Single atom particle statistics

In Faujasites. Bezus et al. (49) reported in 1978 statistical calculations on the low-coverage adsorption thermodynamics of methane in NaX zeolite (Si/Al = 1.48). As for single-atom adsorbates described earlier, the agreement between their calculated values and a range of experimental values was excellent. Allowing for different orientations of the molecule, they calculated a value of 17.9 kJ/mol for the isosteric heat of adsorption at 323 K. Experimental values available for comparison at that time (134-136) ranged from 17.6 to 18.8 kJ/mol. Treating the methane molecule as a hard-sphere particle, with a radius of 2 A, resulted in a far lower heat of adsorption (12.6 kJ/mol). Further calculations (99) yielded heats of adsorption of 19.8 and 18.1 kJ/mol for methane in NaX and NaY zeolites, respectively. 

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. 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

The properties and the behavior of systems composed of many elementary particles, atoms, and molecules, are described by quantum statistics.3-5 Let bl "bN be a complete set of observables of an N-particle system, where b, is a complete observable of the single-particle systems, for example, b1 = r1s1, with r, being the position and 5, being the spin projection. Then the microscopic state is given by a vector bx bN) in the space of states dKN. 

The interaction of light with matter provides some of the most important tools for studying structure and dynamics on the microscopic scale. Atomic and molecular spectroscopy in the low pressure gas phase probes this interaction essentially on the single particle level and yields information about energy levels, state symmetries, and intramolecular potential surfaces. Understanding enviromnental effects in spectroscopy is important both as a fundamental problem in quantum statistical mechanics and as a prerequisite to the intelligent use of spectroscopic tools to probe and analyze molecular interactions and processes in condensed phases. 

Liquid atomization is a process for converting a bulk liquid volume of fluid into a myriad of single particle elements of multiple sizes (drops), which can be statistically described. Therefore, it is worth synthesizing the underlying statistical principles associated with a certain distribution function and the atomization process itself. 

Frenkel(47), by kinetic theory, and Jost(44,56), by statistical mechanics, showed that w Ne ol. Only a fraction of these holes, or corresponding interstitial atoms, will diffuse however, because an activation energy is necessary for diffusion. Thus the number moving is proportional to e o+2Ei)/2BT An approximate value of Z>, the diffusion coefficient, can be deduced if it is assumed that there are six ions around each hole, distant d from its ceptre, and capable of moving with the mean thermal velocity in any one of six directions. Each of the six particles may move in a single direction (to the hole), so the six of them are equivalent to a single particle free to move in all directions. Therefore the diffusion constant for a hole is... 

Hq is the number density of particles in the excited state, no is the number density of particles in the ground state, gq and go are the statistical weights of the corresponding levels, Eq is the excitation energy of the state q, k is Boltzmann s constant (1.38 X 10 erg K) and T is the absolute temperature. In Eq. (19) a relationship is formulated between the temperature and the atom number densities in a single excited state and in the ground state, respectively. As the latter is not constant, the Boltzmann equation can be better formulated as a function of the total number of particles n distributed over all states. Then... 

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