# Temperature Programmed Desorption (TPD)

## A method to study thermodynamics and kinetics of desorption from solid surfaces.

- Reference: "Principles of Adsorption and Reaction on Solid Surfaces" by Richard I. Masel [Wiley, 1996]

## Contents:

- TPD Experiments
- Redhead Model of TPD
- Redhead Exercise with solution
- Exercises and Problems
- Extended Redhead Model of TPD

## TPD Experiments

**TPD experiments** begin with a gas or mixture of gases adsorbed onto a
cold crystal surface (often a metal crystal). This surfaces is then heated at a controlled
rate (programmed rate). The adsorbates will then react as they are heated and the reaction
products desorb from the surface. A mass spectrometer is used to monitor the desorption
products. The results of the experiment are the desorption rate of each product species
versus the temperature of the surface, the **TPD spectrum**.

McCabe and Schmidt studied the TPD spectrum of carbon monoxide adsorbed on platinum
[R.W. McCabe, L. D. Schmidt, Surface Sci. **66**, 101 (1977)]. Their Fig. 3,
reproduced below, shows the measured TPD spectrum for CO adsorbed at 300K on the (110)
crystal face of Pt (solid line). The dashed line is calculated according to the Redhead
model used to interpret the spectrum. The temperatures are in Kelvin and the rate of
desorption is shown on an arbitrary scale.

Using these data and this model, McCabe and Schmidt found the activation energy for
desorption of CO from (110) Pt to be 26.0 kcal mol^{-1}.

## Redhead Model of TPD

Theoretical models of TPD are used to interpret the TPD spectra with the goal of
determining themodynamic and kinetic parameters. A 1963 paper by Redhead shows how this
may be done [P.A. Redhead, *Vacuum* **12**, 203 (1963)]. Redhead's
model is based on the assumption that the desorption process follows a simple power rate
equation:

r_{d}/N_{s} = -dq_{A}/dt = k_{0}
exp(-e_{A}/kT) q_{A}^{n}.

One views the right hand side as the product of an Arrhenius rate constant, the
pre-exponential k_{0} and the temperature dependent exponential factor, with a
power of the coverage. The symbols are defined as follows.

- r
_{d}is the rate of desorption of species A (measured by mass spectrometry) - N
_{S}is the concentration of surface adsorption sites. - q
_{A}is the coverage of species A on adsorption sites (fraction of sites occupied by A molecules). - t is the time (the independent variable of the rate process).
- k
_{0}is a pre-exponential factor for the rate constant (depends on atomic masses and bond strengths in species A). - T is the temperature in Kelvin (K).
- n is the "order of the desorption reaction (the power of the rate law). First order: n=1, etc.
- E
_{A}is the activation energy (per mole) for the desorption of A. Alternatively, e_{A}(per molecule). - k or k
_{B}is the Boltzmann constant (per molecule), 1.38066 10^{-23}J K^{-1}. - Alternatively, R (=N
_{A}k_{B}) per mole, 1.31451 J K^{-1}mol^{-1}. - NA is Avogadro's number, 6.02214 10
^{23}.

As TPD experiments are commonly performed, the temperature is controlled by a computer
to increase linearly with time: T = T_{0} + b_{H}
t. Here, b_{H} is the heating rate or "ramp
rate". Finally, the rate of desorption, r_{d}, is proportional to the
intensity of the mass spectometric peak at the mass of species A being desorbed. One
monitors this MS peak versus time as the temperature of the surface/adsorbate increases.

To use the Redhead model, one solves the rate expression for q_{A}(t),
coverage versus time. As given, this is a first order ODE that yields to separation of
variables. Instead, we will illustrate a solution using numerical integration that can be
readily extended to more complicated and realistic models.

## Redhead Exercise

To illustrate the Reahead 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) = 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. You can set up and solve the rate equation for the given
parameter values by any computational method you prefer. When you are finished, you can
compare with the method and solution found here (click on the graph): .

## Exercises and Problems:

**Supplemental exercise**: The peak in the TPD spectrum plays an important
role in interpreting the desorption process. In particular, the peak temperature helps to
determine the activation energy for desorption.

- Find the maximum rate, r
_{p}, the corresponding peak temperature, T_{p}, and the surface coverage at the peak, q_{p}, according to the Redhead model TPD spectrum found for the Redhead exercise above. [Ans.: r_{p}= 0.197, T_{p}= 400K, q_{p}= 0.261.] - Use the values found in (a) to check Redhead's relation between activation energy and
peak temperature and coverage: E
_{A}/RT_{p}= ln[(k_{0}T_{p}n q_{p}^{n-1})/b_{H}] - ln[E_{A}/(R T_{p})].

**Problems**

- If the desorption process requies that two adsorbed molecules must collide on the
surface before one molecule of species A is desorbed, then the order or power of the rate
law is expected to be n = 2. Calculate and plot the rate (and coverage) verus T for n=2;
use the same activation energy, pre-exponential, and initial conditions as before. Compare
with the n=1 results. [Is the peak temperature greater or smaller (why?)? Does Redhead's E
_{A}(T_{p},q_{p}) relation still work?] - When the Redhead model is used to determine the activation energy, E
_{A}is uncertain because measurement errors propagate into E_{A}. Let the relative error in peak temperature be dT_{p}/T_{p}, and that in peak coverage by dq_{p}/q_{p}. The desorption rate law order and pre-exponential factor are also uncertain, dk_{0}/k_{0}and dn/n.

(a) Show that the relative error in activation energy is given by

(dE_{A}/E_{A})^{2}= (1+(2 RT_{p}/E_{A}))2 (dT_{p}/T_{p})^{2}+ (RT_{p}/E_{A})^{2}(dk_{0}/k_{0})^{2}+ (RT_{p}/E_{A})^{2}(n-1)^{2}(dq_{p}/q_{p})^{2}+ (RT_{p}/E_{A})^{2}(n+n(ln(q_{p})))^{2}(dn/n)^{2}

(b) Suppose that the values and uncertainties in measured and assumed parameters are: T_{p}= 400K±10K, q_{p}= 0.2±0.02, log(k_{0}) = 13±2, and n = 2±1. Calculate the approximate uncertainty in activation energy. Also, determine which measurement or assumption introduces the biggest (smalles) uncertainty to E_{A}.

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