Consider a bio$-$film that is associated with a skin infection. Bio$-$films, which are

colonies of bacteria that can cling to living and inert surfaces, can cause wide array of

human infections. Infections caused by bacteria living within bio$-$films are often

chronic, because antibiotics that are applied to the surface of the bio$-$films have

difficulty penetrating through the film thickness.

An antibiotic $($species A$)$ is applied at the top layer of a bio$-$film $($species B$)$ so that a fixed

concentration of medication CAo$=4$x$10-3$ kmol$/$m$3$

exists at the upper surface of the biofilm. The diffusion coefficient of the antibiotic within the biofilm, DAB$=2$x$10-12$ m$2$

$/$s$.$ The

antibiotic is consumed by biochemical reactions within the film and the consumption rate

depends on the local concentration of A: RA$=$ k$1$ CA ; k$1=0\mathrm{.}1$ s$-1$

$.$ To eradicate the bacteria

the antibiotic must be consumed at least at the rate of $($RA$=)0\mathrm{.}2$x$10-4$ kmolm$-3$

s

$-1$

$,$ since at

smaller absolute consumption rates, the bacteria will be able to grow back faster than it is

destroyed.

What is the maximum thickness of the biofilm that can be successfully treated by the

antibiotic mentioned above? Show the necessary derivation.