1. Chemical Kinetics And The Time-Dependent Diffusion Equation
Consider the block below showing the diffusion of a species from left to right.
The gradient in color represents the time-
dependent concentration
C(x),t
C(x+Δx,t)
C(x-Δx,t) Ci
Ci+1
Ci-1
Consider the situation at fixed time, t. Given
,, we can determine the concentrations
to the left and right side of x, by expanding about
C(x,t) in a Taylor series.

2. Chemical Kinetics And The Time-Dependent Diffusion Equation
Addition of these expansions yield, A
Using chemical reaction rate theory we can also write an expression for the time-
dependent concentration at position xi, in terms of 1st order kinetics: B
where the forward and reverse “reactions” have identical rate constants, kf=kb=k.
We have also changed the notation since;

3. Chemical Kinetics And The Time-Dependent Diffusion Equation
Rearranging Eq. B,
Comparison of the RHS of this equation to the LHS of equations A shows that,
In this expression we can identify the coefficient of the spatial derivative as the
diffusion coefficient, Here, k has units of sec-1 and (Δx)2 has units
of (length)2.