How Can the Diffusion Coefficient Be Related to Molecular Quantities

The diffusion coefficient D has appeared in both the macroscopic (Section 4.2.2) and the atomistic (Section 4.2.6) views of diffusion. How does the diffusion coefficient depend on the structure of the medium and the interatomic forces that operate To answer this question, one should have a deeper understanding of this coefficient than that provided hy the empirical first law of Tick, in which D appeared simply as the proportionality constant relating the flux / and the concentration gradient dc/dx. Even the random-walk intapretation of the diffusion coefficient as embodied in the Einstein-Smoluchowski equation (4.27) is not fundamental enough because it is based on the mean square distance traversed by the ion after N stqis taken in a time t and does not probe into the laws governing each stq) taken by the random-walking ion. 

This search for the atomistic basis of the diffusion coefficient will commence from the picture ofrandom-walking ions (see Sections 4.2.4 to 4.2.6). It will be recalled that a net diffusive transport of ions occurs in spite of the completely random zigzag dance of individual ions, because of unequal numbers of ions in different regions. 

Consider one of these random-walking ions. It can be proved (see Appendix 4.1) that the mean square distance jP 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)  

It has further been shown (Section 4.2.6) that in the case of a one-dimensional random walk, x depends on time according to the Einstein-Smoluchowski equation 

Equation (4.108) shows that the diffusion depends on how far, on average, an ion jumps and how frequently these jumps occur. 

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