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In this online lecture (click "View Presentation"), Prof. Allam discusses the solution of a diffusion-capture problem.

On slide 8, he formulates two equations

[tex]I = C_0 (\rho_0 - \rho_S)\\

I = A \frac{dN}{dt} = A k_F N_0 \rho_S[/tex]

from which he arrives at an expression for [tex]N(t)[/tex], namely [tex]N(t)=\rho_0 t\left[\frac{A}{C_0} + \frac{1}{k_F}\right]^{-1}.[/tex]

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

Can someone point me out on how to arrive at the expression for [tex]N(t)[/tex]? Would be greatly appreciated.

The work is further discussed in Nair et al., APL 88, 233120, 2006.