Question

The time dependent concentration of solute particles in a solution is given by

c(r, t) = [n/8(pi D t)^(3/2)] e^(-r^2 /4 D t)

where r^2 = x^2 + y^2 + z^2 with x, y, and z being the values of three components of the position in the Cartesian coordinate system.

a) Show that the above distribution satisfies the following three dimensional diffusion equation:

(partial/partial t)c(r,t) = D (partial^2 / partial x^2 + partial^2 / partial y^2 + partial^2 / partial z^2) c(r, t)

b) Find out the expression for average distance <r> as a function of time, where <...> means average over c(r, t)

c) Find out the experession for average mean-square distance <r^2>