Mean free path speed dependence

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 ... 

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). 

The type of flow of a fluid through a tube depends on the rate of flow. At low speeds it is laminar, where all the particles are moving parallel to each other as in a gently flowing stream. As the speed increases a point is reached when the flow is turbulent. The point at which this occurs is determined by the shape of the containing vessel or tube. The above discussion refers to a fluid, where collisions between the particles themselves are more important than collisions between the particles and the vessel or tube walls, and applies to a gas where the mean free path is small compared... 

The molecular djffusivity (or diffusion coefficient) D. defined as the proportionality constant between the diffusive flux and the negative of the composition gradient, is therefore proportional to the product of the mean molecular speed and the mean distance between collisions. If the simple kittalic theory expressions for the mean molecular speed and the mean free path of like molecules are used, one finds a modast temperature and pressure dependence of the diffusivity. 

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 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) ... 

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)... 

An accurate calculation of mean free path requires computing the average distance a molecule travels between collisions, a quantity which depends upon the speed of the molecule. The mean free path of stationary molecules is zero since they do not move until struck by other molecules. To very fast molecules the remainder of the gas appears as immobile target molecules and the calculation of the previous section is exact. We thus expect (2.4) to provide an upper bound for the speed-dependent mean free path. 

Fig. 2.3. (jmd )X a), the ratio of the speed-dependent mean free path to its value at infinite speed, in a one-component system as a function of the scaled speed a. 

It is possible to modify (2.31) to account for the distribution of molecular speeds and thereby to consider the speed dependence of the mean free path. However, such modification does not incorporate the effect of macroscopic gradients on the distribution function. Furthermore, it is still restricted to billiard ball dynamics. Thus, even though it is conceptually more in tune with the previous analysis we do not pursue it here. 

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