Mean free path of molecules

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). 

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... 

In a shock wave, as is known, a change of state occurs on the mean free path of molecules of the gas. The outward analogy between the theory of detonation and the shock wave theory prompted many authors to consider that a detonation front is just as sharp as the front of a shock wave. Jouguet spoke in favor of instantaneous reaction. 

Figure 7. Ratio of mean free path of molecules to diameter of nozzle along length of nozzle for zinc source. (Reproduced with permission from reference 7. Copyright 1985 American Institute of Physics.)...

Wave length of monochromatic light Mean free path of molecules Mean free path of particles in Brownian motion Capillary conductivity (Gardner)... 

The Mean Free Paths of Molecules. There is an interesting dependence of the rate of effusion of a gas through a small hole on the molecular weight of the gas. The speeds of motion of different molecules are inversely proportional to the square roots of their molecular weights. If a small hole is made in the wall of a gas container, the gas molecules will pass through the hole into an evacuated region outside at a rate determined by the speed at which they are moving (these speeds determine the probability that a molecule will strike the hole). 

As the pore dimension decreases (down to fractions of a micron) or the mean free path of molecules increases, which can be achieved by lowering the pressure or raising the temperature, the molecules collide more frequently with the pore walls of the membrane than with one another. When the so-called Knudsen flow (Fig. 9) is achieved. 

The value of A is determined as follows. Under standard atmospheric conditions the mean free path of molecules by the order of magnitude is /free l/(lVaOg), where NA = 2.68 x 1019cm-3 is the number of molecules per cubic centimeter and erg 10 Mcm2 is the mean cross section of molecules collision. For pressure in the range 1-3 bar this gives /free 10 6cm. 

This process governs the rate of deposition of the molecules of nonvolatile compounds on the surface of gas ducts, and contributes to broadening of the chromatographic zones. Being of the order of 0.1 pm at STP, the mean free path of molecules, which is inversely proportional to pressure, reaches 1 cm only at about 0.01 mmHg. In dense enough gas, in the absence of convective flow, the macroscopic picture of migration of molecules (as well as of aerosol particulates) is described by the equations of diffusion. The mean squared diffusional displacement z2D of molecules, the time of diffusion t and the mutual diffusion coefficient >i 2 are related by ... 

As the particle size dp reduces and approaches the region where the fluid loses its continuum (mean free path of molecules = X), Stokes law needs to corrected by the Cunningham correction factor, Co-... 

Another feature of aerosols is related to the fact that the particle size of dispersed matter, r, is comparable to the mean free path of molecules in a gas, Am. The choice of a particular method used to adequately describe the particle motion depends on the ratio between r and AM, which is given by the Knudsen number, Kn=AM/2r [1-4]. When Kn<10 2 one can apply the laws of mechanics (aerodynamics) of continuous media, and in particular the Stokes law, eq. (V.4) ... 

Fig. 4.2. A schematic showing the presence of gas at (a) higher and (b) lower pressures. The pressure is a function of the number of collisions with the container by the gas molecules and the average force that each collision carries. The molecules also collide with each other, and the mean free path of molecules in the gas is also a function of the gas molecule density (and therefore pressure).

Fortunately, it is not necessary to trace the complex individual paths which molecules are forced to take in diffusing down a pore. It will be sufficient to determine the net rate of flow of molecules past a given cross-section of the pore when the concentration gradient at this point is known. The particular form of this rate equation depends on three factors (a) The magnitude of the pore radius as compared to the length of the mean free path of molecules between intermolecular collisions, (b) The presence or absence of total pressure differences along the pore which can lead to a mass flow of molecules into or out of the pore, (c) Under certain specialized conditions it appears the presence of physically adsorbed layers of the pore wall may affect the rate of transport via two dimensional surface diffusion. 

Moreover, there must be some lower limit of microchannel hydraulic diameter beyond which the fluid behavior begins to deviate from behavior observed at the macroscale. At some point, the characteristic length scale of the flow will become so small that the continuum hypothesis must break down. However, the mean free path of molecules in liquid flows in microchannels is on the order of nanometers, and thus, this lower limit is likely to be significantly smaller than 1 pm. Experiments will have to be performed to determine what this lower limit is and also to determine the characteristics of flow in this regime. 

If the size of the bubble reactor is small compared with the mean free path of molecules in the circulation flow, the diffusion supply of reagents cannot be a limiting factor in the kinetics of synthesis. All these factors together are the reasons for the activation of chemical processes, and control particle size in the range of a few nanometers, i.e. one order of magnitude smaller than that without cavitation. 

Mean free path of molecules Molecular length Efficiency of heat exchanger Efficiency factor Heaviside function Molecular time scale Dimensionless residence time... 

Knudsen diffusion [20,30,32,37 ] depending on gas pressure and mean free path in the gas phase applies to pores between 10 A and 500 A in size however, there are examples in the literature where it was observed for much larger pores [41 ]. In this region, the mean free path of molecules in gas phase A is much larger than the pore diameter d. It is common to use the so-called Knudsen number K = X d o characterize the regime of permeation through pores. When 1, viscous (PoiseuiUe) flow is realized. The condition for Knudsen diffusion is 1. An intermediate regime is realized when A), 1. 

Qe is the energy transferred per imit total area of the particle normal to the direction of heat transfer. The effective thermal conductivities of catalyst pellets are remarkably low because of the pore structure. The contribution of the thermal conductivity of the solid skeleton is little, since the extremely small heat transfer areas existing at solid-solid contact points offer substantial resistance to heat transfer. The gas phase filling the void spaces in the pores also participates in hindering heat conduction experimental results indicate that decreases as Gp increases. At low pressures, when the mean free path of molecules is greater than or equal to pore size, increases with total pressure since free-molecule conduction starts to dominate. There are no general correlations for predicting Ae from the physical properties of the solid and fluid phases involved. An approximate correlation based on the thermal conductivities of the individual phases and the porosity of the particle has been proposed ... 

The Effusion and Diffusion of Gases the Mean Free Paths of Molecules... 

Modern gas-diffusion medium in low-temperature fuel cells is typically a highly porous carbon paper with porosity in the range of sgdl = 0.6-0.8 and with the mean pore radius in the order of 10 pm (10 cm). By the order of magnitude, the mean free path of molecules in atmospheric pressure air is = l/(A LO-fci ), where Nl = 2.686 10 cm is the Loschmidt number (number of molecules in a cubic centimetre of atmospheric pressure gas at standard temperature) and akin — 10 cm is the molecular cross-section for kinetic collisions. With this data we get 3 10 cm, or 3 10 pm. Obviously, mean pore radius in the GDL is nearly 3 orders of magnitude greater than I f and the physical mechanism of molecule transport is binary molecular diffusion. 

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