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.

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;

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.