library(tidyverse)
knitr::opts_chunk$set(tidy.opts=list(width.cutoff=60),tidy=TRUE, echo = TRUE, message=FALSE, warning=FALSE, fig.align="center")Starting with 1D Diffusion over time (t) and infinite space (x) the concentration in the system is described by: \[C(x,t) = \frac{M_0}{A \sqrt{4 \pi D t}}e^{\frac{-x^2}{4 D t}}\]
For a no-flux boundary at x=0, and evaluating at x=0:
\[C_{x=0}(t) = \frac{2 M_0}{A_e \sqrt{4 \pi D_m t}}\]
Substituting in from the equation for \(I_{swv}\) gives:
\[I(t) = \frac{2 I_0 V_e}{A_e \sqrt{4 \pi D_m t}}\]
Since \(A_e\) is the area of the electrode, and \(V_e\) is the volume of solution probed by the electrode \(V_e / A_e\) depends on the diffusion layer at the electrode. \[\frac{V_e}{A_e} = \sqrt{D_{ap} t_s}\]
So, our model can be described as: \[I(t) = \frac{I_0 \sqrt{D_{ap} t_s}}{\sqrt{\pi D_m t}}\]
Fitting the data to the equation, \[y = \frac{a}{\sqrt{t}} + b\]
Allows us to write down \(a\) as:
\[ a = \frac{I_0 \sqrt{D_{ap} t_s}}{\sqrt{\pi D_m}}\]
Therefore \(D_m\) is:
\[D_m = \frac{I_0^2 D_{ap} t_s}{\pi a^2}\]
\[y = m x + b\]