The electronic partition function

This is the value of the rotational partition function for unsymmetrical linear molecules (for example, heteronuclear diatomic molecules). Using this value of we can calculate the values of the thermodynamic functions attributable to rotation. 

For nonlinear molecules with principal moments of inertia, 7, Ip, and ly, the value of the rotational partition function at high temperature is 

Using Eq. (29.30a) we can write the electronic partition function in the form. 

If these conditions are not met, we simply add as many higher terms as are required by the particular case. In many practical cases Eq. (29.63) is all that is needed. 

As we will see below, if we are to use the expression in Eq. (29.63) in the calculation of an equilibrium constant f or a chemical reaction, we must choose a common energy zero for all the species involved in the reaction. The condition of separated atoms at rest in the gas phase (that is, at T = 0 K) is usually the most convenient choice of energy zero. The situation is illustrated for a diatomic molecule in Fig. 29.2. The depth of the minimum is the value of Cell thus = —D. This position is the origin for the vibrational energy of the molecule. Thus, if the dissociation energy is Dq (a positive number), then 

---

The electronic partition function of the transition state is expressed in terms of the activation energy (the energy of the transition state relative to the electronic energy of the reactants) E as ... 

With this set of energy levels, the electronic partition function is given by... 

We have seen that for the electronic partition function there is no closed form expression (as there is for translation, rotation, and vibration) and one must know the energy and degeneracy of each state. That is. 

Usually, we would choose the separate atoms in their ground state as the zero energy. The electronic partition function is then... 

Choosing the separate atoms as the zero energy, the electronic partition function of the hydrogen molecule is... 

The various contributions to the energy of a molecule were specified in Eq. (47). However, the fact that the electronic partition function was assumed to be equal to one should not be overlooked. In effect, the electronic energy was assumed to be equal to zero, that is, that the molecule remains in its ground electronic state. In the application of statistical mechanics to high-temperature systems this approximation is not appropriate. In particular, in the analysis of plasmas the electronic contribution to the energy, and thus to the partition function, must be included. 

In practice, it proves more convenient to work within a convention where we define tire ground state for each energy component to have an energy of zero. Thus, we view 1/eiec as the internal energy that must be added to I/q, which already includes Eeiec (see Eq. (10.1)), as the result of additional available electronic levels. One obvious simplification deriving from this convention is that the electronic partition function for the case just described is simply eiec = 1, Inspection of Eq. (10.5) then reveals that the electronic component of the entropy will be zero (In of 1 is zero, and the constant 1 obviously has no temperature dependence, so both terms involving eiec are individually zero). 

In most cases, excited electronic energy levels lie high above the ground-state energy relative to ksT, and the population in the upper levels is negligibly small. In these cases the electronic partition function reduces to one term ... 

As a numerical example, consider calculation of the electronic partition function for the H atom, using explicit evaluation of the summation in Eq. 8.50 (truncated after two terms)... 

Thus, even in this very high temperature example, excited electronic energy levels make a negligible contribution to the electronic partition function. 

The simplest QCE model incorporates environmental effects of cluster-cluster interactions by (1) approximate evaluation of the excluded-volume effect on the translational partition function >trans (neglected in Section 13.3.3) and (2) explicit inclusion of a correction A oenv) for environmental interactions in the electronic partition function qiQiec. Secondary environmental corrections on rotational and vibrational partition functions may also be considered, but are beyond the scope of the present treatment. 

As in (13.89), a nonzero amf brings complex nonlinearity into the electronic partition function,... 

The electronic partition function can be evaluated by summing over spectroscopically determined electronic states, but as the electronic energy-level separations are large, the number of molecules in excited electronic states is negligibly small at ordinary temperatures and the electronic partition function is unity and will be ignored henceforth. 

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. 

Finally, the electronic partition function is considered. The zero of energy is chosen as the electronic ground-state energy. The spacings between the electronic energy levels are, normally, large and only the first term in the partition function will make a significant contribution that is,... 

Crystalline l. The partition function for the crystalline state of I2 consists solely of a vibrational part the crystal does not undergo any significant translation or rotation, and the electronic partition function is unity for the crystal as it is for the gas. 

We have all along neglected the electronic partition function. The reason has been that the energy separation of electronic states is usually so great that only one electronic state, the lowest, is ever occupied at most temperatures. The partition function for electronic states is simply if only the lowest state is occupied, lor... 

In reactions of free radicals or atoms to form molecules the electronic partition function may not be negligible, since atoms or radicals generally have odd numbers of electrons and hence a multiplicity of electronic states, while the molecules will not. 

Subscript designates ligancy of atom, i.e., number of atoms bonded to central atom. The entropy contribution must be corrected by the addition of any electronic entropy, R In qny where qn is the electronic partition function. The quantity R n a must also be subtracted from the total entropy to correct for symmetry. symmetry number of the final species. Values taken from S. W. Benson and J. H. Buss, J. Chem, Phys., 29 (1958). These quantities are not to be used for cyclic structures such as benzene compounds. Estimates of Cp and are good to about 2 cal/mole- K for most species but may be poorer for heavily substituted species such as neopentane. They may also be poorer for very simple H-containing species such as NH3 and CH4. 

For most molecules it is reasonable to set the electronic partition function to the statistical weight of the ground electronic state. 

Sig and Cg. Use of constants based on the " ll state is consistent with the assumption that this state dominates the electronic partition function at high temperatures. is calculated from r. ... 

For some atoms one or more electronic states above the ground state are appreciably occupied even at moderate temperatures, and hence the appropriate terms must be included in the partition function. For example, in the lowest state of the chlorine atom, i.e., when te is zero, the value of 7 is f not very far above this is another state in which 7 is The electronic partition function for atomic chlorine at ordinary temperatures is therefore given by equation (16.17) as... 

SO that from equations (16.18) and (16.19) the electronic partition function for atomic chlorine is found to be... 

---