 

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