Statistical thermodynamics Maxwell-Boltzmann distribution

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. 

Ludwig Boltzmann (1844-1906), the Austrian physicist, is famous for his outstanding contributions to heat transfer, thermodynamics, statistical mechanics, and kinetic theory of gases. Boltzmann was a student of Josef Stefan and received his doctoral degree in 1866 under his supervision. The Stefan-Boltzmann law (1884) for black body radiation is the result of the associated work of Josef Stefan and Boltzmann in the field of heat transfer. Boltzmann s most significant works were in kinetic theory of gases in the form of Maxwell-Boltzmann distribution and Maxwell-Boltzmann statistics in classical statistical mechanics. 

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). 

In practice, EPR samples consist of collections of many paramagnetic species, and not single isolated paramagnetic centers. If the population of radicals is in thermodynamic equilibrium, its statistical distribution is described by the Maxwell-Boltzmann equation... 

The zeroth moment of a distribution is 1, the first moment is < i>, the second moment is < P>, etc. The higher moments of a distribution hence compute successively higher averages of the distributions of the independent variable for example, in classical statistical thermodynamics the mean square velocity is the second moment of the Maxwell-Boltzmann speed distribution for an ideal gas, and is directly related to average kinetic energy < KE > = m < v >/2, and hence to temperature [= 3k TI2 for a monatomic gas]. 

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. 

---