Mean Free Path Considerations

In addition to the failure of the free electron gas model to properly account for the observed temperature behavior of resistivity of metals, there are serious problems when other properties are considered. First, let us make a few simple calculations to illustrate one of its major difficulties. Multiplying the collision time of 2.5 x 10 s by the thermal velocity 1 x 10 m/s, one gets a mean free path of 2.5 nm, which is on the order of a few lattice spacing. This seems incredibly small, especially considering that Cu is one of the 

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If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Due to the dependence on mean free path as described in Eq. (4.40), the thermal conductivity of heterogeneous systems is impossible to predict on heat capacity alone. As in previous sections, we do know that disorder tends to decrease thermal conductivity due to mean free path considerations, and this is indeed the case for fillers with high thermal conductivities, such as copper and aluminum in epoxy matrices (see Table 4.12). The thermal conductivity of the epoxy matrix increases only modestly due to the addition of even high percentages of thermally conductive fillers. 

Let us consider the limiting case, namely the absence of collisions, assuming that icoi = 0 in (3.40), in other words that the mean free path considerably exceeds the characteristic diameter of the beam. 

The coupling of an ICP ionization source to a TOF-MS was first reported by Myers and Hieftje [24]. A schematic diagram of their instrument is provided as Fig. 12.9. Similarly to other ICP-MS instruments, it utilized a three-stage differentially pumped interface in order to realize the microtorr pressures required for TOF-MS from mean-free path considerations. This instrument was based on an orthogonal-extraction geometry and utilized a 1.6-m total flight path, which, at a 2-kV acceleration potential, dictated a maximum spectral repetition frequency of 20 kHz. 

E. Whalley and E. R. S. Winter, Trans. Faraday Soc., 46,517 (1950), have succeeded in obtaining an equation similar to Chapman s and derived from the simple mean free path considerations developed here. The work is based on the earlier theory of Fiirth, Proc. Roy. Soc. London), A179, 461 (1942), and permits evaluation of the coefficients kt in excellent agreement with experimental data. 

The strong point of AES is that it provides a quick measurement of elements in the surface region of conducting samples. For elements having Auger electrons with energies hr the range of 100-300 eV where the mean free path of the electrons is close to its minimum, AES is considerably more surface sensitive than XPS. 

Siace the pores ia an aerogel are comparable to, or smaller than, the mean free path of molecules at ambient conditions (about 70 nm), gaseous conduction of heat within them is iaefficient. Coupled with the fact that sohd conduction is suppressed due to the low density, a siUca aerogel has a typical thermal conductivity of 0.015 W/(m-K) without evacuation. This value is at least an order of magnitude lower than that of ordinary glass and considerably lower than that of CFC (chloro uorocarbon)-blown polyurethane foams (54). 

The functional relation ia equation 53 or 54 cannot be determined by dimensional analysis alone it must be suppHed by experiments. The significance is that the mean-free-path problem is reduced from an original relation involving seven variables to an equation involving only three dimensionless products, a considerable saving ia terms of the number of experiments required ia determining the governing equation. 

In the relations given earlier, it is assumed that the fluid can be regarded as a continuum and that there is no slip between the wall of the capillary and the fluid layers in contact with it. However, when conditions are such that the mean free path of the molecules of a gas is a significant fraction of the capillary diameter, the flowrate at a given value of the pressure gradient becomes greater than the predicted value. If the mean free path exceeds the capillary diameter, the flowrate becomes independent of the viscosity and the process is one of diffusion. Whereas these considerations apply only at very low pressures in normal tubes, in fine-pored materials the pore diameter and the mean free path may be of the same order of magnitude even at atmospheric pressure. 

We, therefore, decided to study the reaction of a molecular stream passing through an exhausted enclosure which is permeated by radiation corresponding to a temperature that would be sufficient under ordinary circumstances to produce chemical reaction. Since, in the molecular stream, the gas is at so low a concentration that the mean free path is considerably longer than the length of the stream, the number of collision is negligible. This method seems, therefore, to offer a direct means of investigating the reaction due to radiation alone. 

Laminar flow In circular tubes with parabolic velocity distribution Is known as Poiseuille flow. This special case is found frequently in vacuum technology. Viscous flow will generally be found where the molecules mean free path is considerably shorter than the diameter of the pipe X d. 

Thus, a susceptibility that depends on frequency and wave vector implies that the relation between P(x, t) and E(x, t) is nonlocal in time and space. Such spatially dispersive media lie outside our considerations. However, spatial dispersion can be important when the wavelength is comparable to some characteristic length in the medium (e.g., mean free path), and it is well at least to be aware of its existence it can have an effect on absorption and scattering by small particles (Yildiz, 1963 Foley and Pattanayak, 1974 Ruppin, 1975, 1981). 

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