Molecular travel mean free path

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. 

The change from a viscous to a molecular flow regime occurs when the mean free path L of the gas molecules in the system exceeds the minimum physical dimensions of the system. The mean free path is a measure of the average distance a molecule travels between collisions. The derivation of L involves a number of assumptions about the ideality of the gas and the nature of the collisions and by definition some 63.2% of the molecules in a particular gas collide with other molecules within the distance L. The mean free path for any gas can be calculated from Equation (1.1)... 

Maxwell distribution of molecular speeds The formula for calculating the percentage of molecules that move at any given speed in a gas at a specified temperature, mean free path The average distance that a molecule travels between collisions. 

The molecular diffusivity D must be replaced by an effective diffusivity De because of the complex internal structure of the catalyst particle which consists of a multiplicity of interconnected pores, and the molecules must take a tortuous path. The effective distance the molecules must travel is consequently increases. Furthermore, because the pores are very small, their dimensions may be less than the mean free path of the molecules and Knudsen diffusion effects may arise Equation 10.170 is solved in Volume 1 to give equation 10.199 for a catalyst particle in the form of a flat platelet... 

We have already likened the macroscopic transport of heat and momentum in turbulent flow to their molecular counterparts in laminar flow, so the definition in Eq. (5-60) is a natural consequence of this analogy. To analyze molecular-transport problems (see, for example. Ref. 7, p. 369) one normally introduces the concept of mean free path, or the average distance a particle travels between collisions. Prandtl introduced a similar concept for describing turbulent-flow phenomena. The Prandtl mixing length is the distance traveled, on the average, by the turbulent lumps of fluid in a direction normal to the mean flow. 

The mean free path is defined as the average distance a molecule will travel in a gas before it collides with another molecule. This is related to molecular spacing but takes into account the fact that all molecules are in a constant state of motion and thus are more widely separated than they would be if they were firmly bound to each other. Mean free path can be estimated by using the following simple argument. 

If a molecule travels 1 cm, it sweeps out an imaginary volume of ira2(l). With n molecules per unit volume, the number of molecules struck per centimeter is tto n, and the mean free path is then the reciprocal, or l/(mra2). If it is assumed that the molecular velocities are distributed according to maxwellian theory rather than having a sin-... 

Since aerosol particles are continually undergoing molecular bombardment, their paths are smooth curves rather than segments of straight lines. It still is possible to define an apparent mean free path for the aerosol particles (Fuchs, 1964). This is the distance traveled by an average particle before it changes its direction of motion by 90°. The apparent mean free path represents the distance traveled by an average particle in a given direction before particle velocity in that direction equals zero. But this is just the stop distance. 

Another very important concept in kinetic theory is the average distance a molecule travels between collisions—the so-called mean free path. On the basis of a very simple conception of molecular collisions, the following equation for the mean free path A can be derived ... 

LIQUID DIFFUSIVITIES. The theory of diffusion in liquids is not as advanced or the experimental data as plentiful as for gas diffusion. The diffusivities in liquids are generally four to five orders of magnitude smaller than in gases at atmospheric pressure. Diffusion in liquids occurs by random motion of the molecules, but the average distance traveled between collisions is less than the molecular diameter, in contrast to gases, where the mean free path is orders of magnitude greater than the size of the molecule. 

The average velocity of a gas molecule is determined by the molecular weight and the absolute temperature of the gas. Air molecules, like many other molecules at room temperature, travel with velocities of about 500 m s"1 but there is a distribution of molecular velocities. This distribution of velocities is explained by assuming that the particles do not travel unimpeded but experience many collisions. The constant occurrence of such collisions produces the wide distribution of velocities. The quantitative treatment was carried out by Maxwell in 1859, and somewhat later by Boltzmann. The phenomenon of collisions leads to the concept of a free path, that is the distance traversed by a molecule between two successive collisions with other molecules of that gas. For a large number of molecules, this concept must be modified to a mean free path which is the average distance travelled by all molecules between collisions. For molecules of air at 25°C, the mean free path X at 1 mbar is 0.00625 cm. It is convenient therefore to use the following relation as a scaling function ... 

The kinetic theory of dilute gases accounts for collisions between spherical molecules in the presence of an intermolecular potential. Ordinary molecular diffusion coefficients depend linearly on the average kinetic speed of the molecules and the mean free path of the gas. The mean free path is a measure of the average distance traveled by gas molecules between collisions. When the pore diameter is much larger than the mean free path, collisions with other gas molecules are most probable and ordinary molecular diffusion provides the dominant resistance to mass transfer. Within this context, ordinary molecular diffusion coefficients for binary gas mixtures are predicted, with units of cm /s, via the Chapman-Enskog equation (see Bird et al., 2002, p. 526) ... 

The diffusion of molecules in pores can be classified in a number of different regimes depending on the pore diameter (Fig. 2). For large pore diameters of the order of 1 (xm or larger, usually called macropores, collisions between the molecules occurs much more frequently than collisions with the wall, and molecular diffusion is the dominant mechanism. Typically, the diffusion constants of gases are around lO m s. As the size of the pores decreases, the number of collisions with the wall increases imtil the diffusion length finally becomes smaller than the mean free path (the average distance traveled... 

Whenever the mean free path is large compared to the pore diameter, no new phenomena appear since the forced flow will be Knudsen flow. This is indistinguishable from Knudsen diffusion (which we have already discussed) since molecules travel completely independently and are not affected by total pressure differences. However, when the mean free path is small compared to the pore radius, Poiseuille (stream-lined) flow will occur in the presence of total pressure differences. The total flow of a particular molecular si>ecies will then be the diffusive flow due to the partial pressure gradient of that species plus a contribution due to the stream-lined flow in which molecules are carried in proportion to their concentration. Thus, when ordinary diffusion and Poiseuille flow are both occurring the total flow rate of a molecular species A through a pore cross-section is ... 

An intrinsic length scale in dilute gases is the mean free path, which measures the average distance the gas molecules travel between coflisitMis. The ratio of the molecular mean free path of gas A to a flow characteristic length scale L is defined as the Knudsen number Kn ... 

MOLECULAR EFFUSION AND DIFFUSION (SECTION 10.8) It Mows from kinetic-molecular theory that the rate at which a gas undergoes effusion (escapes through a tiny hole) is inversely proportional to the square root of its molar mass (Graham s law). The diffusion of one gas through the space occupied by a second gas is another phenomenon related to the speeds at which molecules move. Because moving molecules undergo frequent collisions with one another, the mean free path—the mean distance traveled between collisions—is short. Collisions between molecules limit the rate at which a gas molecule can diffuse. 

We learned in Chapter 10 that gas molecules move at some average speed that depends inversely on their molar mass, in a straight line, until they collide with something. The mean free path is the average distance molecules travel between collisions, oco (Section 10.8) Recall also that the kinetic-molecular theory of gases assumes that gas molecules are in continuous, random motion. oco(Section 10.7)... 

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