Chemical Kinetics And The Time-Dependent Diffusion Equation
Consider the block below showing the diffusion of a species from left to right.
Ci-1 Ci Ci+1
The gradient in color represents the timedependent concentration
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.
C(x),t
C(x-Δx,t)
C(x+Δx,t)
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 timedependent 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;
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.