Mean jump distance

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 ... 

Combining the PFG self diffusion measurement with a measurement of the correlation time provides a means of determining directly the mean jump distance. 

Consider one of these random-walking ions. It can be proved (see Appendix 4.1) that the mean square distance traveled by an ion depends on the number N of jumps the ion takes and the mean jump distance I in the following manner (Fig. 4.36) ... 

To go further than Eq. (4.108), one has to examine the factors that govern the mean jump distance / and the jump frequency k. For this, the picture of a liquid (in which diffusion is occurring) as a structureless continuum is inadequate. In reality, the liquid has a structure—ions and molecules in definite arrangements at any one instant... 

Leonard-Jones equation, 45 Mean jump distance, 464... 

The NMR gradient diffusion experiments yield values of D directly. To obtain the mean jump distance , xc must be determined usually from the intermolecular contribution to 1/T x or 1/Tlp. 

X is the mean jump distance. The difference in concentration is given as... 

There exists another mechanism of field-enhanced mobility, but it needs a far higher applied voltage than EHD motion, and consequently has never been observed. Indeed, a classical rate-process approach of ionic diffusion and migration (see for instance Conway, 1970) shows that the drift velocity of an ion in the direction of the field is proportional to slnh (El/2 U), with 1 the mean jump distance of the ion from one site of low energy to another. If E 2 U/1, the drift velocity is proportional to E if E 2 U/1, however, it grows exponentially. 1 being about a few... 

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