The concentration
at distance
from the center of the spherical reactor satisfies the diffusion equation
if
and 0 if
, in spherical coordinates, where
is the diffusion coefficient,
is the radius of the spherical reactor, and
is the reaction rate coefficient. The sphere is in an unbounded fluid. In the units used here, the concentration approaches
far from the sphere.
The reaction inside the sphere is first-order kinetics, meaning the reaction rate is proportional to the concentration: the number of molecules removed from the fluid per unit time and per unit volume is
at a position with distance
from the center of the sphere. The removal of the chemical in the reactor creates a concentration gradient and hence a diffusion of the chemical from the surrounding fluid into the sphere.
For comparison, the fully absorbing sphere satisfies
for
with boundary condition
.
The fully absorbing sphere gives the maximum possible rate a sphere can absorb the diffusing chemical. The reactor gets close to this maximum by consuming arriving molecules before they have a chance to diffuse back out of the reactor. Achieving this requires the typical distance a molecule in the reactor diffuses before being consumed,
, to be small compared to the radius of the sphere,
. In this case, most of the consumption occurs close to the surface of the sphere.
H. C. Berg,
Random Walks in Biology,
ed., Princeton: Princeton University Press, 1993.
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