Kinetics effusion

The results on effusion can apparently be accounted for quite easily in terms of the well known kinetic theory expression for the rate of incidence of molecules on unit area of a wall bounding a region occupied by gas, namely... 

As a consequence of these simple deductions, Graham s experiments c effusion through an orifice came to be regarded as one of the earliest direct experimental checks on the kinetic theory of gases. However, a closer examination of his experimental conditions reveals that this view is mistaken. As mentioned earlier, his orifice diameters ranged upwards from 1/500 in., while the upstream pressure was never very much less thai atmospheric. Under these circumstances the molecular mean free path len ... 

According to Ktiudsen if a small circular orifice of diameter less than the mean free path of the molecules in a container, is opened in the wall of the container to make a connection to a high vacuum sunounding the container, the mass of gas effusing tlnough the orifice, of area A, is given by an equation derived from the kinetic theoty, where tire pressure is in amiospheres. 

As we have implied, diffusion is a rather complex process so far as molecular motion is concerned. Effusion, the flow of gas molecules at low pressures through tiny pores or pinholes, is easier to analyze using kinetic theory. 

Gopinath, C. S. and Zaera, F. (2000) Transient kinetics during the isothermal reduction of NO by CO on Rh(lll) as studied with effusive collimated molecular beams , J. Phys. Chem. B, 104, 3194. 

Urrre= root-mean-square speed KE= kinetic energy r = rate of effusion M= molar mass jc= osmotic pressure /= van t Hoff factor... 

The inverse relationship between the rate of effusion and the square root of the mass follows directly from the connection between temperature and kinetic energy described in the previous section. Because temperature is a measure of average kinetic energy and is independent of the gas s chemical identity, different gases at the same temperature have the same average kinetic energy ... 

The connection between temperature and kinetic energy obtained from the kinetic-molecular theory makes it possible to calculate the average speed of a gas particle at any temperature. An important practical consequence of this relationship is Graham s law, which states that the rate of a gas s effusion, or spontaneous passage through a pinhole in a membrane, depends inversely on the square root of the gas s mass. 

Example 2.1. To estimate the number of gas molecules hitting the liquid surface per second, we recall the kinetic theory of ideal gases. In textbooks of physical chemistry the rate of effusion of an ideal gas through a small hole is given by [12]... 

This should make sense. The rate of effusion depends on the speed of the molecules. The average kinetic energy is the same for two different gases at the same temperature. The speed of the gas molecules depends on the kinetic energy 1/2 mv1. 

The kinetic theory of gases was briefly discussed. It enables the mean or thermal velocity (c) of gas molecules at a given temperature to be obtained and gas flux to be calculated. From the latter, effusion rates, area-related condensation rates and conductances under molecular flow can be determined (see Examples 1.5 and 1.7-1.10). Calculation of collision frequency (obtained from c, n and the collision cross-section of molecules), enables the mean free path (f) of particles to be determined. The easily obtained expression for Ip is a convenient way of stating the variation of / withp (Examples 1.11-1.15). 

We shall consider in detail the predictions of the hard-sphere model for the viscosity, thermal conductivity, and diffusion of gases indeed, the kinetic theory treatment of these three transport properties is very similar. But first let us consider the simpler problem of molecular effusion. 

Only (c) The average kinetic energies of their molecules must be the same, must be true. The average speeds and their effusion rates depend on the molar masses (Graham s law). Their pressures and nnmbers of moles (and thus molecules) are governed by the ideal gas law. 

This equation is Graham s law, and thus the kinetic molecular model fits the experimental results for the effusion of gases. 

The effusion method originally suggested by Knudsen is essentially a method of pressure measurement which utilizes the fact that pressure is the effect of the bombardment of the walls of the containing vessel by the molecules. If a small part of the wall is replaced by a hole leading to an evacuated space, then the molecular shower will pass through the hole, and the rate at which molecules do this depends only on the mean component of velocity of the gas molecules and the number present, and may be calculated by kinetic theory to be apj 2nmkT) molecules per second, where a is the area of the hole, p is the pressure, m is the mass of a molecule, k is Boltzmann s constant, and T is the absolute temperature 2 19 In the derivation of this formula it is assumed that ... 

A schematic diagram for the enrichment of by gaseous diffusion of UFe through an effusion barrier is shown in Figure 5, which also illustrates the counter-current flow and cascade principles. The limiting separation factor a is given by the kinetic theory of gases... 

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