Statistical thermodynamics energy levels, calculations from

Equation 18.60 shows that statistical thermodynamics can calculate temperature-dependent equilibrium constants from partition functions. Because the partition functions themselves are ultimately determined from the energy levels of the chemical species, we see once again how a knowledge of energy levels—obtained from spectroscopy—helps us make thermodynamic predictions about chemical reactions. 

In summary, Eq. (86) is a general expression for the number of particles in a given quantum state. If t = 1, this result is appropriate to Fenni-rDirac statistics, or to Bose-Einstein statistics, respectively. However, if i is equated torero, the result corresponds to the Maxwell -Boltzmann distribution. In many cases the last is a good approximation to quantum systems, which is furthermore, a correct description of classical ones - those in which the energy levels fotm a continuum. From these results the partition functions can be calculated, leading to expressions for the various thermodynamic functions for a given system. In many cases these values, as obtained from spectroscopic observations, are more accurate than those obtained by direct thermodynamic measurements. 

The main objective of statistical mechanics is to calculate macroscopic (thermodynamic) properties from a knowledge of microscopic information like quantum mechanical energy levels. The purpose of the present appendix is merely to present a selection1 of the results that are most relevant in the context of reaction dynamics. 

Hence, in the light of our both accounts of causality, the molecular dynamics model represents causal processes or chains of events. But is the derivation of a molecule s structure by a molecular dynamics simulation a causal explanation Here the answer is no. The molecular dynamics model alone is not used to explain a causal story elucidating the time evolution of the molecule s conformations. It is used to find the equilibrium conformation situation that comes about a theoretically infinite time interval. The calculation of a molecule s trajectory is only the first step in deriving any observable structural property of this molecule. After a molecular dynamics search we have to screen its trajectory for the energetic minima. We apply the Boltzmann distribution principle to infer the most probable conformation of this molecule.17 It is not a causal principle at work here. This principle is derived from thermodynamics, and hence is statistical. For example, to derive the expression for the Boltzmann distribution, one crucial step is to determine the number of possible realizations there are for each specific distribution of items over a number of energy levels. There is no existing explanation for something like the molecular partition function for a system in thermodynamic equilibrium solely by means of causal processes or causal stories based on considerations on closest possible worlds. 

A completely separate branch of science is statistical mechanics which is concerned with microscopic properties. This subject makes use of what quantum mechanics tells us about the energy levels of molecules, and allows us to calculate macroscopic properties on the basis of this information. The area of overlap between statistical mechanics and thermodynamics is known as statistical thermodynamics which allows us, for example, to calculate equilibrium constants for chemical reactions using the molecular properties obtained from quantum mechanics. 

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. 

Equation 17.54 is a useful conclusion. The (translational) partition function, originally defined as an infinite sum of negative exponentials of the energy levels, is equal to an expression in terms of the mass of the gas particles, the absolute temperature, the system volume, and some fundamental universal constants. This expression lets us calculate explicit values for q, which can then be used to determine values for energy, entropy, heat capacity, and so on. These calculated values—determined from a statistical rather than a phenomenological perspective—can then be compared to experimental values. We will thus get the first chance to see how well a statistical approach to thermodynamics compares with experiment. 

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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