TPD Fundamentals

# 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]

## 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:

rd/Ns = -dqA/dt = k0 exp(-eA/kT) qAn.

One views the right hand side as the product of an Arrhenius rate constant, the pre-exponential k0 and the temperature dependent exponential factor, with a power of the coverage. The symbols are defined as follows.

• rd is the rate of desorption of species A (measured by mass spectrometry)
• NS is the concentration of surface adsorption sites.
• qA 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).
• k0 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.
• EA is the activation energy (per mole) for the desorption of A. Alternatively, eA (per molecule).
• k or kB is the Boltzmann constant (per molecule), 1.38066 10-23 J K-1.
• Alternatively, R (=NA kB) per mole, 1.31451 J K-1 mol-1.
• NA is Avogadro's number, 6.02214 1023.

As TPD experiments are commonly performed, the temperature is controlled by a computer to increase linearly with time: T = T0 + bH t. Here, bH is the heating rate or "ramp rate". Finally, the rate of desorption, rd, 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 qA(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:

• 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) = 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. 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.

1. Find the maximum rate, rp, the corresponding peak temperature, Tp, and the surface coverage at the peak, qp, according to the Redhead model TPD spectrum found for the Redhead exercise above. [Ans.: rp = 0.197, Tp = 400K, qp = 0.261.]
2. Use the values found in (a) to check Redhead's relation between activation energy and peak temperature and coverage: EA/RTp = ln[(k0 Tp n qpn-1)/bH] - ln[EA/(R Tp)].

Problems

1. 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 EA(Tp,qp) relation still work?]
2. When the Redhead model is used to determine the activation energy, EA is uncertain because measurement errors propagate into EA. Let the relative error in peak temperature be dTp/Tp, and that in peak coverage by dqp/qp. The desorption rate law order and pre-exponential factor are also uncertain, dk0/k0 and dn/n.
(a) Show that the relative error in activation energy is given by
(dEA/EA)2 = (1+(2 RTp/EA))2 (dTp/Tp)2 + (RTp/EA)2 (dk0/k0)2 + (RTp/EA)2 (n-1)2 (dqp/qp)2 + (RTp/EA)2(n+n(ln(qp)))2 (dn/n)2
(b) Suppose that the values and uncertainties in measured and assumed parameters are: Tp = 400K±10K, qp = 0.2±0.02, log(k0) = 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 EA.

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