Mean free path estimation

For the systems considered the gradients are very small. Thus, in any region of the gas, concentration, temperature, and mean velocity are well-defined experimental quantities. There is no objection, either in principle or practice, to inserting a probe which can measure the local values of these macroscopic parameters (as, for example, one might measure temperature at different points in a room). As the gas is not at equilibrium the velocity distribution is positionally dependent but the mean molecular properties change little over distances comparable with the mean free path (estimated in Section 2.2 to be between 30 and 300 nm for a gas at 300 K and 1 atm). In any region, extending over many mean free paths, the properties of the gas can be characterized by the local values of the macroscopic observables. 

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. 

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. 

In small pores and at low pressures, the mean free path of the gas molecule (or atom) is significantly greater than the pore diameter Jpore- Rs magnitude may be estimated from... 

The work on gas theory had many extensions. In 1865 Johann Josef Loschmidt used estimates of the mean free path to make the first generally accepted estimate of atomic diameters. In later papers Maxwell, Ludwig Boltzmann, and Josiah Willard Gibbs extended the rrratherrratics beyorrd gas theory to a new gerreralized science of statistical mechanics. Whenjoined to quantum mechanics, this became the foundation of much of modern theoretical con-derrsed matter physics. 

With the knowledge of g, we can estimate the inverse mean free path of a phonon with frequency co. As done originally within the TLS model, the quantum dynamics of the two lowest energies of each tunneling center are described by the Hamiltonian //tls = gcTz/2 + Aa /2. This expression, together with Eqs. (15) and (17), is a complete (approximate) Hamiltonian of... 

Based on the attenuation of the iron 2p signal and assuming a mean free path for the iron electrons of 1.5 to 2 nanometers, it is estimated that the carbon overlayer is at least 1.8 to 2.5 nano-... 

The deposition rate increases upon increasing the pressure. This is explained by noting that the impingement rate per unit area, r,, of molecules on the filament is linearly dependent on the pressure as r, = pj 2nksT, with the gas temperature. However, as the pressure becomes higher, the collisional mean free path of the silane becomes smaller, and the silane supply to the filaments becomes restricted. Moreover, the transport of deposition precursors to the substrate is restricted as well. The mean free path of silane was estimated to be 2.5 cm at a pressure of 0.02 mbar [531]. i.e.. the mean free path about equals the distance between filament and substrate. Indeed, a maximum in deposition rate is observed at this pressure. This corresponds to a value of pdk of 0.06 (cf. [530]). The microstructure parameter plotted as a function of pd has a minimum around Ms = 0.06 0.02 [530]. 

In high-mobility liquids, the quasi-free electron is often visualized as having an effective mass m different fron the usual electron mass m. It arises due to multiple scattering of the electron while the mean free path remains long. The ratio of mean acceleration to an external force can be defined as the inverse effective mass. Often, the effective mass is equated to the electron mass m when its value is unknown and difficult to determine. In LRGs values of mVm 0.3 to 0.5 have been estimated (Asaf and Steinberger,1974). Ascarelli (1986) uses mVm = 0.27 in LXe and a density-dependent value in LAr. 

One can go a step further and use the /P//s ratio for a quantitative estimate of the dispersion. Through the years, several methods have been proposed to predict XPS intensity ratios for supported catalysts. Angevine et al. [29] modeled their catalyst with crystallites on top of a semi-infinite support, as sketched in Fig. 3.9a. However, as the inelastic mean free path of, for example, Si02 is 3.7 nm, photoelectrons coming from particles inside pores as deep as 10 nm below the surface still contribute to the XPS signal and the assumption of a semi-infinite support is probably too simple. Indeed, the model predicts /P//s ratios that may be a factor of 3 too high [30],... 

Thus, using the estimated turbulent mean free path in the atmosphere of 500 m, the above criterion gives Ei s 10% if z 0.913/ 450 m. 

The highly toxic pesticide, methyl bromide or CHsBr, is diffusing through air. Estimate the mean velocity and mean free path of the methyl bromide molecules at an atmospheric pressure of 1 atm and a room temperature of 20° C. 

The mean free path X, of a molecule in air can be calculated from the sizes of the molecules involved. The most probable collision partners for a trace molecule (such as CFC-12) in air are molecular nitrogen (N2) and oxygen (02). The trace molecule i is hit whenever its center gets closer to the center of an air molecule than the critical distance, rcrit = r, + rair (Fig. 18.8). Picturing the molecules as spheres, the molecular radius r, can be estimated from the collision cross-section A listed in chemical handbooks such as the Tables of Physical and Chemical Constants (Longman, London, 1973) ... 

Note that in this procedure the effect of molecular mean free path, that is, of molecular size is neglected. As an example we estimate diffusivity of toluene (Mk) uene = 92 gmol-1) from diffusivity of benzene (Afbenzene = 78 gmol-1) and get Dtoluene a (0.096 cm s-1) [92/ 78]-1/2 = 0.088 cm s-1. The experimental value is 0.086 cm s 1 (Gilliland, 1934). 

Illustrative Example 18.2 Estimating Molecular Diffusivity in Air Problem Estimate the molecular diffusion coefficient in air, >,a, ofCFC-12 (see Illustrative Example 18.1) at 25°C (a) from the mean molecular velocity and the mean free path, (b) from the molar mass, (c) from the molar volume, (d) from the combined molar mass and molar volume relationship by Fuller (Eq. 18-44), (e) from the molecular diffusivity of methane. 

The validity of our description can be checked in Fig. 3 where we compare the actual position of the acquired spectra to the position extracted from the fit, in units of the relevant characteristic length. The data points follow a monotonous behavior, but with a significant scattering. From the slope of the mean line we can draw through the data points on the N side (top part of Fig. 3), we can extract the value = 94 nm. This corresponds exactly to the estimation based on the gap A and the measured mean free path of 16 nm in Au. On the S side (bottom part of Fig. 3), the estimated length is s = 50 nm. Taking into account the reduced gap A, it corresponds to a mean free path of 4.5 nm which is half the value estimated from transport properties of similar samples. In fact, it should be considered more as a property of the Nb-Au layer at its border than a property of the bare Nb film. 

The symbol A represents the estimated mean free path for photoelectrons of the appropriate energy. 

The ideal gas relation was derived under the assumption that each molecule travels undisturbed from wall to wall, which is certainly not true at common pressures and temperatures. To see this, we need to get some estimate of the mean free path k (the mean distance a molecule travels before it undergoes a collision), and the mean time between collisions. 

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