Molecular partition functions translation

Table 6.2 Forms for translational, rotational, and vibrational contributions to the molecular partition function...

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

We have thus reduced the problem from finding the ensemble partition function Q to finding the molecular partition function q. In order to make further progress, we assume that the molecular energy e can be expressed as a separable sum of electronic, translational, rotational, and vibrational terms, i.e.,... 

We see that the partition function of a molecule is the product of the contributions of the translational, rotational, vibrational, and electronic partition functions, which we can calculate separately, as discussed next. We will see in Section 8.5 that any thermodynamic quantity of interest can be derived from the molecular partition function. Thus it is important to be able to evaluate q. 

Here we use the label i to denote a molecular energy level, which may denote at once the specific translational (t), rotational (r), vibrational (u), and electronic (e) energy level of the molecule. From Eq. 8.46 and the definition of the molecular partition function q,... 

Using the previously derived expressions for q, we can now obtain expressions for each of the entropy terms. Equation 8.59 gives the molecular partition function for three-dimensional translational motion of a gas. Substituting this qtrans into Eq. 8.102, we obtain... 

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

The translational contribution to the molecular partition function, which is calculated using Eq. 8.59, clearly makes the largest contribution. (In obtaining this value, we also made use of the ideal gas law to calculate the volume V = 0.02479 m3 of a mole of gas at this temperature and pressure.) The rotational partition function is evaluated via Eq. 8.67, and the vibrational partition function for each mode is found via Eq. 8.71. Only the very... 

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). 

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. 

Each molecular partition function qt may in turn be factored into contributions from translational (g transX vibrational (qiyib), rotational (q Yib), and electronic (qi iec)... 

The information needed to evaluate the molecular partition functions qb (13.63), may in principle be obtained from experimental spectroscopic measurements or theoretical calculations on each molecule i. Each type of energy contribution to qt (translational, rotational, vibrational, electronic) in principle requires associated quantum energy levels... 

The evaluation of the molecular partition function can be simplified by noting that the total energy of the molecule may be written as a sum of the center-of-mass translational energy and the internal energy, E = Etrans + Emt, which implies... 

For p we use Eq. (38) of Chapter 5 with U0 as the desorption energy and the molecular partition function written as a product of a translational partition function [Eq. (72) of Chapter 5] and qint, which accounts for the internal degrees of freedom of the molecule ... 

The partition function of the system Q is related to the molecular partition function of the individual molecules in the system. In our development of rate constants we make use of the molecular partition functions. The molecular partition function per unit volume for an ideal gas is the product of the translational, rotational, vibrational and electronic energy states in the molecule... 

In the statistical-mechanical evaluation of the molecular partition function of an ideal gas, the translational energy levels of each gas molecule are taken to be the levels of a particle in a three-dimensional rectangular box see Levine, Physical Chemistry, Sections 22.6 and 22.7. 

In the molecular-motion contribution, the molecular partition function is the product of translational, rotational, vibrational, and electronic partition functions. If the molecule in solution is assumed to have the entire volume of the solution available to it, the ratio of gas-phase and solution-phase translational partition functions equals one. Likewise, the electronic partition function ratio will be one. It is unclear what one should use for the rotational partition function in solution, but if this is assumed to have the same form as that in the gas phase, the rotational partition function ratio (which involves the moments of inertia) will be very close to one, since structural changes from gas to solution are slight. Significant contributions to the vibrational partition function are made only by the low-frequency vibrational normal modes, and these modes sometimes show substantial changes in frequency on going from the gas phase to solution. If a vibrational calculation is done in the gas phase and in solutitm, one can calculate AG°oiv m, but the most common procedure is to omit it, assuming that its contribution is negligible. 

Although we have derived this equation assuming that the reaction coordinate for atom transfer corresponds to translational motion, the same expression is obtained if the reaction coordinate is assumed to be vibrational motion. According to Eq. (37), the reaction rate constant may be calculated using the relevant molecular partition functions, known from statistical mechanics, remembering that Oahb clude the translational motion... 

TST predicts the trend of decreasing Arrhenius pre-exponential factor with increasing reactant size and molecular complexity that is revealed by experimental measurements of rate coefficients, and that SCT explained away by invoking the steric factor. This trend arises in TST through the internal degrees of freedom, which are accounted for in the partition functions, and which are not present in the structureless point masses of SCT. If electronic, vibrational, rotational, and translational dfs are independent, the molecular partition function factors into electronic, vibrational, rotational, and translational contributions, i.e., Q = e v r t-The orders of magnitude of the partition functions per df are v l-10 per vibrational or internal rotational df, per overall... 

In particular, the microscopic formulation of the transition state theory, that is, by writing in terms of molecular partition functions, allows accounting for differences in translational, rotational and vibrational degrees of freedom between the species involved as reactant and the transition state. 

In equation 85, Q represents the molecular partition function for the transition state species and reactants A and B. It is constructed from contributions related to translational, rotational, and vibrational degrees of freedom as follows ... 

Molecular partition function including three translational degrees of freedom m ... 

In this equation k is called the transmission coefficient and is taken to be equal to unity in simple transition-state theory calculations, but is greater than imity when tunneling is important (see below), c° is the inverse of the reference volume assumed in calculating the translational partition function (see below), m is the molecularity of the reaction (ie, m = 1 for unimolecular, 2 for bimolecular, and so on), is Boltzmann s constant (1.380658 x 10 J molecule K ), h is Planck s constant (6.6260755 x 10 J s), Eq (commonly referred to as the reaction barrier) is the energy difference between the transition structure and the reactants (in their respective equiUbriiun geometries), Qj is the molecular partition function of the transition state, and Qi is the molecular partition function of reactant i. 

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