# Biosensor diffusion equation solution

1. Oct 11, 2012

### mzh

Hi
In this online lecture (click "View Presentation"), Prof. Allam discusses the solution of a diffusion-capture problem.
On slide 8, he formulates two equations
$$I = C_0 (\rho_0 - \rho_S)\\ I = A \frac{dN}{dt} = A k_F N_0 \rho_S$$
from which he arrives at an expression for $$N(t)$$, namely $$N(t)=\rho_0 t\left[\frac{A}{C_0} + \frac{1}{k_F}\right]^{-1}.$$

Here, $$I, C_0, \rho_0, \rho_S$$ indicate a flux of the species with concentration $$N$$ over a surface $$A$$ and $$\rho_0, \rho_S$$ are the concentrations in equilibrium and at the surface of the sensor where the capture with the rate $$k_F$$ happens.

Can someone point me out on how to arrive at the expression for $$N(t)$$? Would be greatly appreciated.
The work is further discussed in Nair et al., APL 88, 233120, 2006.