Speed, molecular mean

This important result is used to find the root mean square speeds of the gas-phase molecules at any temperature (Fig. 4.25). We can rewrite this equation to emphasize that, for a gas, the temperature is a measure of mean molecular speed. From... 

From basic statistical-thermodynamics arguments [60] the mean molecular speed in a gas can be determined approximately from the relationship between pressure and mass density as... 

In the free molecular flow regime, where the molecular mean free path X is much larger than the orifice diameter, the speed of molecules exiting the reactor can be taken as the mean molecular speed, (v) = SRT/nMW, and the flux, FM, is given by... 

The speed of sound in a gas is related to mean molecular speed v through the relation... 

Here, D is the diffusivity of the reagent gas in the gaseous medium, V the mean molecular speed of a Maxwell-Boltzmann gas, and a the mass accommodation coefficient. The standard solution is readily obtained (3) as follows ... 

Where A is the mean free path and v is the mean molecular speed. The Knudsen diffusivity is proportional to the mean pore radius rt ... 

A mechanical pump providing even lower vacuum levels is the turbomolecular pump, in which one or more balanced rotors (turbine blades) spin at 20,000 to 50,000 rpm. At these rotation rates, the periphery moves at a speed that exceeds the mean molecular speeds of most molecules, and gas-rotor collisions impart a momentum component to the gas in the direction of the exhaust. Compression ratios up to 10 can be achieved as long as the outlet pressure is kept below about 0.1 Torr by a forepump. 

A spherical embryo of radius r is bombarded by vapor molecules at a rate 4wrWc/4 where N(= P/k ) is the molecular number density of the vapor and C (= VSksTl n-m) is the mean molecular speed. As a result of using kinetic theory of gasses for the mean molecular speed we can write the condensation rate, CR, as... 

YaNcS as the number of wall collisions per second, where c is the mean molecular speed, S is the surface, and N the number of molecules per unit volume. The average time for capture of a molecule by collision with the wall is NVy the total number of molecules, divided by the rate of capture, or tew = 4F//Sc , where represents the probability of being captured on collision with the wall. 

The mean molecular speed, u, of the reactive species is given by... 

U Deposition Uniformity U = fjf = f ) f) Reactant Mean Molecular Speed X Transverse Position z Normalized Transverse Position (z = x/a) a Ratio of Diffusivities P Deposition Modulus A Reactant Mean Free Path Normalized Preform Thickness p Total Molar Density o Reactive Species Molecular Diameter ( ) Reaction Probability / Net Molar Yield of Deposition Reaction... 

A = 2rs/3TRT, B = Tip s/32tRT, and v = iRT/uMf are the gas mean molecular speed L is the mean free path at unit pressure... 

SRT/ nMif = Vi is the mean molecular speed Dij is the ordinary diffusion coefficient M, is the molecular weight of component i... 

The diffusion constant has units of m s. It is proportional to the mean free path, A, and to the mean molecular speed u, but the proportionality constant is difficult to calculate in general. For the simplest case, a single-component gas, the proportionality constant is so... 

In literature, the temperature dependence has been occasionally simplified. For example, the mean molecular speed was taken constant equal to that at 7 over the entire range (7s, 7a). Then ... 

FYom this formula it is seen that to calculate /i we need to determine the mean molecular speed ( ci )m- For real systems the average molecular speed is difficult to determine. Assuming that the system is sufficiently close to equilibrium the velocity distribution may be taken to be Maxwellian. For molecules in the absolute Maxwellian state the peculiar velocity equals the microscopic molecular velocity, i.e., Ci = ci, because the macroscopic velocity... 

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. 

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