Maxwell molecules

Further evaluation of the integrals for the general case of a v-power law will be discussed in Section 1.17 the calculation for v = 5, the Maxwell molecule force law, gives typical results, and will be completed here. 

The collision integral for Eq. (1-87) may be developed in the same manner and yields for Maxwell molecules ... 

Eqs. (1-76) show that these values of the coefficients produce the Navier-Stokes approximations to pzz and qz [see Eq. (1-63)] the other components of p and q may be found from coefficient equations similar to Eq. (1-86) and (1-87) (or, by a rotation of coordinate axes). The first approximation to the distribution function (for this case of Maxwell molecules) is ... 

This is formally true only for interactions between particles involving hard-sphere potentials. In fact, under these circumstances, the interacting particles perceive each others presence only at the point of contact. When the interactions are governed by a smooth potential, theoretically collisions should not be treated as point processes. Very often, however, even in these cases collisions are described as discontinuous processes occurring at the shortest distance between interacting particles, as in the case of the Maxwell molecules, as explained in the original work of Maxwell (1867). Eor more details readers are referred to Chapter 6. 

The values of the dimensionless parameters 2 and CO for the most classic collision models are given in Table 1. The Maxwell molecules (MM) model assumes a linear relationship between viscosity and temperature, although for the hard sphere (HS) model, the viscosity is proportional to the square root of the temperature. These models could be roughly considered as limits for the real behavior of gases, and the variable hard sphere (VHS) model proposed by Bird [2] is much more accurate. Another sophistication has been proposed by Koura and Matsumoto who developed the variable soft... 

Most of the previous developments have proceeded from Maxwell s boundary conditions which are known now to be in error. WALDMANN and co-workers have derived thermodynamically consistent boundary conditions for higher-order constitutive equations. For monatomic gases and Maxwell molecules, these constitutive equations reduce to Grad s 13 moment equations. However, WALDMANN has pointed out that Grad s boundary conditions are thermodynamically inconsistent. For the drag force problem, VESTNER and WALDMANN derived... 

For the thermal-force problem VESTNER and WALDMANN [2.103] find the following expression for the thermal force using thermodynamically consistent boundary conditions and constitutive relations which reduce for monatomic gases and Maxwell molecules to Grad s 13 moment equations... 

The kinetic theory of gases has had a long and rich history.t In its modem form, kinetic theory begins with the work of D. Bernoulli, Qausius, and most importantly Maxwell, who first used statistical methods to compute the properties of gases, recognizing that the random motions of the gas molecules could be best described by a distribution function. Besides giving the form of this distribution function for a gas at equilibrium, Maxwell derived equations for the transport of mass, momentum, and energy in a dilute gas. For a gas composed of so-called Maxwell molecules, which interact with repulsive forces... 

The spectrum of L(V) depends on the intermolecular potential of the gas molecules. For hard-sphere molecules the spectrum of L has a discrete and a continuous part, with the continuum ranging from — m to —00 with m a positive constant. If the potential is repulsive and varies with r as (r) = Kr with 5 >2, then the spectrum is discrete. For the special case s = 4, the so-called Maxwell molecules, the eigenvalues and eigenfunctions of L are completely known. However, nothing is known about the spectrum of L... 

Then, after inserting these expressions in the integral equations (102a) and (102b) we obtain an infinite set of equations for determining the coefficients and br. In the case of Maxwell molecules only and bo do not vanish, since the... 

Since Eq. (138a) is a linear equation for it is much easier to solve than the nonlinear equation. In spite of this simplification, it is still difficult to produce explicit solutions to this equation unless the operator L has a simple form and the geometry of the boundaries is simple enough. The operator L can be characterized by its sp>ectrum, and only for Maxwell molecules is this spectrum known explicitly. For this case the spectrum is discrete and the corresponding eigenfunctions can be expressed in terms of Sonine polynomials and spherical harmonics in y/ 8-52.6i-64) however, even for this special potential, it is still difficult to solve the linearized Boltzmann equation. 

In the special case of Maxwell molecules with uniform shear flow one can obtain an exact set of kinetic equations for the average position and velocity, as well as their fluctuations, from Eqs. (l)-(3). 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

The Maxwell-Boltzmann distribution function (Levine, 1983 Kauzmann, 1966) for atoms or molecules (particles) of a gaseous sample is... 

The Maxwell-Boltzmann velocity distribution function resembles the Gaussian distribution function because molecular and atomic velocities are randomly distributed about their mean. For a hypothetical particle constrained to move on the A -axis, or for the A -component of velocities of a real collection of particles moving freely in 3-space, the peak in the velocity distribution is at the mean, Vj. = 0. This leads to an apparent contradiction. As we know from the kinetic theor y of gases, at T > 0 all molecules are in motion. How can all particles be moving when the most probable velocity is = 0 ... 

As stated earlier, within C(t) there is also an equilibrium average over translational motion of the molecules. For a gas-phase sample undergoing random collisions and at thermal equilibrium, this average is characterized by the well known Maxwell-Boltzmann velocity distribution ... 

Examining transition state theory, one notes that the assumptions of Maxwell-Boltzmann statistics are not completely correct because some of the molecules reaching the activation energy will react, lose excess vibrational energy, and not be able to go back to reactants. Also, some molecules that have reacted may go back to reactants again. 

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. 

It is one of the wonders of the history of physics that a rigorous theory of the behaviour of a chaotic assembly of molecules - a gas - preceded by several decades the experimental uncovering of the structure of regular, crystalline solids. Attempts to create a kinetic theory of gases go all the way back to the Swiss mathematician, Daniel Bernouilli, in 1738, followed by John Herapath in 1820 and John James Waterston in 1845. But it fell to the great James Clerk Maxwell in the 1860s to take... 

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... 

Using cm as unit surface and seconds as unit time, n is the number of molecules falling on 1 cm /sec. The number n thus denotes the number of molecules striking each cm of the surface every second, and this number can be calculated using Maxwell s and the Boyle-Gay Lussac equations. The number n is directly related to the speed of the molecules within the system. It is important to realize that the velocity of the molecules is not dependent on the pressure of the gas, but the mean free path is inversely proportional to the pressure. Thus ... 

Worse was to come. Boltzmann in 1872 made the same weird statistical equality hold for every mode in a dynamical system. It must, for example, apply to any internal motions that molecules might have. Assuming, as most physicists did by then, that the sharp lines seen in the spectra of chemical elements originate in just such internal motions, any calculation now of Cp/C would yield a figure even lower than 1.333. Worse yet, as Maxwell shatteringly remarked to one student, equipartition must apply to solids and liquids as well as gases Boltzmann has proved too much. ... 

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