Mean-free-path, dependence

The rate of dissociation increases rapidly above 2000°C. It also increases with decreasing pressure.P" ] The rate of recombination (i.e., the formation of the molecule) is rapid since the mean-free-path dependent half-life of atomic hydrogen is only 0.3 sec. 

Ceramic membrane is the nanoporous membrane which has the comparatively higher permeability and lower separation fector. And in the case of mixed gases, separation mechanism is mainly concerned with the permeate velocity. The velocity properties of gas flow in nanoporous membranes depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collision. The Knudsen number Kn Xydp is characteristic parameter defining different permeate mechanisms. The value of the mean free path depends on the length of the gas molecule and the characteristic pore diameter. The diffusion of inert and adsorbable gases through porous membrane is concerned with the contributions of gas phase diffusion and sur u e diffusion. 

For gas molecules, the heat capacity is a constant equal to C = (n/2)pkB where n is the number of degrees of freedom for molecule motion, p is the number density, and kB is the Boltzmann constant. The rms speed of molecules is given as v = V3kBTlm, whereas the mean free path depends on collision cross section and number density as = (pa)-1. When they are put together, one finds that the thermal conductivity of a gas is independent of p and therefore independent of the gas pressure. This is a classic result of kinetic theory. Note that this is valid only under the assumption that the mean free path is limited by inter-molecular collision. 

The velocity relevant for transport is the Fermi velocity of electrons. This is typically on the order of 106 m/s for most metals and is independent of temperature [2], The mean free path can be calculated from i = iyx where x is the mean free time between collisions. At low temperature, the electron mean free path is determined mainly by scattering due to crystal imperfections such as defects, dislocations, grain boundaries, and surfaces. Electron-phonon scattering is frozen out at low temperatures. Since the defect concentration is largely temperature independent, the mean free path is a constant in this range. Therefore, the only temperature dependence in the thermal conductivity at low temperature arises from the heat capacity which varies as C T. Under these conditions, the thermal conductivity varies linearly with temperature as shown in Fig. 8.2. The value of k, though, is sample-specific since the mean free path depends on the defect density. Figure 8.2 plots the thermal conductivities of two metals. The data are the best recommended values based on a combination of experimental and theoretical studies [3],... 

As the temperature is increased, electron-phonon scattering becomes dominant. The mean free path for such scattering varies as 71"" with n larger than unity. The mean free path of electrons at room temperature is typically on the order of 100 A. The mean free path depends on the material but is independent of the sample, since electron-phonon scattering is an intrinsic process. As a result of electron-phonon scattering, thermal conductivity of metals decreases at higher temperatures. 

It will seen from Eqs. (10.1) and (10.2) that the value of the mean free path depends on the value of the radius of the nucleus. The reciprocal of the mean free path, K, the attenuation coefficient, (sometimes referred to in the literature as an absorption coefficient or an opacity coefficient) for a proton inside a nucleus is ... 

Now consider k, -. it depends upon the mean free path of the dissolved gas in the liquid and the mean free path depends upon liquid density. We can, therefore, consider Atl to be constant within the variability of liquid density. Thus, 1/A l is the slope of the resistance equation describing a semi-batch process. In summary, all the proposed functions relating k a to a variety of independent variables are actually functions relating l to those same independent variables. This knowledge allows us to refocus our effort to improve a semibatch process, i.e., to improve Aroveraii-... 

The slip flow near the boundary surface can be analyzed based on the type of fluids, i.e., gas and Newtonian and non-Newtonian liquids. The sUp flow in gases has been derived based on Maxwell s kinetic theory. In gases, the concept of mean free path is well defined. Slip flow is observed when characteristic flow length scale is of the order of the mean free path of the gas molecules. An estimate of the mean free path of ideal gas is /m 1/(Vlna p) where p is the gas density (here taken as the number of molecules per unit volume) and a is the molecular diameter. The mean free path / , depends strongly on pressure and temperature due to density variation. Knudsen number is defined as the ratio of the mean free path to the characteristic length scale... 

The continuum model assumes continuous and indefinitely divisible matter. For gases the continuum model is valid when the mean free pathlength of the molecules K is much smaller than the characteristic length of the flow I. The mean free path, depending on pressure and temperature of a gas molecule modeled as a rigid sphere. 

Randrianalisoa and Baillis (2008) used Monte Carlo simulation to model heat conduction in porous Si. In their method, an original 3D pore network that reproduces the morphology of mesoporous Si was developed. The nonlinear phonon dispersion curves of Si and the phonon mean-free path dependent on temperature, frequency, and polarization were also considered. The model of steady-state phonon transport through the pore network was simulated. Their results were compared with experimental results of porous Si thin films on a p" -type Si substrate. 

Note that in Eq. (3.34), Aioverain kc> and ki are constants, k denotes the reaction rate that occurs in unit time, which depends upon the pressure and temperature at which we operate the semibatch reactor. Since, for most reactions, we operate semibatch reactors at constant pressure and temperature, Atrxd does not change, ka and Atl depend upon the mean free path of a molecule through the gas phase and through the liquid phase, respectively. Both mean free paths depend upon the density of their respective phases. If the phase densities are constant, then kc and ki will be constant. Therefore, any fluctuation in /coveraii must arise from fluctuations in ac and... 

A lrf depend upon the mean free path of a molecule in the liquid and the mean free path depends upon the liquid density, which, in our case, we assume is constant. Therefore, any fluctuations in Aioveraii must arise from fluctuations in brf or ulrf or both. 

The mean free path depends on the ratio T/p. What value of T/p is necessary for the mean free path of He (d = 65 pm) to equal 130 pm How feasible do you think It will be to attain this ratio ... 

The temperature dependence of the upper critical field in the presence of RE-impurities is determined by the following pair-breaking mechanisms a) the effect of the magnetic field acting on the electron orbits. This leads to Abrikosov flux lines in type II superconductors. This effect is mean free path dependent ... 

Because a set of binding energies is characteristic for an element, XPS can analyse chemical composition. Almost all photoelectrons used in laboratory XPS have kinetic energies in the range of 0.2 to 1.5 keV, and probe the outer layers of tire sample. The mean free path of electrons in elemental solids depends on the kinetic energy. Optimum surface sensitivity is achieved with electrons at kinetic energies of 50-250 eV, where about 50% of the electrons come from the outennost layer. 

A more accurate calculation will account for differences in the energy dependent mean free paths of the elements and for the transmission characteristics of the electron analyser (see [7]). 

Using cm as unit surface and seconds as unit time, n is the number of molecules falling on 1 cm /sec. The number n thus denotes the number of molecules striking each cm of the surface every second, and this number can be calculated using Maxwell s and the Boyle-Gay Lussac equations. The number n is directly related to the speed of the molecules within the system. It is important to realize that the velocity of the molecules is not dependent on the pressure of the gas, but the mean free path is inversely proportional to the pressure. Thus ... 

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