Molecules electronic partition functions

We explicitly excluded molecules in our earlier treatment of the electronic partition function. Let us consider Select for molecules now, starting with a diatomic molecule and generalizing the result to other molecules. 

The key to getting an electronic partition function for molecules depends on how we define the zero position for energy. Virtually all numerical scales have benchmarks that are used to define certain numerical values. For example, for atoms we defined the zero point as the ground electronic state. Vibrations and rotations have well-defined minimum-energy points that serve as starting points. But what about electronic energy  

FIGURE 18.1 The electronic potential energy diagram for a hypothetical diatomic molecule. In the ground state, some of the lower vibrational energy levels are indicated. How is the zero point of energy defined for a molecule that has electronic energy with this behavior  

For giggt of diatomic molecules, the benchmark for electronic energy is the dissociation limit. This means that the dissociation limit is arbitrarily assigned a value of zero 

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PROBLEM 5.3.12. Show that in a CE, in the absence of degeneracy, the single-molecule electronic partition function is... 

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. 

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. 

In some instances, multiple electronic states of a molecule are possible, and an electronic partition function must be evaluated. One would use an explicit summation over electronic energy levels, as in Eq. 8.50. No simple general theory is available to account for all the manifold of electronic energy states. 

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. 

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 ... 

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. 

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

In nitric oxide, which is an exception among stable diatomic molecules, each level has a multiplicity of two (A-type doubling), so that the electronic partition function is actually 4.0. 

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 electronic partition function involves a sum over electronic quantum states. These are the solutions to the electronic Schrodinger equation, i.e. the lowest (ground) state and all possible excited states. In almost all molecules, the energy difference between the ground and excited states is large compared with kT, which means that only the first term (the ground state energy) in the partition function summation (eq. (13.11)) is important. 

D17.1 An approximation involved in the derivation of all of these expressions is the assumption that the contributions from the different modes of motion are separable. The expression = kT/hcB is the high temperature approximation to the rotational partition function for nonsymmetrical linear rotors. The expression q = kT/hcv is the high temperature form of the partition function for one vibrational mode of the molecule in the haimonic approximation. The expression (f- =g for the electronic partition function applies at normal temperatures to atoms and molecules with no low lying excited electronic energy levels. 

Electronic Contribution. If a molecule has an electronic ground state of degeneracy go a first excited state at an energy Cei with a degeneratgrgj, then the electronic partition function will be... 

Because Dq is typically large and is a positive exponential, the first term in the equation above typically dominates, and we can approximate the diatomic molecule s electronic partition function as... 

The hydrogen molecule has a of 457.8 kj/mol and a vibrational frequency of 1.295 X 10 s Calculate H2 s electronic partition function at 298 K. Assume that the ground electronic state is singly degenerate. Hydrogens first excited electronic state lies at 1.822 X 10 J above the ground state and has a degeneracy of 1. 

Choose a diatomic gas and compute its translational, rotational, vibrational, and electronic partition functions at 298.15 K and 1.000 bar, looking up parameters as needed in Table A.22 in Appendix A or in some more complete table. Unless you choose NO or a similar molecule with an unpaired electron, assume that only the ground electronic state needs to be included. 

In most atoms and molecules, the ground electronic level is nondegenerate and the first excited level is sufficiently high in energy that at room temperature the second and further terms in the sum can be neglected. In this case the electronic partition function is very nearly equal to and... 

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