Maxwell-Boltzmann partition

To understand how collision theory has been derived, we need to know the velocity distribution of molecules at a given temperature, as it is given by the Maxwell-Boltzmann distribution. To use transition state theory we need the partition functions that follow from the Boltzmann distribution. Hence, we must devote a section of this chapter to statistical thermodynamics. 

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. 

Maxwell-Boltzmann particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles are indistinguishable. For example, individual electrons in a solid metal do not maintain positional proximity to specific atoms. These electrons obey Fermi-Dirac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

It is noted that the right-hand side is the ratio of the translational partition functions of products and reactants times the Boltzmann factor for the internal energy change. In the derivation of this expression we have only used that the translational degrees of freedom have been equilibrated at T through the use of the Maxwell-Boltzmann velocity distribution. No assumption about the internal degrees of freedom has been used, so they may or may not be equilibrated at the temperature T. The quantity K(fhl, ij) may therefore be considered as a partial equilibrium constant for reactions in which the reactants and products are in translational but not necessarily internal equilibrium. 

The excess contribution can be calculated from the crystal field energies according to the following equations where Q is the partitioning function described by the Maxwell-Boltzmann distribution law in equation (4), T the temperature, R the universal gas constant, the energy of the level i, and g, its degeneracy ... 

From a( j=0,n- -n ), the BCRLM rate coefficient can be obtained by Maxwell-Boltzmann averaging over these cross sections. As is shown in the Chapters by Bowman, Walker and Poliak, a more realistic rotation-ally averaged BCRLM (RBCRLM) rate coefficient is obtained by Maxwell-Boltzmann averaging the rotationally accumulated cross section a(n- n ) over translational energy, and dividing by the rotational partition function... 

In thermal equilibrium the partition of the total energy into the different modes is governed by the Maxwell-Boltzmann distribution, so that the probability p q) that a mode contains the energy qhv is... 

The calculation of the heat capacity from the partition function is nearly always a rather difficult task which can be mastered only in approximation. The reason has to be seen in the fact that the mechanical many-body problem can be solved only in approximation and that often even in rather sbnple cases no closed expression for the partition function can be found (see. e. g. Section II.3.4). Considerable simplifications can only be reckoned with if the Maxwell-Boltzmann statistics are valid [Eqs. (57) and (68)] and the Hanultonian of the system can be split in additive terms which are independent of each other. 

The Maxwell-Boltzmann Statistics, 282. The Fermi-Dirac Statistics, 285. The Bose-Einstein Statistics, 287. Relation of Statistical Mechanics to Thermodynamics, 289. Approximate Molecular Partition Functions, 292. An Alternative Formulation of the Distribution Law, 296. 

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