Maxwellian molecules

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 ... 

The thermal diffusion factor a is proportional to the mass difference, (mi — mo)/(mi + m2). The thermal diffusion process depends on the transport of momentum in collisions between unlike molecules. The momentum transport vanishes for Maxwellian molecules, particles which repel one another with a force which falls off as the inverse fifth power of the distance between them. If the repulsive force between the molecules falls off more rapidly than the fifth power of the distance, then the light molecule will concentrate in the high temperature region of the space, while the heavy molecule concentrates in the cold temperature region. When the force law falls off less rapidly than the fifth power of the distance, then the thermal diffusion separation occurs in the opposite sense. The theory of the thermal diffusion factor a is as yet incomplete even for classical molecules. A summary of the theory has been given by Jones and Furry 15) and by Hirschfelder, Curtiss, and Bird 14), Since the thermal diffusion factor a for isotope mixtures is small, of the order of 10", it remained for Clusius and Dickel (8) to develop an elegant countercurrent system which could multiply the elementary effect. 

The Maxwellian molecules are useful in exploratory calculations in which a differentiable potential function is needed. For these molecules [66] the in-termolecular force between pairs at a distance r apart is of the form where K is a constant. Adopting this particular potential representation the solution of the equation of transfer reduces to a feasible problem, thereby Maxwell [66] obtained analytical expressions for the transport coefficients as mentioned earlier. 

The collision number we are seeking may now be obtained by direct substitution into equation (1-8) using Cr instead of c. For the total number of collisions between Maxwellian molecules A and B, the collision number, B) is determined by... 

To simulate the slow-pressure phenomena, a stochastic mixing and reorientation model can be used. It describes the dissipation and the observed tendency of the Reynolds stresses to become isotropic. The stochastic mixing model is similar to that used for the scalar dissipation, whereas the reorientation process is described in analogy with collisions between Maxwellian molecules, i.e., conserving both momentum and energy. Pope [1985] developed an alternative model for the conditional acceleration term, based on the generalized Langevin model. 

Maxwell found that he could represent the distribution of velocities statistically by a function, known as the Maxwellian distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was... 

The flow of gas molecules striking the hole area is simply the product of the hole area A and the intensity of the Maxwellian stream given by Eq. (43). However, since the gas samples are characterized by different pressures, Pi and l, as well as by different temperatures, T and T2, the flow of gas sample 1 is not equal to the flow of gas sample 2, and there exists a net flow through the hole,... 

However, because of the Maxwellian distribution function for molecular speeds, not all H2 molecules will be faster than all 02 molecules and some H2 molecules will have velocities near 0 m/s. Some 02 molecules will be moving faster than the average speed of H2 molecules. 

It should be noted that a Maxwellian form of fie) is a reasonable approximation to the actual distribution at low electron energies. This observation is indicated in Figure 5 i24). However, the first ionization potential of most atoms and molecules is above eV. Thus, many of the important homogeneous processes that occur in glow discharges, such as ionization, take place as a result of high energy electrons in the "tail" of the distribution. These electrons are precisely the ones that are not adequately described by a Maxwellian distribution function. 

A sufficiently rarefied gas, or a mixture of gases, consists of a number of neutral molecules of species 1 and 2 (which may or may not be the same). We may assume a distribution of velocities (measured in the laboratory frame), fi ( ) d3u, that may be modeled by a Maxwellian distribution function, with i = 1 or 2, as long as the duration of the average collision is short compared to the time between collisions. For binary collisions, one usually transforms from laboratory coordinates, Vj, to relative ( >12) and center-of-mass (1>cm) velocities,... 

The random velocities of atoms and molecules are described by velocity distribution functions which can often be approximated by a Maxwellian distribution (as in Eq. 2.10). If radiating atoms have such a distribution, the resulting line profile is a Gaussian,... 

The mechanism by which equilibrium is attained can only be visualized in terms of microscopic theories. In the kinetic sense, equilibrium is reached in a gas when collisions among molecules redistribute the velocilies lor kinetic energies) of each molecule until a Maxwellian distribution is reached for the whole bulk. In the case of the trend toward equilibrium for two solid bodies brought into physical contact, we visualize the transfer of energy by means of free electrons and phonons (lattice vibrations). 

All the experiments described in this chapter were performed on molecules in gas cells, and, unlike in beam experiments, the results obtained are averaged over all possible orientations of the molecules with respect to the relative velocity vector, and over the Maxwellian distribution of the relative velocities of the colliding particles. Nevertheless, the experimental data stimulated interest in the development of theoretical models of elastic... 

The relations for Z and z2 require that the gas phase behaves ideally (an excellent approximation for the conditions of interest), and that the dimensions of the catalytic surface be small relative to the mean free path we also require that the distribution of velocities be Maxwellian. The rate of adsorption of molecules per unit area of surface is s2Z2. We recall that... 

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