A very short pulse injection of tracer is made into a column. When the center

of mass has traveled $2$ m$,$ the standard deviation is $0\mathrm{.}1$ m$.$ The molecular diffusion

coefficient, Dw$,$ of this tracer is $10-5$ cm$2/$s$.$ The porosity is $0\mathrm{.}5.$ The tortuosity is $2.$

$($a$)$ What is the longitudinal dispersion coefficient $($or mechanical dispersion coefficient$),$DL $($or Dm$),$ if the average velocity of pore water is $1$ mm$/$s$?$ What is the effective diffusion coefficient, Deff? Is an assumption that Deff $<<$ DL valid for this tracer?

$($b$)$ What is the dispersivity, $\backslash$alpha $?$

$($c$)$ How does DL vary with velocity? How does $\backslash$alpha vary with velocity?

$($d$)$ If the tracer were replaced by a different tracer having much lower molecular

diffusivity $($perhaps something like latex microspheres having an effective molecular

diffusivity of $10-8$ cm$2/$s$),$ would the tracer distribution be different at this point in the column? $($Assume the spheres move freely through the porous medium and do not get filtered out or stuck.$)$