Diffusion jumps distance

Neutron-scattering and dielectric relaxation studies [23] both indicate that the water molecules solvating monovalent exchangeable cations on montmorillonite are a little less mobile, in respect to translational and reorientational motion, than are water molecules in the bulk liquid. For example, as with vermiculite, neutron-scattering data show that no water molecule is stationary on the neutron-scattering time scale. In the one-layer hydrate of Li-montmorillonite, the residence time of a water molecules is about six times longer than in the bulk liquid, with a diffusive jump distance of about 0.35 nm, and a water molecules reorients its dipole axis about half... 

Ionic transport in solid electrolytes and electrodes may also be treated by the statistical process of successive jumps between the various accessible sites of the lattice. For random motion in a three-dimensional isotropic crystal, the diffusivity is related to the jump distance r and the jump frequency v by [3] ... 

The list below shows the last position reached, in units of the jump step a, during a random walk for 100 atoms, each of which makes 200 jumps. If the jump time is 10-3 s and the jump distance, a, is 0.3 nm, estimate the diffusion coefficient (a) in units of a2 s-1 and (b) in units of m2 s-1 ... 

Figure 6.3 Potential barrier to be surmounted by a diffusing ion in the presence of an electric field schematic. The distance a represents the jump distance between stable sites, and Agm is the average height of the potential barrier.

Figure S5.6 Two adjacent planes, 1 and 2, in a crystal, separated by the jump distance of the diffusing atom, a. The number of diffusing atoms on planes 1 and 2 are N1 and N2, respectively.

The deviations from Gaussian behaviour were successfully interpreted as due to the existence of a distribution of finite jump lengths underlying the sublinear diffusion of the proton motion [9,149,154]. A most probable jump distance of A was found for PI main-chain hydrogens. With the model... 

Fig.4.19 Tseif(Q) obtained for a all the protons in PVE empty MD simulations,/ /// NSE, /=0.55) and b the main chain (filled circle, /=0.66) and the side group hydrogens (empty circle, /=0.51), both from the MDS. Dotted lines are expected Q-dependence from the Gaussian approximation in each case. Solid lines are description in terms of the anomalous jump diffusion model. Insets Chemical formula of PVE (a) and distribution functions obtained for the jump distances (b)...

Thus, microscopically, the diffusion coefficient may be interpreted as one-sixth of the jumping distance squared times the overall jumping frequency. Since / is of the order 3 x 10 ° m (interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D m /s such as Mg diffusion in... 

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. 

Figure 4.43 Diffusion in a potential gradient A/x, where AG I is the height of the activation energy barrier and X is the jump distance. From W. D. Kingery, H. K. Bowen, and D. R. Uhlmann, Introduction to Ceramics. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

The macroscopic diffusion coefficient ) is defined in terms of the mean jump distance a and mean time between jumps r as ... 

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. 

Since the presentation of this model new data have appeared which allow various tests and new conclusions. The diffusion coefficients of Karger (14), together with Equation 1 and the median jump time from the relaxation data at room temperature yield a jump distance of 2.7 A for the zeolitic water as compared with 2.2 A in bulk water (see Table III for a data summary). One might be tempted to explain the jump distance in terms of some geometrical constant of the zeolite structure such as the distance between Sn and Sm ionic sites (40), but with the cages full of... 

Figure 5. Median jump frequencies (Sr ) 1 for water adsorbed on NaX to saturation, for water on charcoal at saturation, and that expected for bulk water (from NMR relaxation times) dashed curve marked diff diffusion coefficients by magnetic field gradient technique normalized to (Sr) 1 by choice of jump distance of 2.7 A + dielectric relaxation times of Jansen...

Fig. 32a, b. Temperature variation of a. the geometrical parameters jump distance r and distance L between neighbouring chains, b. Residence time x and diffusion constant D... 

Consider the diffusion of a randomly walking diffusant in the h.c.p. structure, which is composed of close-packed basal planes stacked in the sequence ABABA. The lattice constants are a and c. The probability of a first-nearest-neighbor jump within a basal plane (jump distance = a) is p, and the probability of a jump between basal planes (jump distance = /a2/3 + c2/4) is 1 — p. If axes X and X2 are located in a basal plane, derive the following expressions for the diffusivities Du and D33 ... 

For nuclei that are coherent with the surrounding crystal, the lattice is continuous across the a//3 interface. The jumps controlling the /3C frequency factor will then be essentially matrix-crystal jumps and /3C will be equal to the product of the number of solute atoms surrounding the nucleus in the matrix, zcXg, and the solute atom jump rate, T, in the a crystal. The jump frequency can reasonably be approximated by T Di/a2 (see Eq. 7.52, where D is the solute tracer diffusivity and a is the jump distance). Therefore,... 

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