Maxwell-Boltzmann kinetic theory

James Clerk Maxwell first conceived the statistical basis of the second law of thermodynamics in 1871, and he is considered the founder of statistical thermodynamics. He was also famous for his electromagnetic wave theory and kinetic theory of gases. Maxwell derived the Maxwell Relations which were mathematical formulations used for advanced study of thermodynamics. Ludwig Boltzmann continued with Maxwell s kinetic theory of gases and arrived at important conclusions regarding dissipation of energy and increase in entropy. 

Translational energy, which may be directly calculated from the classical kinetic theory of gases since the spacings of these quantized energy levels are so small as to be negligible. The Maxwell-Boltzmann disuibution for die kinetic energies of molecules in a gas, which is based on die assumption diat die velocity specuum is continuous is, in differential form. 

The standard theories of chemical kinetics are equilibrium theories in which a Maxwell-Boltzmann distribution of reactants is postulated to persist during a reaction.68 The equilibrium theory first passage time is the TV -> oo limit in Eq. (6), Corrections to it then are to be expected when the second term in this equation is no longer negligible, i.e., when N is not much greater than e — e- )-1. The mean first passage time and rate of activation deviate from their equilibrium value by more than 10% when... 

Hendrik Antoon Lorentz, from Leyden (Holland), presided the conference, whose general theme was the Theory of Radiation and the Quanta. The conference5 was opened with speeches by Lorentz and Jeans, one on Applications of the Energy Equipartition Theorem to Radiation, the other on the Kinetic Theory of Specific Heat according to Maxwell and Boltzmann. In their talks, the authors explored the possibility of reconciling radiation theory with the principles of statistical mechanics within the classical frame. Lord Rayleigh, in a letter read to the... 

In the kinetic theory of gases, the molecules are assumed to be smooth, rigid, and elastic spheres. The only kinetic energy considered is that from the translational motion of the molecules. In addition, the gas is assumed to be in an equilibrium state in a container where the gas molecules are uniformly distributed and all directions of the molecular motion are equally probable. Furthermore, velocities of the molecules are assumed to obey the Maxwell-Boltzmann distribution, which is described in the following section. 

We now proceed to develop a specific expression for the rate constant for reactants where the velocity distributions /a( )(va) and /B(J)(vB) for the translational motion are independent of the internal quantum state (i and j) and correspond to thermal equilibrium.4 Then, according to the kinetic theory of gases or statistical mechanics, see Appendix A.2.1, Eq. (A.65), the velocity distributions associated with the center-of-mass motion of molecules are the Maxwell-Boltzmann distribution, a special case of the general Boltzmann distribution law ... 

The previously described theory in its original form assumes that the classical kinetic theory of gases is applicable to the electron gas, that is, electrons are expected to have velocities that are temperature dependent according to the Maxwell-Boltzmann distribution law. But, the Maxwell-Boltzmann energy distribution has no restrictions to the number of species allowed to have exactly the same energy. However, in the case of electrons, there are restrictions to the number of electrons with identical energy, that is, the Pauli exclusion principle consequently, we have to apply a different form of statistics, the Fermi-Dirac statistics. 

Criticism of the Stosszahlansatz and its corollaries arose as soon as it was recognized as paradoxical that the completely reversible gas model of the kinetic theory was apparently able to explain irreversible processes, i.e., phenomena whose development shows a definite direction in time. These nonstationary,51 irreversible processes were brought into the center of interest by the //-theorem of Boltzmann. In order to show that every non-Max-wellian distribution always approaches the Maxwell distribution in time, this theorem synthesizes all the special irreversible processes (like heat conduction and... 

In 1738, Daniel Bernoulli derived Boyle s Law from Newton s laws of motion applied to gas molecules. This derivation was the basis for an extensive mathematical development of the kinetic-molecular theory more than a century later by Clausius, Maxwell, Boltzmann, and others. Although we do not need to study the detailed mathematical presentation of this theory, we can gain some insight into its concepts from the reasoning behind Bernoulli s derivation. Here we present that reasoning based on proportionality arguments. 

More quantitative results have been obtained by Prigogine16 and co-workers, who adopted a kinetic method of approach and who treated this problem by the modern methods of the kinetic theory of gases. The integro-differential Maxwell-Boltzmann equation was extended to the case of inelastic collisions to get the velocity distribution functions /y, in terms of which the reaction rate may be written... 

The classical rate theory due to Arrhenius proceeds on the Maxwell-Boltzmann distribution of the velocity, and thereby the kinetic energy, of molecules or particles their average kinetic energy equals (2/2)k-QT. If two molecules collide with a kinetic energy larger than an activation energy Ea for a reaction between them to proceed, they are assumed to react. The... 

The findings of Maxwell, Boltzmann, and others resulted in a number of generalizations about gas behavior that have since been known as the kinetic molecular theory of gases, or simply the kinetic theory of gases. Central to the kinetic theory are the following assumptions ... 

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