Partition function free rotational

Here qrot,s and qvib,s are the microscopic partition functions for rotation and vibration of M in the solution while Hm.s and Am, indicate the related numeral densities and the momentum partition functions, respectively. The functional (G ) has now the meaning of the free energy of the entire solute-solvent system, at the temperature T, with respect to a reference state given by non-interaction nuclei and electron, supplemented by the imperturbed pure liquid S, at the same temperature. Then, the fundamental energetic quantity connected with the insertion of the solute in the solvent, i.e. the free energy of solvation can be obtained as... 

In the case of water, the situation is complicated because of the anisotropic nature of the potential. Thus, we have effective harmonic potential for translation, rotation, and librational motions. Each is characterized by a force constant and contributes to the partition function, free energy, and entropy. Furthermore, a water molecule can be categorized by the number of HBs it forms. Since these quantities can be considered as thermodynamic, they make a contribution as the entropy of mixing, also known as the cratic contribution. 

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. 

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.

Figure 2.15 Microscopic pictures of the desorption of atoms and molecules via mobile and immobile transition states. If the transition state resembles the ground state, we expect a prefactor of desorption on the order of 1013 s. If the adsorbates are mobile in the transition state, the prefactor goes up by one or two orders of magnitude. In the case of desorbing molecules, free rotation in the transition state increases the prefactor even further. The prefactors are roughly characteristic of atoms such as C, N and O and molecules such as N2, CO, NO and 02. See also the partition functions in Table 2.2 and the prefactors for CO desorption in Table 2.3.

Rotational and vibrational partition functions can be computed from the geometry and vibrational frequencies that are calculated for a molecule or TS. The entropy can then be obtained from these partition functions. Thus, electronic structure calculations can be used to compute not only the enthalpy difference between two stationary points but also the entropy and free energy differences. 

If the guest molecule is assumed to be confined, in the thermodynamic sense, within a particular cavity, if the internal vibrational and rotational states of the molecule are unaltered by occlusion, and if the potential field within a cavity is sufficiently uniform so that the movement of the molecule within the cavity can be represented as three-dimensional translation within the free volume of the cavity (), then the appropriate expression for the ratio of partition functions becomes simply... 

Higher pre-exponential factors result if the molecule rotates or moves in the transition state. For example, suppose that CO has free rotation in a plane perpendicular to the surface. As rotational partition functions are usually large (see... 

The vaporization of a pure liquid or the reverse process, the condensation of the liquid, provides an interesting test of this delayed equilibration hypothesis. Thus as a hydrogen-bonded molecule, which vibrates in the liquid, separates from the surface it frees itself from the potential energy restrictions which prevented rotation. However, in so far as the evaporating molecule has insufficient collisions with neighbors to equilibrate to the free rotational partition function, fgy of the gas, it will retain substantially the partition function, fb of the condensed phase even in the activated complex. Consequently for the condensation process the usual... 

It is important to note in Figure 1 that both curves show a decrease with temperature, and it should be possible to fit B smoothly onto A by multiplying by a suitable scale factor, possibly as shown by the dashed line. To explain the data shown in Figure 1 the temperature dependence of fs/fg is needed. The rotational partition function for a diatomic molecule that is free is... 

In the computation of the rotational entropies of SCW and NSCW near an ion, the rotation is restricted to libration about the axis perpendicular to the dipole. The third rotation, i.e., about the dipole axis, does not change the orientation of the dipole and may be better calculated as if it were a free rotation. The partition function for this is... 

To relate these thermodynamic quantities to molecular properties and interactions, we need to consider the statistical thermodynamics of ideal gases and ideal solutions. A detailed discussion is beyond the scope of this review. We note for completeness, however, that a full treatment of the free energy of solvation should include the changes in the rotational and vibrational partition functions for the solute as it passes from the gas phase into solution, AGjnt. ... 

We assume that there is no free internal rotation in the molecule, and the contribution from the torsional oscillation (v. = 182.5 cm" ) is included in the vibrational partition function. Extended Huckel calculations (7) show that the potential... 

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