Wall collisions

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

We are going to carry out some spatial integrations here. We suppose that tire distribution function vanishes at the surface of the container and that there is no flow of energy or momentum into or out of the container. (We mention in passing that it is possible to relax this latter condition and thereby obtain a more general fonn of the second law than we discuss here. This requires a carefiil analysis of the wall-collision temi The interested reader is referred to the article by Dorfman and van Beijeren [14]. Here, we will drop the wall operator since for the purposes of this discussion it merely ensures tliat the distribution fiinction vanishes at the surface of the container.) The first temi can be written as... 

Many optical studies have employed a quasi-static cell, through which the photolytic precursor of one of the reagents and the stable molecular reagent are slowly flowed. The reaction is then initiated by laser photolysis of the precursor, and the products are detected a short time after the photolysis event. To avoid collisional relaxation of the internal degrees of freedom of the product, the products must be detected in a shorter time when compared to the time between gas-kinetic collisions, that depends inversely upon the total pressure in the cell. In some cases, for example in case of the stable NO product from the H + NO2 reaction discussed in section B2.3.3.2. the products are not removed by collisions with the walls and may have long residence times in the apparatus. Study of such reactions are better carried out with pulsed introduction of the reagents into the cell or under crossed-beam conditions. 

It turns out that the CSP approximation dominates the full wavefunction, and is therefore almost exact till t 80 fs. This timescale is already very useful The first Rs 20 fs are sufficient to determine the photoadsorption lineshape and, as turns out, the first 80 fs are sufficient to determine the Resonance Raman spectrum of the system. Simple CSP is almost exact for these properties. As Fig. 3 shows, for later times the accuracy of the CSP decays quickly for t 500 fs in this system, the contribution of the CSP approximation to the full Cl wavefunction is almost negligible. In addition, this wavefunction is dominated not by a few specific terms of the Cl expansion, but by a whole host of configurations. The decay of the CSP approximation was found to be due to hard collisions between the iodine atoms and the surrounding wall of argons. Already the first hard collision brings a major deterioration of the CSP approximation, but also the role of the second collision can be clearly identified. As was mentioned, for t < 80 fs, the CSP... 

When a pure gas flows through a channel the accompanying fall in pressure is accounted for partly by acceleration of the flowing stream and partly by momentum transfer to the stationary walls. Since a porous medium may be regarded as an assembly of channels, similar considerations apply to flow through porous media, but in the diffusional situations of principal interest here accelerational pressure loss can usually be neglected. If more than one molecular species is present, we are also interested in the relative motions of the different species, so momentum transfers by collisions between different types of molecules are also important. 

Now encounters between molecules, or between a molecule and the wall are accompanied by momentuin transfer. Thus if the wall acts as a diffuse reflector, molecules colliding wlch it lose all their axial momentum on average, so such encounters directly change the axial momentum of each species. In an intermolecuLar collision there is a lateral transfer of momentum to a different location in the cross-section, but there is also a net change in total momentum for species r if the molecule encountered belongs to a different species. Furthermore, chough the total momentum of a particular species is conserved in collisions between pairs of molecules of this same species, the successive lateral transfers of momentum associated with a sequence of collisions may terminate in momentum transfer to the wall. Thus there are three mechanisms by which a given species may lose momentum in the axial direction ... 

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

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. 

The first form of aerosol modifier is a spray chamber. It is designed to produce turbulent flow in the argon carrier gas and to give time for the larger droplets to coalesce by collision. The result of coalescence, gravity, and turbulence is to deposit the larger droplets onto the walls of the spray chamber, from where the deposited liquid drains away. Since this liquid is all analyte solution, clearly some sample is wasted. Thus when sensitivity of analysis is an issue, it may be necessary to recycle this drained-off liquid back through the nebulizer. 

Alternatively, ions of any one selected m/z value can be chosen by holding the magnetic field steady at the correct strength required to pass only the desired ions any other ions are lost to the walls of the instrument. The selected ions pass through the gas cell and are detected in the singlepoint ion collector. If there is a pressure of a neutral gas such as argon or helium in the gas cell, then ion-molecule collisions occur, with decomposition of some of the selected incident ions. This is the MS/MS mode. However, without the orthogonal TOF section, since there is no further separation by m/z value, the new ions produced in the gas cell would not be separated into individual m/z values before they reached the detector. Before the MS/MS mode can be used, the instrument must be operated in its hybrid state, as discussed below. 

As a result, collisions with the wall occur more frequently than with other molecules. This is referred to as the Knudsen mode of diffusion and is contrasted with ordinary or bulk diffusion, which occurs by intermolecular collisions. At intermediate pressures, both ordinaiy diffusion and Knudsen diffusion may be important [see Eqs. (5-239) and (5-240)]. 

Vacuum Flow When gas flows under high vacuum conditions or through very small openings, the continuum hypothesis is no longer appropriate if the channel dimension is not very large compared to the mean free path of the gas. When the mean free path is comparable to the channel dimension, flow is dominated by collisions of molecules with the wall, rather than by colhsions between molecules. An approximate expression based on Brown, et al. J. Appl. Phys., 17, 802-813 [1946]) for the mean free path is... 

Diffusivity and tortuosity affect resistance to diffusion caused by collision with other molecules (bulk diffusion) or by collision with the walls of the pore (Knudsen diffusion). Actual diffusivity in common porous catalysts is intermediate between the two types. Measurements and correlations of diffusivities of both types are Known. Diffusion is expressed per unit cross section and unit thickness of the pellet. Diffusion rate through the pellet then depends on the porosity d and a tortuosity faclor 1 that accounts for increased resistance of crooked and varied-diameter pores. Effective diffusion coefficient is D ff = Empirical porosities range from 0.3 to 0.7, tortuosities from 2 to 7. In the absence of other information, Satterfield Heterogeneous Catalysis in Practice, McGraw-HiU, 1991) recommends taking d = 0.5 and T = 4. In this area, clearly, precision is not a feature. 

Radicals destroyed by collision with the walls of the reactor... 

---