Jump length, calculation

In a typical liquid, molecules jump randomly to one side or another about 1010 times per second in steps of about 3 A length. Calculate the diffusion coefficient for such molecules. (For liquids, D commonly ranges from 10 6 to 2 x 10-5 cm2/s.)... 

Using this, the tracer atom is described as if it forms a pair with the vacancy on one of the bonds adjacent to its original site, it walks on the bond lattice, and at the end of the walk (which happens after each move with probability prec) it is released with equal probability at either end of the last visited bond. Results for the probabilities of the different jump lengths (beginning-to-end vectors of these trajectories) are shown in Fig. 9. Note, that the model calculations in Fig. 9 contain no adjustable parameters. 

Our measurements indeed show the expected logarithmic increase in mean square jump length with increasing distance away from a step, and provide confirmation that steps are indeed the sole sources and sinks for surface vacancies. Having established the role of the steps as sources and sinks for vacancies we speculate on how the vacancies are formed at the steps. One can envision several different mechanisms at steps that lead to the formation of a surface vacancy. All of them involve removing an atom from a kink, a step or (least likely) from the terrace itself. Since the attachment and detachment of atoms from kinks is often the energetically least costly way to detach an atom from a step, this appears to be the most likely initial process in the formation of a surface vacancy. Calculations on this problem have been performed in the context of the observations on Mn/Cu(00 1) that were mentioned previously [48-50]. In these calculations it was indeed found that the formation of step vacancies at kinks and the subsequent diffusion of these vacancies along a step and their release into the terrace is the most likely scenario for the creation of surface vacancies in a terrace. 

Subsequently, the random jump length, which is distributed logarithmically, is calculated by ... 

It has been demonstrated that the combined application of various NMR techniques for observing molecular rotations and migrations on different time scales can contribute to a deeper understanding of the elementary steps of molecular diffusion in zeolite catalysts. The NMR results (self-diffusion coefficients, anisotropic diffiisivities, jump lengths, and residence times) can be correlated with corresponding neutron scattering data and sorption kinetics as well as molecular dynamics calculations, thus giving a comprehensive picture of molecular motions in porous solids. 

Before calculating this first passage time density, we first demonstrate the validity of Eq. (100) by means of a simulation the results of which are shown in Fig. 14. Random jumps with Levy flight jump length statistics are performed, and a particle is removed when it enters a certain interval of width w around the sink in our simulations we found an optimum value w 0.3. As seen in Fig. 14 (note that we plot gtp(t) ) and for analogous results not shown here, relation (100) is satisfied for 1 < a < 2, whereas for larger w, the slope increases. 

The corresponding jump length projected to the column wall of a cylindrical tube is calculated as ... 

An apparent diffusional jump length using such a cylindrical activation volume can be calculated. Assuming the total free volume for a diffusional jump to be... 

Note. Noble gas radius from Zhang and Xu (1995). Molecular diffusivity from Jahne et al. (1987) except for Ar (Cussler, 1997). For SFe, the radius is based on S-F bond length of 1.56 A plus the radius of F- (1.33 A), and the diffusivity is from King and Saltzman (1995). The jumping distance is calculated from Equation 3-136e using pure water viscosity of 0.89 mPa s at 25°C. 

A computer simulation of diffusion via the vacancy mechanism is performed on a square lattice of screen pixels with a spacing of a = 0.5 mm. The computer performs the calculations so that the vacancy jumps at a constant rate of T = 1000 s-1. The simulation cell is a square of edge length 5 cm containing 10,000 pixels. There is just one vacancy in the simulation cell, and as it moves by nearest-neighbor jumps, it remains within the cell (by using periodic boundary conditions or reflection at the borders). 

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