Partition function vibrational motion

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

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. 

We have seen earlier that for a linear polyatomic molecule, the vibrational motions can be divided into (3rj — 5) fundamentals, where rj is the number of atoms. For a nonlinear molecule (3rj - 6) fundamentals are present. In either case, each fundamental vibration can be treated as a harmonic oscillator with a partition function given by equations (10.100) and (10.101). Thus. 

Values of and Qb can be calculated for molecules in the gas phase, given structural and spectroscopic data. The transition state differs from ordinary molecules, however, in one regard. Its motion along the reaction coordinate transforms it into product. This event is irreversible, and as such occurs without restoring force. Therefore, one of the components of Q can be thought of as a vibrational partition function with an extremely low-frequency vibration. The expression for a vibrational partition function in the limit of very low frequency is... 

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. 

The function g is the partition function for the transition state, and Qr is the product of the partition functions for the reactant molecules. The partition function essentially counts the number of ways that thermal energy can be stored in the various modes (translation, rotation, vibration, etc.) of a system of molecules, and is directly related to the number of quantum states available at each energy. This is related to the freedom of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy change is related to the ratio of the partition functions ... 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

The partition function of a molecule also contains torsional motions and the construction of such a function requires the knowledge of molecular mass, moments of inertia, and constants describing normal vibration modes. Several of these data may be acquired from infrared and Raman spectra (67SA(A)891 85JST( 126)25), but the procedure has not yet been extensively applied owing to experimental limitations. To characterize the barrier one also needs to know more than one constant, and these are often not available from... 

The partition function for vibrational motion in mode i is given by Eq. 8.71. Therefore the derivative with respect to temperature is... 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

The Boltzmann distribution of the populations of a collection of molecules at some temperature T was discussed in Section 8.3.2. This distribution, given by Eq. 8.46 or 8.88, was expressed in terms of the quantum mechanical energy levels and the partition function for a particular type of motion, for instance, translational, vibrational, or rotational motion. It is useful to express such population distributions in other forms, particularly to obtain an expression for the distribution of velocities. The velocity distribution function basically determines the (translational) energy available for overcoming a reaction barrier. It also determines the frequency of collisions, which directly contributes to the rate constant k. 

Hard-Sphere Collision Limit It is interesting to evaluate the behavior of Eq. 10.9 when both reactants A and B are atomic species. In this situation the only degrees-of-freedom contribution to the molecular partition functions are from translational motion, evaluated via Eq. 8.59. The atomic species partition functions have no vibration, rotational, or (for the sake of simplicity) electronic contributions. 

The vibrational frequency for motion perpendicular to the surface (z direction), denoted here va, may be associated with the desorption attempt frequency, which is a time constant or pre-exponential associated with the surface desorption process. The partition function for this degiee-of-ffeedom is then... 

The internal motion partition function of the guest molecule is the same as that of an ideal gas. That is, the rotational, vibrational, nuclear, and electronic energies are not significantly affected by enclathration, as supported by spectroscopic results summarized by Davidson (1971) and Davidson and Ripmeester (1984). 

Note that the 1 /N term is assigned to the translational partition function, since all gases have translational motion, but only molecular gases have rotational and vibrational degrees of freedom. The electronic partition function is usually equal to one unless unpaired electrons are present in the atom or molecule. 

The partition functions can be factorized into contributions corresponding to the various forms of motion when they are uncoupled (see Appendix A.l), and it is advantageous to rewrite the expression for the rate constant in terms of partition function ratios for the translational, rotational, vibrational, and electronic motion ... 

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

The quantized energy ej can be of electronic, vibrational, rotational or translational type, readily calculated from the quantum laws of motion. In a macrosystem the sum over all the quantum states for the complete set of molecules, the sum over states defines the canonical partition function ... 

Boltzmann6 proposed that at the temperature T = 0, all thermal motion stops (except for zero-point vibration), and the entropy function S can be evaluated by a statistical function W, called the thermodynamic probability W (or, as we will learn in Section 5.2, the partition function Q for a microcanonical ensemble) ... 

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