Statistical thermodynamics vibrational energy levels

Given that we can calculate the total energy of a molecule as a function of its geometry, we can then calculate the electronic and vibrational states (Wj) associated with the nuclear motions. The energies of the vibrational states are usually calculated in the harmonic approximation. Using statistical mechanics, we can now start to evaluate thermodynamic properties. If we have a set of N particles or molecules distributed over a set of energy levels Wj then the population of level j is... 

The internal partition function for molecules having inversion may be factored, to a good approximation, into overall rotational and vibrational partition functions. Although inversion tunnelling results in a splitting of rotational energy levels, the statistical weights are such that the classical formulae for rotational contributions to thermodynamic functions may be used. The appropriate symmetry number depends on the procedure used to calculate the vibrational partition function. 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

For atomic and molecular systems, we actually have such expressions They come from the application of quantum mechanics to the translations, rotations, vibrations, and electronic states of atoms and molecules. Admittedly, Boltzmann didn t have quantum mechanics, because he developed the rudiments of statistical mechanics about 50 years before quantum mechanics was formulated. In fact, some ofhis expressions are incorrect by not including Planclfs constant (Boltzmann was unaware of its existence for most ofhis life). But in the calculation of thermodynamic values, the Planck s constants cancel. Their omission was, ultimately, unnoticed. However, in the material to come, we will use the quantum-mechanical basis of energy levels. 

We have seen how statistical thermodynamics can be applied to systems composed of particles that are more than just a single atom. By applying the partition function concept to electronic, nuclear, vibrational, and rotational energy levels, we were able to determine expressions for the thermodynamic properties of molecules in the gas phase. We were also able to see how statistical thermodynamics applies to chemical reactions, and we found that the concept of an equilibrium constant presents itself in a natural way. Finally, we saw how some statistical thermodynamics is applied to solid systems. Two similar applications of statistical thermodynamics to crystals were presented. Of the two, Einstein s might be easier to follow and introduced some new concepts (like the law of corresponding states), but Debye s agrees better with experimental data. 

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