Maxwell-Boltzmann statistics, 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. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

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 frequency with which the transition state is transformed into products, iT, can be thought of as a typical unimolecular rate constant no barrier is associated with this step. Various points of view have been used to calculate this frequency, and all rely on the assumption that the internal motions of the transition state are governed by thermally equilibrated motions. Thus, the motion along the reaction coordinate is treated as thermal translational motion between the product fragments (or as a vibrational motion along an unstable potential). Statistical theories (such as those used to derive the Maxwell-Boltzmann distribution of velocities) lead to the expression ... 

In the general approach to classical statistical mechanics, each particle is considered to occupy a point in phase space, i.e., to have a definite position and momentum, at a given instant. The probability that the point corresponding to a particle will fall in any small volume of the phase space is taken proportional to die volume. The probability of a specific arrangement of points is proportional to the number of ways that the total ensemble of molecules could be permuted to achieve the arrangement. When this is done, and it is further required that the number of molecules and their total energy remain constant, one can obtain a description of the most probable distribution of the molecules in phase space. Tlie Maxwell-Boltzmann distribution law results. 

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. 

In most physical applications of statistical mechanics, we deal with a system composed of a great number of identical atoms or molecules, and are interested in the distribution of energy between these molecules. The simplest case, which we shall take up in this chapter, is that of the perfect gas, in which the molecules exert no forces on each other. We shall be led to the Maxwell-Boltzmann distribution law, and later to the two forms of quantum statistics of perfect gases, the Fermi-Dirac and Einstein-Bose statistics. 

We can show, as we did with the Fermi-Dirac statistics, that the distribution (6.5) approaches the Maxwell-Boltzmann distribution law at high temperatures. It is no easier to make detailed calculations with the Einstein-Bose law than with the Fermi-Dirac distribution, and on account of its smaller practical importance we shall not carry through a detailed... 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

Statistical mechanics is a theory which discusses probabilities and distributions and so is relevant to a discussion of situations like that of the ionic atmosphere, and the Maxwell-Boltzmann distribution wiU feature heavily in the theoretical development. 

Step 3 involved smearing out of the discrete ions of the ionic atmosphere into a continuous cloud of charge so that the Poisson equation could be used. The problem in this step, however, is to find the actual distribution of the discrete ions of the ionic atmosphere around the central y -ion. Statistical mechanics in the form of the Maxwell-Boltzmann distribution is used. The derivation of the Maxwell-Boltzmann distribution automatically involves an implicit averaging process, but this statistical mechanical averaging is different from that used for the Poisson equation when smearing out has been done. 

What is perhaps important to emphasise is that the statistical mechanical derivation does not involve the use of the Poisson equation and hence the treatment does not involve the ambiguities involved in the combination of the Maxwell-Boltzmann distribution with the Poisson equation. 

The Debye-Hiickel theory is a study of the equilibrium properties of electrolyte solutions, where departures from ideal behaviour are considered to be a result of coulombic interactions between ions in an equilibrium situation. It is for this reason that equilibrium statistical mechanics can be used to calculate an equilibrium Maxwell-Boltzmann distribution of ions. 

Boltzmann s first significant contribntion to physics was the generalization of James Clerk Maxwell s distribntion of velocities and energies for a sample of gaseons atoms. Althongh Maxwell had deduced this distribution, he provided no physical basis for it. Boltzmann showed that as atoms move toward equilibrium they assume the Maxwell distribution—later known as the Maxwell-Boltzmann distribution—and further that this is the only statistically possible distribution for a system at equilibrium. 

The equilibrium state for a gas of monoatomic particles is described by a spatially uniform, time independent distribution function whose velocity dependence has the form of the Maxwell-Boltzmann distribution, obtained from equilibrium statistical mechanics. That is,/(r,v,0 has the form/ (v) given by... 

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 distribution does not apply and the behaviour of the particles is governed by quantum statistics. Systems in which the particles are degenerate are called degenerate gases examples are the conduction electrons in a metal (or degenerate semiconductor), the electrons In a white dwarf, and the neutrons In a neutron star. See also degeneracy pressure. 

For the case n = 30, TF is shown as a function of r in Fig. 16. W(r)r dr, like the well-known Maxwell-Boltzmann distribution curve, is thus asymmetrical it also has greater breadth on account of the smaller number of statistically independent elements. The result is that we cannot regard the most probable value as the only one likely to be present, as we would in the ordinary statistical treatment of gases. We must, rather, consider neighboring values and attach much greater significance to fluctuation phenomena than in gas statistics. We shall return later to this point, when the occurrence of x-ray interferences in stretched rubber are discussed. 

This relation is known as Boltzmann distribution (or Maxwell-Boltzmann distribution) because it is classically established on statistical arguments. 

The Gaussian distribution is derived from the binomial distribution for large N [5]. It is important for statistics, error analysis, diffusion, conformations of polymer chains, and the Maxwell Boltzmann distribution law of gas velocities. 

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