To illustrate the Readhead model for TPD we first choose realistic values for adsorption parameters:

- k
_{0}= 10^{13}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(-e_{A}/kT) or exp(-E_{A}/RT). - E
_{A}= 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. - b
_{H}= 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 q_{A}
versus time (and hence temperature). Then compute and plot the rate, r_{d}, versus
T; this is the model TPD spectrum. In this exercise, assume n = 1, first order rate
process, and T_{0} = 100K, the initial temperature; and q_{0}
= 0.667, the initial coverage.

We have to solve the following ODE

dq_{A}/dt = - k_{0} exp(-E_{A}/RT) q_{A}^{n} ,

in which T = T_{0} + b_{H}t. Equivalently, the
temperature can be given as a differential: T' = b_{H}.
The values of parameters are given above, although you may wish to change one or more of
them.

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, E_{A}/R, k_{0} 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.

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

A mathcad template for solving the elementary Redhead model exercise is available: redhead_1.mcd. You
can download this template to run using MathSoft's MathCad
software. This solution sets the parameter values as shown above, defines the initial
temperature, T_{0} = 100K, and initial coverage, q_{0}
= 0.667, and the rate law power, n = 1. The solution vector of initial values is set: y =
(T_{0}/K, q_{0}) [Mathcad labels components
starting with zero, y = (y_{0}, y_{1}), and the vector of time
derivatives: D(t,y) = (b_{H}, -k_{0} exp(-E_{A}/k_{B}*y_{0})
*y_{1}^{n}). A time interval t=40 and number of integration points,
npts=5000, are selected. Then mathcad's Rkadapt integration function is invoked: S =
Rkadapt(y,0,t,npts,D). The solution at each step of the interval is returned in columns of
S: t is the first column, T the second, and q_{A} is
the third. The rate is found by finite differences r_{d} = - (q_{Aj+1}
- q_{Aj})/(t_{j+1} - t_{j}). Finally
the calculated model TPD spectrum is plotted r_{d} versus T. Supplemental
exercises are also discussed in the mathcad solution: analysis of peak temperature and
coverage to find activation energy, and verification of Redhead's relation between peak
and E_{A}.