Mean free path, definition

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

For a flying height around 2 nm, collisions between the molecules and boundary have a strong influence on the gas behavior and lead to an invalidity of the customary definition of the gas mean free path. This influence is called a "nanoscale effect" [46] and will be discussed more specifically in Chapter 6. 

As described above, the magnitude of Knudsen number, Kn, or inverse Knudsen number, D, is of great significance for gas lubrication. From the definition of Kn in Eq (2), the local Knudsen number depends on the local mean free path of gas molecules,, and the local characteristic length, L, which is usually taken as the local gap width, h, in analysis of gas lubrication problems. From basic kinetic theory we know that the mean free path represents the average travel distance of a particle between two successive collisions, and if the gas is assumed to be consisted of hard sphere particles, the mean free path can be expressed as... 

The correction of mean free path, hi by the nanoscale effect function results in a smaller mean free path, or a smaller Knudsen number in other word. As a matter of fact, a similar effect is able to be achieved even if we use the conventional definition of mean free path, / = irSn, and the Chapmann-Enskog viscosity equation, /r = (5/16)... 

The change from a viscous to a molecular flow regime occurs when the mean free path L of the gas molecules in the system exceeds the minimum physical dimensions of the system. The mean free path is a measure of the average distance a molecule travels between collisions. The derivation of L involves a number of assumptions about the ideality of the gas and the nature of the collisions and by definition some 63.2% of the molecules in a particular gas collide with other molecules within the distance L. The mean free path for any gas can be calculated from Equation (1.1)... 

Here, k is Boltzmann s constant and mp particle mass. In analogy to a simple kinetic theory of gases, the definition of a mean free path for particles is Ap = vpsi l. The average thermal speed of particle is... 

Calculations of collisions between molecules of a liquid have been made but the postulates on which they rest are not fully established. In fact it is not easy to define a collision between molecules of a liquid or between a solute molecule and a solvent molecule. In gases collision is pictured as a clean-cut process like the collision and rebound of two billiard balls, but in solution the solute molecule is always in contact with a solvent molecule and one might well consider a collision between them as a continuing or sticky collision. The frequency of collision and the mean free path are indefinite. We have no clear picture nor definition and it is not surprising that the mathematical formulas proposed are unsatisfactory. Collisions of one solute molecule with another solute molecule, however, seem to be capable of exact description, at least in some cases. 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

Chemical ionization [5] consists of producing ions through a collision of the molecule to be analysed with primary ions present in the source. Ion-molecule collisions will thus be induced in a definite part of the source. In order to do so, the local pressure has to be sufficient to allow for frequent collisions. We saw that the mean free path could be calculated from Equation 1 (see the Introduction). At a pressure of approximately 60 Pa, the free path is about 0.1 mm. The source is then devised so as to maintain a local pressure of that magnitude. A solution consists of introducing into the source a small box about 1 cm along its side as is shown in Figure 1.4. 

XANES Spectroscopy. In the XANES region the photoelectron has, by definition, low kinetic energy. Because low-energy electrons have a long mean free-path and are strongly scattered by the surrounding... 

At any time, a particle may be considered to be moving in a specific direction with a velocity v - /8kT/-nm. From a definition of the stop distance, the pseudo mean free path lB is... 

The definition applies to electrons in metals / is the mean free path, and vF is the electron velocity on the Fermi sphere. 

The following symbols are used in the definitions of the dimensionless quantities mass (m), time (t), volume (V area (A density (p), speed (u), length (/), viscosity (rj), pressure (p), acceleration of free fall (p), cubic expansion coefficient (a), temperature (T surface tension (y), speed of sound (c), mean free path (X), frequency (/), thermal diffusivity (a), coefficient of heat transfer (/i), thermal conductivity (/c), specific heat capacity at constant pressure (cp), diffusion coefficient (D), mole fraction (x), mass transfer coefficient (fcd), permeability (p), electric conductivity (k and magnetic flux density ( B) ... 

Corresponding to E, there exist a critical radius r,., determining the critical volume around the anode wire, within which avalanche formation can take place. At the distance r from the centre, the potential is V. The mean free path, between ionization is, by definition... 

It should, however, be borne in mind that the depth sensitivity varies with the element. Moreover, in the case of a heterogeneous sample, the response to the elements present varies dramatically depending on their depth an clement solely present in the first atomic layer will be seen more than the same element present in greater proportion at a depth of several times the mean free path. This highlights an ambiguity in the definition of the notion of surface which poses significant problems when interpreting quantitative results. 

These definitions rely upon the molecular mean free path in a single-phase flow, surface tension effects and flow patterns in two-phase flow applications. In recent studies in minichannels the hydraulic diameter ranges from 100 /jm to 2-3 mm. The channel cross sections were either circular or rectangular and much of the research concerned boihng. Commonly, classical correlations have been used with or without modifications to predict flow boihng results in minichannels. However agreement was poor and the need for new correlations was evident. 

We therefore call Ithe mean free path of the ray in the gas (Clausius, 1858). The present author, together with E. Bormann, has shown (1921) how it can be determined on the basis of its definition by the exponential law, by measuring the attenuation of a beam of silver atoms on passage through the gas which is at rest (air). The more important case is that in which beam and gas consist of molecules of the same kind the mean free path I is then a property of this gas. 

The scattering of polaritons by disorder in ID structures is very important for the interpretation of the results. The disorder strongly influences the states in ID structures making all states weakly or strongly localized (34). Any imperfections in the chain replace the ID wire by a collection of finite boxes. The sizes of the boxes depend on the disorder and can be different. The strong localization by definition is localization of polaritons with mean free path small in comparison with the polariton wavelength. For such localized states the wavevector is no... 

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