We assume that we have
available sites per unit area on the clean surface.
All sites on the
surface are equivalent and the adsorption energy is independent from
the coverage of the surface, i.e. all the adsorbate-adsorbate
interaction which we have discussed above are turned off.
Figure: The Langmuir model for adsorption: (a) associative
adsorption (first order process); (b) dissociative adsorption
(second order process). (c) simple energy diagram with activation energies for adsorption and desorption ( 
and 
) and the heat of adsorption 
.
where M is the mass of the incoming molecules. The adsorption rate is now given by the product of the incoming flux with a suitably defined sticking coefficient.
Let's assume for the sticking coefficient that
where 
is the sticking coefficient for the clean surface. This
leads to
The factor c takes care of the fact that only a fraction of the
incoming molecules actually adsorb, even if they find a site. They can
also simply bounce back if the can not get rid of their kinetic energy
or if they come in ``turned the wrong way''. The factor
takes the available sites into account where
is the relative coverage and n is the order of the process.
The last factor takes care of a possible activation energy
necessary
to adsorb a molecule. Rewriting the expression using
highlights the sticking coefficient for the clean surface.
Using this model one can already try to analyse a lot of simple
experiments. Assume that we dose gas on a surface and measure the
coverage as a function of dosage. From the slope of the curve we can
work out the sticking coefficient as a function of coverage and
compare it to our simple model. Fig. 
shows that the
results are often not too good.
Figure: Sticking probability of N 
on tungsten as a function
of coverage. After Ref. [23].
). Desorption takes place if a molecule has enough
energy to overcome an activation energy
for desorption.
Alternatively, two atoms or molecules from neighbouring sites
might desorb together, forming a new molecule.
Figure: Desorption in the Langmuir model.
Again, this equation contains a factor such that not all the molecules with sufficient energy desorb, a factor describing the probability of finding two neighbouring sites which are occupied (for n=2) and a factor for the activation energy . Thermal desorption is directly used in a very common experimental technique called Thermal Desorption Spectroscopy (TDS) . In TDS, one prepares an adsorbate layer with a certain coverage on a surface. Then one places this surface in front of a mass spectrometer and measures the partial pressure from a certain mass one is interested in while (linearly) increasing the temperature of the sample. Let's assume
and
The first condition can be realized by an electronically controlled
ramping of the temperature. The second condition is
only valid at very high pumping speed in the UHV vessel. The measured
increase in partial pressure as a function of time can now be fitted
with the model equation
to obtain the relevant parameters,
in particular the desorption energy .
Much information can already be gained just by looking at
desorption curves like in Fig. .
The figure shows two sets of curves for first and second order
desorption. Every set contains curves for different initial coverages.
One finds that the
maximum of the desorption curves in independent from the initial
coverage for
a first order process but not for a second order process.
Figure: Thermal desorption curves for a linear heating rate:
(a) a first order process (n=1) and (b) a second order process n=2.
shows a thermal desorption spectrum
of H from a tungsten surface. An obvious implication of such a
curve is that there are at least four different binding configurations
of hydrogen on the surface.
Figure: Thermal desorption spectrum for H from a tungsten
surface. After Ref. [24].
) equal to () and
obtain
or
where is the heat of adsorption
, i.e. 
(see Fig. ).
It is actually possible to obtain values for using the
Clausius-Clapeyron equation on isostere data, i.e. data where one
measures the pressure to keep a certain surface coverage as a
function of temperature.
where R is the gas constant. The data for the Clausius-Clapeyron analysis can either be taken from a real isostere measurement or the isostere points can be figured out from measured isotherms. One thing one has to keep in mind, though, is that the major trouble with the equilibrium approaches is that they can not be used to study irreversible processes!
