Knudsen equation

The Knudsen equation for vaporization, expressed in the form Ga = pa(Ma/2jiRT) -gives the expression for die flux... 

Total vapor pressures associated with each of the above reactions were calculated by use of the Knudsen equation... 

This equation is commonly designated the Hertz-Knudsen equation. 

The assumptions inherent in the derivation of the Hertz-Knudsen equation are (1) the vapor phase does not have a net motion (2) the bulk liquid temperature and corresponding vapor pressure determine the absolute rate of vaporization (3) the bulk vapor phase temperature and pressure determine the absolute rate of condensation (4) the gas-liquid interface is stationary and (5) the vapor phase acts as an ideal gas. The first assumption is rigorously valid only at equilibrium. For nonequilibrium conditions there will be a net motion of the vapor phase due to mass transfer across the vapor-liquid interface. The derivation of the expression for the absolute rate of condensation has been modified by Schrage (S2) to account for net motion in the vapor phase. The modified expression is... 

The third, fourth, and fifth assumptions inherent in the Hertz-Knudsen equation are also used in the derivation of Eq. (60). Maa (M3) has shown by experimental studies that Eq. (60) provides a good model for predicting the net rate of phase change. Maa (M4) and O Connor (Ol) have shown that if these assumptions are replaced by significantly less restrictive ones, and the derivation repeated, the calculated values of the net rate of phase change differ by less than 10% under the most extreme conditions. 

REGRESSION ANALYSIS OF VAPOR PRESSURE DATA CALCULATED WITH KNUDSEN EQUATION... 

The optimum data are selected in step l by using a precalculated standard deviation, which is directly related to the constancy of temperature. These selected sets of data are used in step 2 for calculation of equilibrium vapor pressures by means of the Knudsen equation. A regression analysis of these vapor pressure data is then carried out in step 3. 

If a surface with a finite number of sites is exposed to a gas, the number of molecules hitting the surface is given by the Hertz-Knudsen equation ... 

The complete Knudsen equation (1.26) will have to be used in the transitional area 10 2 < d p < 6 10 mbar cm. Conductance values for straight pipes of standard nominal diameters are shown in Figure 9.5 (laminar flow) and Figure 9.6 (molecular flow) in Chapter 9. Additional nomograms for conductance determination will also be found in Chapter 9 (Figures 9.8 and 9.9). 

In the case of the adsorption of benzene vapour by mercury examined by Iredale (p. 67) at a pressure of 12-5 mm. at 300° K., 0-564 of the surface is covered with benzene and 0-436 is bare. From the Herz-Knudsen equation it can be calculated that 0 902 x 10 gm. mols of benzene hit this bare surface per second, whilst on the covered surface 0 443 x 10 gm. mols are present. Thus the life of a benzene molecule on the mercury surface is 4 9 x 10 seconds. Over a free benzene surface, if the orientation of the molecules be similar to that on mercury, 1 666 x 10 gm. molecules evaporated per second or the life of a benzene molecule on a benzene surface is 4 7 X 10 seconds. 

The rates of evaporation of water through unimolecular fi has been examined by Hedestrom and by the writer. It is foi that the rate of evaporation of water through unimolecular fi of fatty acids on the surface of water is relatively slow in compari to the high rate of evaporation calculated with the aid of Herz-Knudsen equation. The rate is approximately inveri proportional to the surface film pressure F as is indicated fi the following figures. 

Knudsen flow is characterized by the mean free path (A) of the molecules, which is larger than the pore size, and hence collisions between the molecules and the pore walls are more frequent than intermolecular collisions. A lower limit for the significance of the Knudsen mechanism has usually been set at dp> 20 A [28]. The classical Knudsen equation for diffusion of gas is... 

The non-isothermal Knudsen eflusion technique was used to study vapor pressures of tars. The experimental details have been described previously [Oja and Suuberg 1997, 199S]. About 10 mg of dry tar was placed into a hermetic effusion cell with a small orifice, from which the saturated vapor effuses out into the vacuum outside the cell. To overcome effects caused by changes in tar con iosition during effusion, the experiment involved first a continuous cool-down followed by a continuous heat-up of the sample. From the mass loss data (by talcing into account both cool-down and heat-up as a whole cycle) the vapor pressure was calculated using the Knudsen equation. 

When the number of molecule to wall collisions is strongly dominant Kn 1) the flow of a single gas in a long capillary imder the action of a concentration pressure gradient can be described by the Knudsen equation [15] ... 

According to Turkdogan [5] the maximum value of in Eq.(9.65) is obtained by setting t equal to unity and zero activation (adsorption) energy (Eg = 0). Equation (9.65) then transforms to the classical Hertz-Knudsen equation for the number of moles striking a imit surface area per unit time Rma ... 

During the CVD process, the molecules to be incorporated into the lattice of substrate must be delivered to the surface from the bulk gas. The flux of incident molecules onto the substrate surface is defined by the Hertz-Knudsen equation [3,... 

The method of quantitative vaporization is based on the Hertz Knudsen equation [87]. The mass loss rate dm/dt of a substance A, vaporized at the temperature T in the Knudsen cell, is determined by weighing, in addition to the ion intensity measurement, k is obtained by the equatiop... 

The molar fluxes at isothermal conditions are related by simple Knudsen equation (Knudsen, 1950) ... 

What does this correlation imply concerning the relationship between activation energy for chemisorption and fractional surface coverage Hint Start your derivation with the Hertz-Knudsen equation as given in the text). 

There is no equation comparable to the Knudsen equation for diffusion of liquids in small pores, but the pore walls do limit the movement of molecules and cause a decrease in the diffusivity. Diffusion coefficients can be predicted from the bulk diffusivity and a hindrance factor that depends on the pore size and the solute size. For moderate-molecular-weight solutes, the empirical Wilke-Chang equation is used for the bulk diffusivity [13] ... 

These data can be evaluated using a modified Knudsen equation where the evaporative flux Q is expressed by... 

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