# Solutions of the Redhead Exercise

• k0 = 1013 s-1 , the pre-exponential factor describes the frequency that species A vibrates in the direction leading to desorption. The fraction of such vibrations that lead to desorption increases with temperature as determined by the Arrhenius exponential factor, exp(-eA/kT) or exp(-EA/RT).
• EA = 24 kcal mol-1, the Arrhenius activation energy for the desorption process. This is the minimum energy required to surmount a barrier to desorption. It is the depth of the well in which species A is held to the surface adsorption site.
• bH = 10 K s-1 , the ramp rate gives the rate at which the surface is heated.

Our goal is to compute the model TPD spectrum. First we solve for qA versus time (and hence temperature). Then compute and plot the rate, rd, versus T; this is the model TPD spectrum. In this exercise, assume n = 1, first order rate process, and T0 = 100K, the initial temperature;  and q0 = 0.667, the initial coverage.

We have to solve the following ODE
dqA/dt = - k0 exp(-EA/RT) qAn ,
in which T = T0 + bHt. Equivalently, the temperature can be given as a differential: T' = bH. The values of parameters are given above, although you may wish to change one or more of them.

## DYNASYS solution

Dynasys can solve this model. First, it is helpful to modify the equations:
T' = beta, this the ramp rate equation;
q' = -w, this the rate of desorption (q in place of q for coverage);
w = exp(-eok/T + lnk0)*q^n, this is the desorption rate expression according to Redhead's model.
Dynasys uses the Latin alphabet. For simplicity we reduce the number of constants to four: b, EA/R, k0 and n. But k0 has been absorbed into the exponential: k0 = exp(ln(k0)). You can write your own dynasys solution file and check the result against the following solution file. Dynasys allows you to change parameters (or equations), solve and plot again without clearing any existing plots; this makes comparisons easy.

## Java Applet solution

Here is an applet for solving this ODE. You can change the values of parameters in the applet to solve related problems.