Molecular Electronic Partition Function

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,... 

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. 

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 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 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. 

The molecular partition functions, Q, can be related to molecular properties of reactants and products. The partition function expresses the probability of encountering a molecule, so that the ratio of partition functions for the products versus the reactants of a chemical reaction expresses the relative probability of encountering products versus reactants and, therefore, the equilibrium constant. The partition function can be written as a product of independent factors at the level of various approximations, each of which is related to the molecular mass, the principal moments of inertia, the normal vibration frequency, and the electronic energy levels, respectively. When the ratio of isotopic partition function is calculated, the electronic part of the partition function cancels, at the level of the Born-Oppenheimer approximation, an approximation stating that the motion of nuclei in ordinary molecular vibrations is slow relative to the motions of electrons. 

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... 

The molecular partition function for a system includes terms that relate to different forms of energy nuclear, electronic, vibrational energy of molecules, their rotational energy, their translational energy and interaction energies between different molecules. 

Recognizing that molecules are an important part of chemistry, we will define a molecular partition function, Q, that is the product of partition functions from various energies of a molecule translational, vibrational, rotational, electronic, and nuclear. 

Here, represents the contribution of all other internal motions of the molecule to the molecular partition function (rotations, vibrations, electronic and nuclear spin motions). For atomic liquids, this term can be taken as being equal to 1. 

Normally, the excited electronic energy levels are very large compared to hgT, whence it is usually sufficient to set the electronic molecular partition function as... 

We now apply the expression for U in Eq. (25.3-7) to obtain a formula to compare with Eq. (25.3-8). Atoms have only translational and electronic energy. The degeneracy of an energy level is the product of the translational degeneracy and the electronic degeneracy, and the energy is the sum of the translational and the electronic energy. The molecular partition function is... 

Wehave already determined that the molecular partition function for a dilute monatomic gas is the product of a translational partition function and an electronic partition function. We obtained a formula for the translational partition function in Eq. (25.3-21) ... 

Calculate the value of the molecular partition function of CI2 gas at 298.15 K and a pressure of 1.000 bar. Find the Russell-Saunders term symbol for the ground-state of atomic chlorine and find the degeneracy of this level. Assume that the electronic partition function can be approximated by go> the degeneracy of the ground-level. 

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