Gases molecule-wall collisions

Bulk diffusion that are significant for large pore sizes and high system pressures in which gas molecule-molecule collisions dominate over gas molecule-wall collisions. 

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. 

The Stefan-Maxwell equations have been presented for the case of a gas in the absence of a porous medium. However, in a porous medium whose pores are all wide compared with mean free path lengths it is reasonable to guess that the fluxes will still satisfy relations of the Stefan-Maxwell form since intermolecular collisions still dominate molecule-wall collisions. 

Table 4.2 gives decomposition lifetimes extracted from the flow tube data. Keep in mind that these kinetic data are for conditions in which a combination molecule-buffer gas, molecule-molecule, and molecule-wall collisions occur. The wall surface in the flow tube was quartz, and the slight discoloration observed indicates that there was initially some decomposition on the walls. Note, however, that there was no build-up of material on the walls beyond the initial transparently thin carbonaceous coating, and no products (e.g., polymer fragments) were observed that might be expected from reaction on the walls, or from bimolecular reactions. It appears that the decomposition is dominated by true unimolecu-lar reactions however, collisions with the walls are undoubtedly important in energizing the molecules for dissociation. 

A collision between a gas molecule and a surface sometimes leads to a heterogeneous reaction. We will obtain an expression for the rate of molecule-wall collisions. 

Viscous flow of a pure fluid (Kn 1). The gas acts as a continuum fluid driven by a pressure gradient, and both molecule-molecule collisions and molecule-wall collisions are important. This is sometimes called convective or bulk flow. 

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. 

Basic mechanisms involved in gas and vapor separation using ceramic membranes are schematized in Figure 6.14. In general, single gas permeation mechanisms in a porous ceramic membrane of thickness depend on the ratio of the number of molecule-molecule collisions to that of the molecule-wall collisions. In membranes with large mesopores and macropores the separation selectivity is weak. The number of intermolecular collisions is strongly dominant and gas transport in the porosity is described as a viscous flow that can be quantified by a Hagen-Poiseuille type law ... 

In a gas sample the molecules are very far apart and do not attract one another significantly. Each kind of gas molecule acts independently of the presence of the other kind. The molecules of each gas thus collide with the walls with a frequency and vigor that do not change even if other molecules are present (Figure 12-11). As a result, each gas exerts a partial pressure that is independent of the presence of the other gas, and the total pressure is due to the sum of all the molecule-wall collisions. 

To describe the combined bulk and Knudsen diffusion flrrxes the dusty gas model can be used [44] [64] [48] [49]. The dusty gas model basically represents an extension of the Maxwell-Stefan bulk diffusion model where a description of the Knudsen diffusion mechanisms is included. In order to include the Knudsen molecule - wall collision mechanism in the Maxwell-Stefan model originally derived considering bulk gas molecule-molecule collisions only, the wall (medium) molecules are treated as an additional pseudo component in the gas mixture. The pore wall medium is approximated as consisting of giant molecules, called dust, which are uniformly distributed in space and held stationary by an external clamping force. This implies that both the diffusive ffrrx and the concentration gradient with respect to the dust particles vanish. 

When the mean free path length of the molecule is large compared with the diameter of the pore and molecule-wall collisions dominate over molecule-molecule collision, the flow is called Knudsen diflPusion. Molecules of different species move entirely independent of each other. This occurs at low density of the gas (low pressure) or in fine pores. 

In this type of flow the mean free path of the molecule is very small compared to the pore diameter and molecule-molecule collision dominates over molecule-wall collision. The mixture becomes like a single gas and the driving force for the continuum fluid is the pressure gradient. 

The Knudsen number, Kn, is frequently used to characterize different transport regimes in terms of the ratio of X to the critical apparatus dimension, L (Kn = X/L) (Stechelmacher, 1986). The dimension L is typically chosen such that Kn represents the ratio of intermolecular to molecule-wall collision frequencies. For gas flow between the source and the substrate regions, L is the length of the transport chamber or the... 

Holt et al. [16] measured water and gas flow through the pores of double-walled carbon nanotubes. These tubes had inner diameters less than 2 nm with nearly defect-free graphitic walls. Five hydrocarbon and eight non-hydrocarbon gases were tested to determine flow rates and to demonstrate molecular weight selectivity compared with helium. Water flow was pressure driven at 0.82 atm and measured by following the level of the meniscus in a feed tube. The results for both gas and liquid show dramatic enhancements over flux rates predicted with continuum flow models. Gas flow rates were between 16 and 120 times than expected according to the Knudsen diffusion model in which fluid molecule-wall collisions dominate the flow. 

Thermodynamic macroscopic properties or coordinates are derived from the statistical long time averaging of the observable microscopic coordinates of motion. For example, the pressure we measure is an average over about 10 " molecule-wall collisions per second per square centimeter of surface for a gas at standard conditions. If a thermodynamic property is a state function, its change is independent of the path between the initial and final sfafes, and depends only on the properties of the initial and final states of the system. The infinitesimal change of a state function is an exact differential. 

A microscopic theory may be developed by using a calculational scheme based on following the trajectories (position and velocity) of each molecule in the system. At each molecule-molecule or molecule-wall collision, new trajectories would have to be computed. Such calculations can be performed for limited number of molecules and short periods of time. Such calculations yield the probability distribution of particle velocities or kinetic energies. For example, the temperature of a monoatomic gas could then be computed from the average kinetic energy. Therefore, statistical thermodynamics determine probability distributions and average values of properties when considering all possible states of the molecules consistent with the constraints on the overall system. 

In Fig. 7.6-la a gas molecule A at partial pressure at the entrance to a capillary is diffusing through the capillary having a diameter of d m. The total pressure P is constant throughout. The mean free path X is large compared to the diameter d. As a result, the molecule collides with the wall and molecule-wall collisions are important. This type is called Knudsen diffusion. 

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The pressures exerted by gases demonstrate molecular motion. Gases are collections of molecules, so the pressure exerted by a gas must come from these molecules. Just as the basketball In Figure 2J exerts a force when it collides with a backboard, moving gas molecules exert forces when they collide with the walls of their container. The collective effect of many molecular collisions generates pressure. 

The second part of Figure 14-1 shows a molecular view of what happens in the two bulbs. Recall from Chapter 5 that the molecules of a gas are in continual motion. The NO2 molecules in the filled bulb are always moving, undergoing countless collisions with one another and with the walls of their container. When the valve between the two bulbs is opened, some molecules move into the empty bulb, and eventually the concentration of molecules in each bulb is the same. At this point, the gas molecules are in a state of dynamic equilibrium. Molecules still move back and forth between the two bulbs, but the concentration of molecules in each bulb remains the same. 

The first possibility is that the attractive potential associated with the solid surface leads to an increased gaseous molecular number density and molecular velocity. The resulting increase in both gas-gas and gas-wall collision frequencies increases the T1. The second possibility is that although the measurements were obtained at a temperature significantly above the critical temperature of the bulk CF4 gas, it is possible that gas molecules are adsorbed onto the surface of the silica. The surface relaxation is expected to be very slow compared with spin-rotation interactions in the gas phase. We can therefore account for the effect of adsorption by assuming that relaxation effectively stops while the gas molecules adhere to the wall, which will then act to increase the relaxation time by the fraction of molecules on the surface. Both models are in accord with a measurable increase in density above that of the bulk gas. 

Molecules in a gas are in constant motion at speeds on the order of the speed of a rifle bullet at equilibrium there is no net flow of gas and the motion is random. This motion produces collisions of the molecules with the walls of the vessel containing the gas, with a change in momentum of the gas molecule resulting from each collision. This change in momentum produces a force per unit area, or pressure on the wall. Consider those molecules with the component of velocity in the x direction between the value of vx and vx + dvx. The x direction is defined as the direction normal to the wall. The fraction of molecules with the x component of velocity in this range, denoted dN(vx)/N, is given by the density function, f(vx), where... 

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