Mean free path of gases

Figure 2.36 shows the effect of the ratio r/k on the relative proportions of Knudsen to Poiseuille flow in a cylindrical capillary [57], When r/k is greater than one, Poiseuille flow predominates. Because the mean free path of gases at atmospheric pressure is in the range of 500-2000 A, for Knudsen flow to predominate and a separation to be obtained, the membrane pore radius must be less than 500 A. 

The preceding section assumed that the mass-transport mechanism in a fluid medium is dominated by molecule-molecule collisions. However, the mean free path of gases often exceeds the dimensions of small pores typical of solid catalysts. In this situation, called Knudsen diffusion, molecules collide more often with the pore walls than with other molecules. According to Equation (6.3.1), the Knudsen diffu-sivity of component A, D a, is proportional to r / , but is independent of both pressure and the presence of other species ... 

A criterion for determining whether high vacuum conditions exist is that the mean free path of gases in a system must be of the same order of magnitude as the dimensions of the system, or larger. 

In the case where the diameter of the hole in the diffusion layer is much larger than the mean free path of gases, ordinary diffusion dominates and the limiting current is expressed by Eq. (31). 

Table 8. Mean Free Path of Gases at O C and 1 Atmospheric Pressure [237]...

The mean free path, A is another fundamental scale necessary for describing the dynamical properties of gases. Mean free path is defined as the average distance traveled by the molecule between two successive collisions. Kinetic theory of gases establishes the following two expressions for the mean free path of gases. 

The molecular diffusivity D may be expressed in terms of the molecular velocity um and the mean free path of the molecules Xrn. In Chapter 12 it is shown that for conditions where the kinetic theory of gases is applicable, the molecular diffusivity is proportional to the product umXm. Thus, the higher the velocity of the molecules, the greater is the distance they travel before colliding with other molecules, and the higher is the diffusivity D. 

In the previous section, the molecular basis for the processes of momentum transfer, heat transfer and mass transfer has been discussed. It has been shown that, in a fluid in which there is a momentum gradient, a temperature gradient or a concentration gradient, the consequential momentum, heat and mass transfer processes arise as a result of the random motion of the molecules. For an ideal gas, the kinetic theory of gases is applicable and the physical properties p,/p, k/Cpp and D, which determine the transfer rates, are all seen to be proportional to the product of a molecular velocity and the mean free path of the molecules. 

In bulk diffusion, the predominant interaction of molecules is with other molecules in the fluid phase. This is the ordinary kind of diffusion, and the corresponding diffusivity is denoted as a- At low gas densities in small-diameter pores, the mean free path of molecules may become comparable to the pore diameter. Then, the predominant interaction is with the walls of the pore, and diffusion within a pore is governed by the Knudsen diffusivity, K-This diffusivity is predicted by the kinetic theory of gases to be... 

For polyatomic gases in porous media, however, the relaxation rate commonly decreases as the pore size decreases [18-19]. Given that the relaxation mechanism is entirely different, this result is not surprising. If collision frequency determines the Ti, then in pores whose dimensions are in the order of the typical mean free path of a gas, the additional gas-wall collisions should drastically alter the T,. For typical laboratory conditions, an increase in pressure (or collision frequency) causes a proportional lengthening of T1 so the change in T, from additional wall collisions should be a good measure of pore size. 

Fig. 1.2. Mean free path of some gases as a function of pressure. 

UPS studies of supported catalysts are rare. Griinert and coworkers [45] recently explored the feasibility of characterizing polycrystalline oxides by He-II UPS. A nice touch of their work is that they employed the difference in mean free path of photoelectrons in UPS, V 2p XPS and valence band XPS (below 1 nm, around 1.5 nm, and above 2 nm, respectively) to obtain depth profiles of the different states of vanadium ions in reduced V205 particles [45]. However, the vast majority of UPS studies concern single crystals, for probing the band structure and investigating the molecular orbitals of chemisorbed gases. We discuss examples of each of these applications. 

In situ characterization. Catalysts should preferably be investigated under the conditions under which they are active in the reaction. Various reasons exist why this may not be possible, however. For example, lattice vibrations often impede the use of EXAFS, XRD and Mossbauer spectroscopy at reaction temperatures the mean free path of electrons and ions dictates that XPS, SIMS and LEIS are carried out in vacuum, etc. Nevertheless, one should strive to choose the conditions as close as possible to those of the catalytic reaction. This means that the catalyst is kept under reaction gases or inert atmosphere at low temperature to be studied by EXAFS and Mossbauer spectroscopy or that it is transferred to the vacuum spectrometers under conditions preserving the chemical state of the surface. 

The species diffusivity, varies in different subregions of a PEFC depending on the specific physical phase of component k. In flow channels and porous electrodes, species k exists in the gaseous phase and thus the diffusion coefficient corresponds with that in gas, whereas species k is dissolved in the membrane phase within the catalyst layers and the membrane and thus assumes the value corresponding to dissolved species, usually a few orders of magnitude lower than that in gas. The diffusive transport in gas can be described by molecular diffusion and Knudsen diffusion. The latter mechanism occurs when the pore size becomes comparable to the mean free path of gas, so that molecule-to-wall collision takes place instead of molecule-to-molecule collision in ordinary diffusion. The Knudsen diffusion coefficient can be computed according to the kinetic theory of gases as follows... 

Continuum theory applies when the mean free path of the vapor A,- is small compared with the droplet radius, that is, when the Knudsen number Kn is small (Kn = A,/a 1). From the kinetic theory of gases (Jeans, 1954), the mean free path of the vapor in a binary system is given by... 

The other approach to liquid-state theory is to treat liquids as dense gases typical of high pressures, except that they have much higher densities and lower compressibility. In the van der Waals equation format, the value of (V - b)/V representing the free volume would be of the order of 10%. The mean free path of free flight between collisions becomes much shorter, and is comparable to the molecular diameters. 

The kinetic theory of gases has therefore given us a fairly simple equation for the diffusion coefficient of a molecule. All that remains to be determined is the mean velocity of the molecule and the mean free path of the molecule. 

According to the kinetic theory of gases, the self-diffusivity of a hard-sphere gas is given by DG = (2/5)(u)L, where (u) is the average velocity and L is the mean free path [4]. Because the mean free path of a confined particle in the liquid is about equal to the diameter of its confining volume, the contribution of the confined particle to the self-diffusivity of the liquid may be written... 

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