Mean free path apparent

Since aerosol particles are continually undergoing molecular bombardment, their paths are smooth curves rather than segments of straight lines. It still is possible to define an apparent mean free path for the aerosol particles (Fuchs, 1964). This is the distance traveled by an average particle before it changes its direction of motion by 90°. The apparent mean free path represents the distance traveled by an average particle in a given direction before particle velocity in that direction equals zero. But this is just the stop distance. 

Example 9.7 Compute the apparent mean free paths for unit-density spheres of 0.01-, 0.1-, and 1-pm diameter. Assume T = 20°C. 

Estimate the apparent mean free path of 0.1-pm. unit-density spheres in air at 20°C and 760-mmHg pressure. 

What is the significance (if any) of the observation that the apparent mean free path goes through a minimum value at about 0.2 p,m ... 

FIGURE 9.11 A two-dimensional projection of the path of (a) an air molecule and (b) the center of a 1-pm particle. Also shown is the apparent mean free path of the particle. 

Certain quantities associated with the Brownian motion and the dynamics of single aerosol particles are shown as a function of particle size in Table 9.5. All tabulated quantities in Table 9.5 depend strongly on particle size with the exception of the apparent mean free path A,p, which is of the same order of magnitude right down to molecular sizes, with atmospheric values Xp 10-60nm. 

It is apparent from the values of ps that diffusional charging is primarily effective with very small particles, as might be expected. With particles as large as 10 microns the charge level produced in any reasonable time is small compared with what can be achieved with corona charging. For the 0.1-micron particle, however, the charge level attained by diffusion is of the same order and may be larger if the mean free path correction is allowed for. 

For a nuclide of mass M, abundance sensitivity is the ratio between the signal at mass M+ arising from the same species to the signal at mass M. Off-peak ions are present because of collisions behind the magnetic filter, of reflections on the tube wall, or of space-charge effects. As a result of the collisions, the energy of these ions is different from the energy of the main beam. They alter the apparent peak baseline in a continuous way. Abundance sensitivity decreases with the mean free path of ions, i.e., when pressure near the collector assembly... 

The second most apparent limitation on studies of surface reactivity, at least as they relate to catalysis, is the pressure range in which such studies are conducted. The 10 to 10 Torr pressure region commonly used is imposed by the need to prevent the adsorption of undesired molecules onto the surface and by the techniques employed to determine surface structure and composition, which require relatively long mean free paths for electrons in the vacuum. For reasons that are detailed later, however, this so-called pressure gap may not be as severe a problem as it first appears. There are many reaction systems for which the surface concentration of reactants and intermediates found on catalysts can be duplicated in surface reactivity studies by adjusting the reaction temperature. For such reactions the mechanism can be quite pressure insensitive, and surface reactivity studies will prove very useful for greater understanding of the catalytic process. 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

C104 salt did obey KR approximately. We concluded [23] that there was no qualitative change in magnetoresistance even though the (inelastic) mean free path became as low as 1/250 of a lattice parameter in the c direction. This is the experimental basis for the remarks in Section III about the possible applicability of the Boltzmann formulas even when the inelastic mean free path perpendicular to the chains is very small, and the apparent absence of a minimum metallic conductivity in the transverse directions. 

First reports on the observation of superconductivity of UPt3 already mentioned the extreme sensitivity of this superconducting state to any shortening of the electronic mean free path (22). The superconducting state of UBeia is apparently somewhat more stable but it is also clear that non-magnetic impurities have quite a drastic influence as well. In fig. 8 we show the temperature dependence of Cp below 1 K (24 ) of pure UBeia and of material doped with small amounts of non magnetic impurities introduced on the U sites. For UBei3 we observe a distinct anomaly of Cp at Tq as is expected. 

The great interest of the molecular polaron model is that it can account for the apparent contradiction between the band-like temperature-dependent mobility and a small mean-free path. The weak point of the approach developed by Silinsh and coworkers is that it is phenomenological. It is worth mentioning that a more consistent analytical theory has been formulated by Kenkre and coworkers [22]. The presentation of this model would go to far outside the scope of this chapter and will not be developed here. 

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