Solutions of the Redhead Exercise

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

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.

MathCad solution

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, T0 = 100K, and initial coverage, q0 = 0.667, and the rate law power, n = 1. The solution vector of initial values is set: y = (T0/K, q0) [Mathcad labels components starting with zero, y = (y0, y1), and the vector of time derivatives: D(t,y) = (bH, -k0 exp(-EA/kB*y0) *y1n). 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 qA is the third. The rate is found by finite differences rd = - (qAj+1 - qAj)/(tj+1 - tj). Finally the calculated model TPD spectrum is plotted rd 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 EA.