Molecular diameter from diffusion

To calodate from difiusion coefficients the effective diameter of the iodine molecule in several solvents. 

The integral diffusion coefficient of iodine for diffusion from a given concentration into pure solvent has been measured at 25 °C in several solvents by Stokes, Dunlop, and Hall (Trans. Faraday Soc. 1953, 49, 886). By extrapolation they obtained values of the dif ion coefficient D at infinite dilution. These values together with viscosities of the solvent are given in table 1. 

We see that in all these solvents the iodine molecule has an effective diameter 3.2g 0.1 A. We conclude that any solvation, whatever that may mean, cannot be very specific. 

We use the following notation r the freezing point of pure water, 

A/H the molar heat (enthalpy increase) of fusion of ice to pure water, molar heat capacity of liquid water (constant pressure) at P, molar heat capacity of ice (constant pressure) at P, the molar mass of water, molality of solution. 

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Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

Fig. 8. Variation of activation energy with kinetic molecular diameter for diffusion in 4A 2eohte (A), 5A 2eohte (0)> carbon molecular sieve (MSC-5A) (A). Kinetic diameters are estimated from the van der Waals co-volumes. From ref. 7. To convert kj to kcal divide by 4.184.

Equation (13) can also be used for the calculation of molecular diameters from experimental viscosity data. These can be compared with molecular diameters as determined by other experimental methods (molecular beams, diffusion, thermal conductivity, x-ray crystallographic determination of molecular packing in the solid state, etc.). 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

A good example for reactant shape selectivity includes the use of catalysts with ERI framework type for selective cracking of linear alkanes, while excluding branched alkanes with relatively large kinetic diameters from the active sites within the narrow 8-MR zeolite channels [61, 62]. Here molecular sieving occurs both because of the low Henry coefficient for branched alkanes and because of the intracrystalline diffusion limitations that develop from slow diffusivities for branched alkane feed molecules. 

From data like these for SF6 a collection of diffusion coefficients (Figure 4), activation enthalpies, and pre-exponential factors (Table I) have been assembled. It was necessary to assume a value of the jump distance which seemed intuitively appropriate, the molecular diameter, for use in Equation 1. 

Many heterogeneous reactions give rise to an increase or decrease in the total number of moles present in the porous solid due to the reaction stoichiometry. In such cases there will be a pressure difference between the interior and exterior of the particle and forced flow occurs. When the mean free path of the reacting molecules is large compared with the pore diameter, forced flow is indistinguishable from Knudsen flow and is not affected by pressure differentials. When, however, the mean free path is small compared with the pore diameter and a pressure difference exists across the pore, forced flow (Poiseuille flow see Volume 1, Chapter 3) resulting from this pressure difference will be superimposed on molecular flow. The diffusion coefficient Dp for forced flow depends on the square of the pore radius and on the total pressure difference AP ... 

Figure 3. The ratio D/D as a function of packing fraction for supercritical ethylene. A indicates ratios calculated using hard sphere diameters determined from diffusion data. 0 indicates ratios calculated using hard sphere diameters determined from compressibility data. The solid lines are the molecular dynamics results, extrapolated to infinite systems, of Alder, Gass and Wainwright (Ref. 24).

Micropore diffusion Diffusion within the small micropores of the adsorbent which are of a size comparable with the molecular diameter of the sorbate. Under these conditions the diffusing molecule never escapes from the force field of the solid surface and steric hindrance is important. For zeolites the terms micropore diffusion and intracrystalline diffusion are synonymous. Raffinate Product stream containing the less strongly adsorbed species. 

The primary requirement for an economic adsorption separation process is an adsorbent with sufficient selectivity, capacity, and life. Adsorption selectivity may depend either on a difference in adsorption equilibrium or, less commonly, on a difference in kinetics. Kinetic selectivity is generally possible only with microporous adsorbents such as zeolites or carbon molecular sieves. One can consider processes such as the separation of linear from branched hydrocarbons on a 5A zeolite sieve to be an extreme example of a kinetic separation. The critical molecular diameter of a branched or cyclic hydrocarbon is too large to allow penetration of the 5A zeolite crystal, whereas the linear species are just small enough to enter. The ratio of intracrystalline diffusivities is therefore effectively infinite, and a very clean separation is possible. 

Figure 1. Correlation of diffusion coefficients (D ) of small gaseous molecules in amorphous polyethylene (natural rubber) with their reduced molecular diameters (d-0.5 (Reproduced with permission from Ref. 3. Copyright 1961 John Wiley Sons.)...

Typical values for p are between 0.3 and 0.6, and for tp between 2 and 5. So, a reasonable assumption for the effective diffusion De is that it is Vio of the diffusivity I). This diffusivity D can be calculated from the Knudsen (corresponding to collisions with the wall) and molecular diffusivity (intramolecular collisions). The molecular diffusivity was estimated at 10 5 m2/s, which is reasonable for the diffusion in gases. The Knudsen diffusivity depends on the pore diameter. The exact formulas for the molecular and Knudsen diffusion are given by Moulijn et at1. For zeolites, the determination of the diffusivity is more complicated. The microporous nature of zeolites strongly influences the diffusivity. Therefore, the diffusion... 

The ability to correctly reproduce the viscosity dependence of the dephasing is a major accomplishment for the viscoelastic theory. Its significance can be judged by comparison to the viscosity predictions of other theories. As already pointed out (Section II.C 22), existing theories invoking repulsive interactions severely misrepresent the viscosity dependence at high viscosity. In Schweizer-Chandler theory, there is an implicit viscosity dependence that is not unreasonable on first impression. The frequency correlation time is determined by the diffusion constant D, which can be estimated from the viscosity and molecular diameter a by the Stokes-Einstein relation ... 

Liu and Ruckenstein [Ind. Eng. Chem. Res. 36, 3937 (1997)] studied self-diffusion for both liquids and gases. They proposed a semiem-pirical equation, based on hard-sphere theory, to estimate self-diffusivities. They extended it to Lennard-Jones fluids. The necessary energy parameter is estimated from viscosity data, but the molecular collision diameter is estimated from diffusion data. They compared their estimates to 26 pairs, with a total of 1822 data points, and achieved a relative deviation of 7.3 percent. 

By means of Eq. (34) it is possible to calculate the center-to-center collision distance dxi from a measured diffusion constant D i- If three diffusion constants Dx2, Ms- and Ms can be measured, individual molecular diameters f/j, (I2, and can be obtained if it can be assumed that the collision distances are the arithmetic averages of the molecular diameters involved, as would be the case with hard spheres ... 

If a self-diffusion constant M can be measured approximately through one of the approaches discussed in Exp. 5, a molecular diameter d can be obtained directly from Eq. (35). Conversely, Eqs. (34) and (35) can be used to estimate a diffusion constant if molecular diameters are known from some other source or to calculate self-diffusion constants from binary diffusion constants (or vice versa) by calculating first the molecular diameters. In the latter case, it can be argued that any remaining approximations in the treatment will largely cancel out. 

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