Velocity of gas molecules

Figure 18 illustrates the difference between normal hydrodynamic flow and slip flow when a gas sample is confined between two surfaces in motion relative to each other. In each case, the top surface moves with speed ua relative to the bottom surface. The circles represent gas molecules, and the length of an arrow is proportional to the drift velocity for that molecule. The drift velocity variation with distance is illustrated by the plots on the right. When the ratio of the mean free path to the separation distance between surfaces is much less than unity (Fig. 18a), collisions between gas molecules are much more frequent than collisions of the gas molecules with the surfaces. Here, we have classical fluid flow or viscous flow. If the flow were flow in tubes, Poiseuille s law would be obeyed. The velocity of gas molecules at the surface is the same as the velocity of the surface, and in the case of the stationary surface the mean tangential velocity of the gas at the surface is zero. 

Some radioactive vapours are adsorbed or chemisorbed on surfaces so strongly that the boundary condition is Xo equal to zero. This is also true of particles in the submicrometric size range. The velocity of gas molecules perpendicular to surfaces is of order 100 ms-1, whereas the resistance of the laminar boundary layer to molecular diffusion usually restricts vg to a value of the order 0.1 m s-1 or less. Hence if the accommodation coefficient, the fraction of molecular collisions which entail sorption at the surface, exceeds about 10-3, the surface will act as a perfect sink. 

Problem 7 From kinetic equation of gases, how will you calculate the root mean square velocity of gas molecules under different conditions ... 

Root mean square velocity of gas molecules is always greater than the average velocity under similar conditions. 

The molecules in a gas mixture continually collide with each other, and the diffusion process is strongly influenced by this collision process. The collision of like molecules is of little consequence since both molecules are identical and it makes no difference which molecule crosses a certain plane. The collisions of unlike molecules, however, influence the rate of diffusion since unlike molecules may have different masses and thus different momeniums, and thus the diffusion process is dominated by the heavier molecules. The diffusion coefficients and thus diffusion rales of gases depend strongly on temperature since the temperature is a measure of the average velocity of gas molecules. Therefore, the diffusion rales are higher at higher temperatures. 

To obtain some idea of the order of magnitude of the velocities of gas molecules, we can calculate from the law of distribution the most probable velocity or the average velocity v, or other such mean values (see Appendix I, p. 259) we find, e.g., for the most probable velocity the value = / 2RTIii), so that, e.g., for molecular hydrogen (/X = 2) at 0" C. T = 273° K.)... 

Problem 49-1. Derive Equation 49-5 with the use of the results of Section 14. By equating W to the kinetic energy lAmvi (v being the velocity), derive the Maxwell distribution law for velocities, and from it calculate expressions for the mean velocity and root-mean-square velocity of gas molecules. 

Independently of, and earlier (1860) than Boltzmann, the Scottish mathematician and physicist James Clerk Maxwell (1831-1879) developed a similar theory, which he applied particularly to the distribution of velocities of gas molecules. Because of this, equations (5.211) to (5.215) are sometimes referred to as MaxwelhBoltzmann equations, although this expression is perhaps best reserved for the velocity-distribution equations. 

FIGURE 228. The distribution of velocities of gas molecules at 25°C. Note how much greater the velocities are for ultralight hydrogen molecules and helium atoms. These substances thus escape the earth s atmosphere, unlike the heavier gases. (Adapted from Brown et ah, Chemistry—the Central Science). 

At high E/N values where drift velocities v are not neghgible compared to thermal velocities of gas molecules, the distribution of ion-molecule velocities shifts toward higher values. For moderate v, the new distribution may be approximated by the Maxwell-Boltzmann formula at higher effective temperature, T ... 

Intermolecular collisions are negligible at HV, and evaporation (effusion) is only determined by the free flight of the vapor molecules from the surface of the liquid into the vacuum chamber [52]. The mean velocity of gas molecules, is... 

Berthelot applied a formula deduced by Clausius for the mean velocity of gas molecules ... 

Mean square velocity of gas molecules. The mean velocity of a gas molecule is given by the following equation ... 

Example 10.2 gives another apphcation of the Boltzmann distribution law, the distribution of the velocities of gas molecules. This is the basis for the kinetic theory of gases, a classical model of great historical importance. 

The rates of effusion and diffusion depend on the relative velocities of gas molecules. The velocity of a gas varies inversely with the square root of its molar mass. Lighter molecules move faster than heavier molecules at the same temperature. 

This is the probability function for the velocities of gas molecules in three dimensions. 

In the intermediate range between viscous flow and Knudsen flow, that is, 0.05 pore wall. As a result, the velocity of gas molecules at the wall surface is not zero. This mechanism - combining both viscous flow and Knudsen diffusion - is thus called slip flow. The slip effect is negligibly small when r>> X but becomes significant when r is close to X. A correction has to be applied to the viscous flow with a wall velocity to describe the permeation flux as... 

In the transformation from the microscopic world to the macroscopic one, we also need to eonsider the effect of molecular collision on the distribution of molecular velocity or energy in these systems. The majority of molecules will have a velocity close to the mean value for the molecules, but there are always some molecules with velocity much greater than and others with velocity much lesser than the mean velocity. The distribution of velocities of gas molecules was first described by Maxwell in 1860. The Maxwell distribution of velocities is given by... 

Before we begin our discussion, we review three characteristics of the kinetic theory of gases. First, the actual velocity of gas molecules , is of course proportional to the temperature... 

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