Velocity, mean molecular

Illustrative Example 18.2 Estimating Molecular Diffusivity in Air Problem Estimate the molecular diffusion coefficient in air, >,a, ofCFC-12 (see Illustrative Example 18.1) at 25°C (a) from the mean molecular velocity and the mean free path, (b) from the molar mass, (c) from the molar volume, (d) from the combined molar mass and molar volume relationship by Fuller (Eq. 18-44), (e) from the molecular diffusivity of methane. 

Equation 3.13 fulfills our basic objective in describing average molecular motion. However, inasmuch as we cannot ordinarily measure directly the average velocity of molecules or ions undergoing transport, it is advantageous to transform 0 into the flux density /, a parameter that is more directly observable. The flux density of a component is the number of moles carried through a unit areain unit time. It is related to that component s mean molecular velocity 0 and concentration c by... 

What is the value of the diffusion velocity (the mean molecular velocity caused by a concentration gradient) of a component found at a concentration of 0.010 mol/L and with a concentration gradient of 1.0 mol/L cm The diffusion coefficient is 5.0 x 10 6cm2/s. 

Inasmuch as mean molecular velocity is given by U = Jlc, Eq. 3.14, the velocity induced by thermal diffusion is simply... 

Using physical reasoning, justify the Tm dependence of the diffusion coefficient as shown by Eq. (II-2). Hint Recall that mean molecular velocity is proportional to F 2 and that the density of an ideal gas is inversely proportional to temperature. 

Table 6.24 Molecular velocities ° v2 and mean molecular velocities v of selected gases (at T = 273 K)...

The mean molecular velocity is given by (SRT/irM), while the velocity of sound is given by (yRT/M), both increasing as T. The velocity also increases because of the increase in space velocity due to the heating. 

If the diffusion coefficients of H and O atoms in the gas mixtures can be estimated satisfactorily, then the results of the above treatment can be used also to derive values of the rate coefficients fe2 and k. Using estimated hard sphere values (Dq = Xqc/3 where Xq is the mean free path at unit pressure and c is the mean molecular velocity), Baldwin obtained, at 520 °C... 

In the experimental arrangement shown in Figure 7.1 at steady state, the net flux is equal to the mean molecular velocity multiplied by the total concentration. According to the kinetic theory, the viscosity of a gas is independent of pressure, while it is expected to vary with the gas composition. Substitution of Eq. (7.14) into Eqs. (7.10) and (7.11), with Vd from Eq. (7.12) yields the expression for the fluxes caused by gradients in both composition and total pressure... 

The macroscopic motion of the liquid at the point r can be identified with the mean molecular velocity c0(r) at the given point. The components of the dilatation and distortion at r then are the space derivatives of c0. In the case of the Newtonian liquids, to which the present discussion is confined, the elements of the pressure tensor are proportional to the components of the rate of shear plus contributions from the hydrostatic pressure and the rate of dilatation. Thus we have... 

The thermal conductivity coefficient has been derived from Browniah motion theory by Irving and Kirkwood33 in terms of the equilibrium singlet and pair distribution functions °/a) and °/Thermal conduction under a macroscopic temperature difference involves a gradient in the mean square molecular velocity rather than in the mean molecular velocity. The steady-state radial distribution function then remains spherically symmetric except for a small correction arising from the number density variation with the temperature. As the analysis introduces no new assumptions and is somewhat lengthy, it will not be reproduced here. The resulting equation for the thermal conductivity coefficient x is... 

For Knudsen diffusion molecular gas theory gives that the Knudsen diffusion coefficient is proportional to the pore radius and the mean molecular velocity... 

Under true superpermeable hydrogen flux conditions, the large numbers of molecules predicted to impact upon a membrane surface follow, in part, as a consequence of the very high molecular speeds of gas phase hydrogen relative to the size of reactor vessels. For example, the mean velocity of a hydrogen molecule, H2, in the gas phase at 273 K (0 °C) is 1.7 km s [8]. Mean molecular velocity increases in proportion to the square root of the absolute temperature. In a chemical reactor at 673 K (400 °C), for example, the mean velocity of H2 will increase by a factor of (673 K/273 K) / from 1.7 km s at 273 K to 2.7 km s at 673 K. Mean molecular velocity decreases inversely with the square root of the molecular mass. For deuterium molecules, D2, with a molecular mass approximately twice that of H2, the mean molecular velocity is less than that of H2 by a factor of 2 /, approximately 1.2 km s at 273 K (0 °C) [8]. 

In a pure gas, the mean molecular velocities vm are equal and uncorrelated, hence their mean relative velocity is vm V2 leading to Equation 1.18. For a heteromolecular gas of ions and molecules, one may approximate" ... 

In failing to recognise fully the difference between kinetic and equilibrium conditions, when developing an expression for the distillation rate, the author first made the mistake of assuming that the pressure of a gas is the same in all directions. Of course it is not, when the gas has a large net velocity in one direction. Thus it was necessary to devise a correction to be applied when the gas has a net velocity which is a sizeable fiaction of the mean molecular velocity. This enters into the expressions for evaporation and condensation, as shown below. 

But the rate of collision of gas molecules with a surface is a familiar result of the kinetic theory of gases the number of molecules striking a unit of surface area per unit time is proportional to the mean molecular velocity v and to the number density c of molecules in the gas ... 

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