I am going to describe a technique that is used fairly extensively in heterogeneous catalysis for
characterizing catalysts as well as studying reactions. It is called temperature programmed
desorption. What I want to do is describe it and I will solve the equations that describe
the behavior we observe experimentally. So the idea is pretty simple. We start off with
some gas that is adsorbed on any surface. As we raise the temperature that gas is going
to desorb into the gas phase. So what we want to do is study this desorption and what we
are going to do with the temperature- programmed desportion is measure the rate of desorption
as a function of time or as a function of temperature. So the rate of desorption is
going to be proportional to some pre-exponential factor, some exponential term containing an
activation energy, and then we will consider the case were it is proportional to the concentration.
So this is first order desorption. It does not have to be first order, but we are going
to look at the simple case. This is the rate, and this is the rate that in the actual experiment
we would plot the rate vs temperature as we raise the temperature.So this is a surface
concentration, not a gas phase concentration, but a surface concentration, and the E is
the activation energy for the desorption, and the activation energy for desorption is in many cases just equal to what is often
referred to as the heat of adsorption, where this is a positive value. So this would be
the same minus delta H of adsorption. Since its adsorption is typically exothermic. So
what happens is we raise the temperature. The most common way is a linear ramp, which
means the derivative of temperature with respect to time is a constant. We are going to use
beta to indicate this. So as we raise the temperature. We typically start at low enough
temperature that there is no desorption. So down here we can wait without being concerned
that the molecule is being desorbing at a significant rate. As we raise the temperature the rate
starts increasing exponentially, because of this term, and then what is going to happen
is that it will go through a maximum and it is going to go through a maximum because
we desorbed enough. So this term is increasing as we raise the temperature. This term is
decreasing, and eventually this term dominates, the rate decreases, we get to the point where
now everything has desorbed. So this is what we would observe in a temperature programmed
desorption experiment. The temperature increasing, we often measure this heat temperature as
the indication of how strongly the molecule is adsorbed on the surface, and of course
that temperature depends on how fast we carry out the heating. So let's look at some just some
sample numbers as a way to determine how this curve shape actually looks like. So I have
picked the rate of heating of 10 degrees per second, and in an ultra-high vacuum system
this might be a reasonable rate. We are going to start at 300 K. At this point 300 K. I
am going to use a dimensionless surface concentration. We could certainly do it in terms of molecules
per square cm. It tends to be a large number. So lets just do it dimensionless to make it
easier to see the behavior, and essentially what I am doing is solving two equations.
The mass balance, change in concentration on the surface with respect to time. Is equal
to the rate of desorption, which of course has this exponential dependence on temperature,
and we're using a first order case, and the change in temperature with respect to time is equal
to beta. Then we are going to solve these equations simultaneously. We will do this
numerically, because of this exponential term. We want to plot the change in concentration
of A with respect to time vs the temperature. This is what we referred to temperature-programmed
desorption. This would be what we typically plot. So what I am going to do now is solve
these two different differential equations numerically. I've used POLYMATH, and I will
show you the program and the output. On the left is the POLYMATH program with a number
of comments of what we are doing, but basically we are solving 2 ordinary differential equations
simultaneously, and then plotting the rate of desorption. That is what we measure experimentally as
a function of temperature, and what I am showing in these three plots are for 3 starting
concentrations. So in this case I've used values of 100, 50, and 25. The important thing to notice
is that the curves look identical except scaled by their initial concentration. In other words
if I take this curve and multiply by 2 I get this curve. If I take this curve and multiply
by 4 again. I get this curve. The peak temperature does not change for this first order process.
By this point of course everything has desorbed, and this again is what we would measure experimentally
as a way to characterize how strongly something is adsorbed on a surface. The area under these
curves is proportional to how much is adsorbed on the surface initially. So that is a brief
description of temperature program desorption. Remember this is for first order. The second
order behavior or 0 order have different temperature dependence, but the idea is the
same. We can certainly with the same program change to 2nd order or 0 order and observe
what the behavior is.