Gas translational diffusion

For Knudsen diffusion, d is the pore diameter and the activation energy for diffusion is zero. For activated gas translational diffusion d is the diffusional length between adjacent low-energy sites [37] or the pore diameter [38], and the activation energy is between 10 and 60 kJ/mol [37]. The parameter z, the lattice coordination number, depends on the zeolite type, z is equal to 4 for ZSM-5 and to 6 for 5 A, compared to 3 for Knudsen diffusion. 

The activation energy for gas translational diffusion represents the energy barrier that a molecule has to overcome upon moving from one channel intersection to the other. The order of magnitude can be calculated from the difference in potential field in a channel and in an intersection. The transition from Knudsen to configurational diffusion depends on the size of the molecule with respect to the zeolite channels and the proportions of the molecule (ratio of length to diameter), as is shown in Fig. 11 for ZSM-S and zeolite-A. 

The application of the Maxwell-Stefan theory for diffusion in microporous media to permeation through zeolitic membranes implies that transport is assumed to occur only via the adsorbed phase (surface diffusion). Upon combination of surface diffusion according to the Maxwell-Stefan model (Eq. 20) with activated-gas translational diffusion (Eq. 12) for a one-component system, the temperature dependence of the flux shows a maximum and a minimum for a given set of parameters (Fig. 15). At low temperatures, surface diffusion is the most important diffusion mechanism. This type of diffusion is highly dependent on the concentration of adsorbed species in the membrane, which is calculated from the adsorption isotherm. At high temperatures, activated-gas translational diffusion takes over, causing an increase in the flux until it levels off at still-higher temperatures. 

Figure 15 Combined gas translational diffusion and surface diffusion as a function of temperature. The surface coverage on the feed side is also included. Parameters are A5 = —50 J mol K , = 1 mmol g Q = 25 kJ-mol". Efi — X 10" m -sec" = 15 kJ mol" / = 50 pim ...

The same model was applied to permeation of lighter hydrocarbons (C1-C3) through the silicalite-1 membrane [50]. In the case of methane, ethane, and ethene, some concentration dependence of the Maxwell-Stefan diffusivity was observed. This can be caused either by the importance of interfacial effects, which are not taken into account, or by the contribution of activated-gas translational diffusion to the net flux. The diffusivities calculated from these permeation experiments were, however, in rather good agreement with diffusivity values from the literature, which implies that these zeolitic membranes could also be a valuable tool for the determination of diffusion coefficients in zeolites. 

The temperature dependence of the methane permeation through a silicalite membrane, showing a maximum and a minimum as a function of temperature (Fig. 3 [14]), can not be predicted by using the Maxwell-Stefan description for surface diffusion only. Such a maximum and minimum in the permeation as a function of temperature can be predicted only when the total flux is described by a combination of surface diffusion and activated-gas translational diffusion (Fig. 15). 

At high temperatures, the adsorption phenomena can become negligible and molecules can be considered as being in a quasi-gaseous state even within the constrained environment of the zeolite framework. This state is referred to as gas translational diffusion , activated gaseous diffusion or activated-Knudsen diffusion . 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

From this modelling approach it seems that the combined surface difiusion and activated gas translational difiusion can describe the observed single component permeation behaviour. The interpretation for the latter type of diffusion is not well-crystallized at present. It might have to do with an increasing deformation of the silicalite-1 structure with increasing temperature that causes this apparently activated process. This aspect has not been considered up to now, the silicalite-1 has been considered as a rigid structure. 

Evaluating the not-abundant results on zeolitic membrane modeling, it becomes apparent that the most successful direction is the approach based on two contributions surface diffusion and ga.s translational diffusion, with the former being dominant at low, the latter at high temperatures. The description of the surface diffusion based on the... 

NO2 is a stable paramagnetic gaseous molecule at normal temperatures. The ESR parameters of NO2 trapped in a solid matrix have been well established from single-crystal ESR measurements and have been related to the electronic structure by molecular orbital studies [39]. Thus, the NO2 molecule has potential as a spin probe for the study of molecular dynamics at the gas-solid interface by ESR. More than two decade ago temperature-dependent ESR spectra of NO2 adsorbed on porous Vycor quartz glass were observed [40] Vycor is the registered trademark of Coming, Inc. and more information is available at their website. The ESR spectral line-shapes were simulated using the slow-motional ESR theory for various rotational diffusion models developed by Freed and his collaborators [41]. The results show that the NO2 adsorbed on Vycor displays predominantly an axial symmetrical rotation about the axis parallel to the O—O inter-nuclear axis below 77 K, but above this temperature the motion becomes close to an isotropic rotation probably due to a translational diffusion mechanism. 

An ideal gas has by definition no intermolecular structure. Also, real gases at ordinary pressure conditions have little to do with intermolecular interactions. In the gaseous state, molecules are to a good approximation isolated entities traveling in space at high speed with sparse and near elastic collisions. At the other extreme, a perfect crystal has a periodic and symmetric intermolecular structure, as shown in Section 5.1. The structure is dictated by intermolecular forces, and molecules can only perform small oscillations around their equilibrium positions. As discussed in Chapter 13, in between these two extremes matter has many more ways of aggregation the present chapter deals with proper liquids, defined here as bodies whose molecules are in permanent but dynamic contact, with extensive freedom of conformational rearrangement and of rotational and translational diffusion. This relatively unrestricted molecular motion has a macroscopic counterpart in viscous flow, a typical property of liquids. Molecular diffusion in liquids occurs approximately on the timescale of nanoseconds (10 to 10 s), to be compared with the timescale of molecular or lattice vibrations, to 10 s. 

In analyzing dielectric relaxation spectra in condensed matter, two different concepts of molecular dynamics are of great significance. The first, which is predominantly applicable to liquids, is to consider the liquid as a dense gas with very frequent collisions, the rate of which is so high that the reorientation of a molecule is completely controlled by the rate. Thus, in this case the rotational dynamics becomes similar to the Brownian translational diffusion and it is known as rotational diffusion. The second approach, which is more frequently used to describe dynamics in molecular crystals and intramolecular dynamics, emphasizes reorientation of a molecule in the presence of an orientation-dependent potential with well-defined minima. In this case, the molecule resides for finite time intervals in different potential wells, jumping between them from time to time. Since the time of jump is very short in comparison with the time of residence in the well, the concept is known as reorientation by instantaneous jumps. On infinite increase of the number of potential wells and the corresponding decrease of the angular distance between them, this approach reduces to rotational diffusion. ... 

Oxide movements are determined by the positioning of inert markers on the surface of the oxideAt various intervals of time their position can be observed relative to, say, the centreline of the metal as seen in metal-lographic cross-section. In the case of cation diffusion the metal-interface-marker distance remains constant and the marker moves towards the centreline when the anion diffuses, the marker moves away from both the metal-oxide interface and the centreline of the metal. In the more usual observation the position of the marker is determined relative to the oxide/ gas interface. It can be appreciated from Fig. 1.81 that when anions diffuse the marker remains on the surface, but when cations move the marker translates at a rate equivalent to the total amount of new oxide formed. Bruckman recently has re-emphasised the care that is necessary in the interpretation of marker movements in the oxidation of lower to higher oxides. 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

In contrast to crystalline solids characterized by translational symmetry, the vibrational properties of liquid or amorphous materials are not easily described. There is no firm theoretical interpretation of the heat capacity of liquids and glasses since these non-crystalline states lack a periodic lattice. While this lack of long-range order distinguishes liquids from solids, short-range order, on the other hand, distinguishes a liquid from a gas. Overall, the vibrational density of state of a liquid or a glass is more diffuse, but is still expected to show the main characteristics of the vibrational density of states of a crystalline compound. 

A good example of translational fractionation is one-way diffusion through an orifice that is smaller than the mean-free path of the gas. Related, but somewhat more complex velocity-dependent fractionations occur during diffusion through a host gas, liquid, or solid. In these fractionations the isotopic masses in the translational fractionation factor are often replaced by some kind of effective reduced mass. For instance, in diffusion of a trace gas JiR through a medium, Y, consisting of molecules with mass ttiy. 

Note that x translates into the derivative dB/dn of the lattice gas, and hence measures the increase in coverage as the adsorbate gas pressure increases, assuming there is equilibrium between the layer and surrounding gas. The quantity k Tx is proportional to the peak intensity of the diffuse scattering at the Bragg position corresponding to the overlayer ordering described by the order parameter... 

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