Statistical thermodynamics electronic energy levels

The occupation of the electron energy levels under thermal equilibrium at T > 0 K can be derived from statistical thermodynamics [11]. The reasoning results in the well-known Fermi-Dirac occupation function. The probability that an electron energy level at E is occupied by an electron is given by... 

This is not the T -dependence, as experimental measurements suggest. Therefore, while Einstein s application of statistical thermodynamics to crystals was useful, it has its limitations. (It might be considered similar, in some respects, to Bohr s attempt to describe electron energy levels by assuming quantized angular momentum. It worked in some respects—mostly in application to hydrogen atoms—but had its deficiencies in a more global sense.)... 

It has been said that chemistry itself relies on explanatory concepts loosely defined as valence and chemical bond (Tom, 1985). In fact, as was previously stated the most difficult aspects of chemistry are the explanatory (rhematic) discourses that try to map one level into another. This is done in many and efficient ways, but essentially using models of the particular instances. As chemistry is a huge science it has been developing in separate branches and the concepts used as well as the strategic explanations are very different from one field to another. The explanations most frequently use the already observed coherences within the discourses but they do vary from the electronic explanations of the mechanisms in organic chemistry to the static idea of energy of activation or to the all-powerful idea of free energy of a system in statistical thermodynamics. 

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 existence of discrete electronic states of electrons confined in a small metal cluster has been observed to influence the thermodynamic stability of the system, in particular during the production of sodium clusters in supersonic beams composed of the metal vapor and an inert gas. The statistics of the relative abundances of different particle sizes reveal the existence of magic numbers for the number of atoms in the cluster, A = 8, 20,40,58, 92,... [3.9]. This has been interpreted in terms of the existence of degenerate energy levels in a spherical well with infinite-potential walls. Particularly stable structures are obtained when the number of valence electrons is such that it leads to a closed-shell electronic structure, i. e. a structure with a completely filled energy level and an empty up-... 

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. 

For high-density electron ensembles such as valence electrons in metals, Fermi statistics are applicable. In a thermodynamic sense, the Fermi level, E (defined at 0 K as the energy at which the probability of finding an electron is 0.5) can be regarded as the electrochemical potential of the electron in a particular phase (in this... 

The method of calculation of thermodynamic functions for gaseous RX (R = La-Lu and X = F, Cl) implies that the statistical sums over the rovibrational levels of the ground and excited electronic states of the molecules are equal. This allows one to consider the contribution of excited electronic states in the form of a correction to the contribution of the ground state. As follows from the previous section, due to different values of these corrections, some sets of final data for certain RX compounds noticeably differ from one another. At real temperatures, the difference in reduced Gibbs energy is as large as 4-7 J/(mol K) for some monochlorides and even larger, about 5-9 J/(mol K), for some monofluorides. 

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