Gases Maxwellian velocity distribution

The physical condition of the kinetic theory of gases can be described by elastic collisions of monodispersed spheres with the Maxwellian velocity distribution in an infinite vacuum space. Therefore, for an analogy between particle-particle interactions and molecular interactions to be directly applicable, the following phenomena in gas-solid flows should not be regarded as significant in comparison to particle-particle interactions the gas-particle... 

From the preceding discussions it is evident that at least four different temperatures have to be considered which under laboratory conditions are all equal the excitation temperature Tex of the molecule, defined by the relative populations of the levels, the kinetic temperature Tk, corresponding to the Maxwellian velocity distribution of the gas particles, the radiation temperature Traa, assuming a (in some cases diluted) black body radiation distribution, and the grain temperature 7, . With no thermodynamic equilibrium established, as is common in interstellar space, none of these temperatures are equal. These non-equilibium conditions are likely to be caused in part by the delicate balance between the various mechanisms of excitation and de-excitation of molecular energy levels by particle collisions and radiative transitions, and in part by the molecule formation process itself. Table 7 summarizes some of the known distribution anomalies. The non-equilibrium between para- and ortho-ammonia, the very low temperature of formaldehyde, and the interstellar OH and H20 masers are some of the more spectacular examples. 

At any given instance, a number of chain molecules of equal length will have a random distribution of chain end-to-end distances. This information can be obtained from a derivation analogous to that used to derive the Maxwellian velocity distribution of molecules in an ideal gas. 

At thermal equilibrium, the molecules of a gas follow a Maxwellian velocity distribution. At the temperature T, the number of molecules ni(v-)dv in the level Ei per unit volume with a velocity component between v- and -h di is... 

This relationship should be familiar to the reader—it is the famous maxwellian velocity distribution formula for a gas at equilibrium. 

Aq is the mean free path for energy transfer no change in direction is implied. M is the mass of a rare gas atom. For zero electric field, fo(x) is given as exp(-x) and for f(p) the Maxwellian velocity distribution is obtained. 

In a series of impressive publications. Maxwell [95-98] provided most of the fundamental concepts constituting the statistical theory recognizing that the molecular motion has a random character. When the molecular motion is random, the absolute molecular velocity cannot be described deterministically in accordance with a physical law so a probabilistic (stochastic) model is required. Therefore, the conceptual ideas of kinetic theory rely on the assumption that the mean flow, transport and thermodynamic properties of a collection of gas molecules can be obtained from the knowledge of their masses, number density, and a probabilistic velocity distribution function. The gas is thus described in terms of the distribution function which contains information of the spatial distributions of molecules, as well as about the molecular velocity distribution, in the system under consideration. An important introductory result was the Maxwellian velocity distribution function heuristically derived for a gas at equilibrium. It is emphasized that a gas at thermodynamic equilibrium contains no macroscopic gradients, so that the fluid properties like velocity, temperature and density are uniform in space and time. When the gas is out of equilibrium non-uniform spatial distributions of the macroscopic quantities occur, thus additional phenomena arise as a result of the molecular motion. The random movement of molecules from one region to another tend to transport with them the macroscopic properties of the region from which they depart. Therefore, at their destination the molecules find themselves out of equilibrium with the properties of the region in which they arrive. At the continuous macroscopic level the net effect... 

Here, n is the number density of the gas atoms and /m is the Maxwellian velocity distribution function, Eq. (2.152). We notice that the spatial dependence of the contribution from the arriving atoms follows the spatial variation of the external field, whereas that is not the case for the contribution of the scattered atoms because of the exponential term in the curly brackets. In other words, the optical response of the gas near the surface is nonlocal. 

These latter measurements led only to relative cross-section values. However, by comparison with absolute values of velocity-averaged cross sections, they can be put on an absolute scale. To do this, the absolute values obtained in FA measurements were used because here the velocity distribution is exactly known—a Maxwellian distribution /(t>, T) with the temperature of the buffer gas. Denoting the velocity-dependent relative total ionization cross section, obtained in the beam experiment, by oKl(v) and the absolute total ionization rate constant obtained in the FA experiment by R(T), then a normalization k may be determined by... 

In this case no detailed collision kinetics are involved. The collision parameters rnm and the parameters in the Maxwellian post-collision velocity distribution f m are derived from experimentally determined gas viscosity or diffusivity, and the collisional invariants, respectively. Usually this term is negligible in present experiments, but exceptions exist [16]. In particular for ITER, and the high collisionality there, these terms are expected to become more relevant. However, due to the BGK-approximations made, their implementation into the models does not require further discussion here. 

E. Collision Frequency between Maxwellian Molecules. Finally, we can calculate the average number of collisions made by a molecule going through a Maxwellian gas if the molecule does not have a fixed velocity V, but has instead a velocity distribution which is itself Maxwellian. This may be done by multiplying Zc [Eq. (VII.8D.4)] by the Maxwellian distribution function and averaging over all values of Vx ... 

In order to calculate the flow we must know something about the distribution of molecular velocities in the gas. Since the gas is not at equilibrium but only in a steady state, we cannot say that we have an equilibrium distribution. However we can make the approximation of assuming that the velocity distribution is flocally Maxwellian, i.e., that the molecules at any given point distant Z from the fixed plate have the normal distribution of velocities with respect to an average which is not zero but is given by the macroscopic stream velocity at that point. Thus at a point Z from the fixed plate the distribution is to be taken as... 

If the gas is in equilibrium then the velocity distribution function is given by the absolute Maxwellian... 

For flow with high Knudsen number, the number of molecules in a significant volume of gas decreases, and there could be insufficient number of molecular collisions to establish an equilibrium state. The velocity distribution function will deviate away from the Maxwellian distribution and is non-isotropic. The properties of the individual molecule then become increasingly prominent in the overall behavior of the gas as the Knudsen number increases. The implication of the larger Knudsen number is that the particulate nature of the gases needs to be included in the study. The continuum approximaticui used in the small Knudsen number flows becomes invalid. At the extreme end of the Knudsen number spectrum is when its value approaches infinity where the mean free path is so large or the dimension of the device is so small that intermolecular collision is not likely to occur in the device. This is called collisionless or free molecular flows. 

From statistical mechanics, it follows that temperature is weU defined when the velocity distribution is Maxwellian. Systems for which this condition is fulfilled are complex reactions where the rate of elastic collisions is larger than the rate of reactive collisions. This is generally true for reactions in not too rarefied media and for many biological and transport processes. It may be noted that molecular collisions are responsible for attainment of Maxwellian distribution. Normally, significant deviations from the Maxwellian distribution are observed only under extreme conditions. Distribution is perturbed when physical processes are very rapid. Thus for a gas, local equilibrium assumption would not be valid when the relative variation of temperature is no longer small within a length equal to mean free path. 

Spectral lamps that emit discrete spectra are examples of nonthermal radiation sources. In these gas-discharge lamps, the light-emitting atoms or molecules may be in thermal equilibrium with respect to their translational energy, which means that their velocity distribution is Maxwellian. However, the population of the different excited atomic levels may not necessarily follow a Boltzmann distribution. There is generally no thermal equilibrium between the atoms and the radiation field. The radiation may nevertheless be isotropic. 

It should be emphasized again, that, for a gas-fluidized system, the results based on the two Maxwellian distributions, which correspond to the EMMS-featured dilute and dense phases, respectively, are the same as those on the bimodal distribution. In greater detail, the total velocity distribution reads... 

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