Here we're gonna look at the Langmuir adsorption isotherm.
This is an equilibrium isotherm.
And the idea is we have some solids,
in this case, in the solid,
if it's adsorbing something, is referred to as an adsorbent
and then we have some component B out here in the gas phase
and some of those B molecules adsorb on the surface.
So B would be called the adsorbate.
And of course adsorption is used in a number of processes
for separations, also important in catalysis.
And the Langmuir isotherm is one
of the simpler representations
of how the number of molecules adsorbed,
which is on the Y axis.
It also could be number of grams adsorbed
per gram of adsorbent, as a function of pressure
if we're looking at gas fades adsorption.
If it was liquid phase, we looking at concentration.
And so the form of the Langmuir isotherm is the amount
of B adsorbed, a constant
an absorption equilibrium constant,
and the pressure B.
And so this form means that low enough pressures down here
XB would just be linear in the pressure B.
And if we get to high enough pressures,
high enough so that this term is large relative to one,
then XB would just be equal to alpha.
In other words, you're gonna saturate out
at some value at high enough pressures.
Now, the Langmuir isotherm is a function of temperature
and we'll look at a simple simulation now to see what
that temperature dependence is.
Well, here we're looking at a simulation
for the Langmuir isotherm.
Now this happens to be case
where we could have two adsorbates,
but essentially made one of them zero.
So we can look at this total pressure
then really it's just the pressure B
and I'm showing a plot of isotherm
and what I can do in this simulation,
I can lower the temperature
and those isotherm now
at lower temperature will approach saturation
at significantly lower pressure.
And I can also raise the temperature.
So over this pressure range
it's pretty much linear.
Correspondingly, instead of lowering that temperature,
I could increase how strongly B is adsorbed
on the surface and that then creates an isotherm
that is the same behavior as if I lowered the temperature.
So the Langmuir isotherm results
by a fairly simple assumption
and doesn't mean it's going to apply
in most cases, but it's a good first case.
And that's just saying an equilibrium,
of course, the rate of adsorption is the rate
of desorption.
That's going to be true.
What the assumption is that we can write the rate
of adsorption as some adsorption rate constant,
the pressure of component B times one minus the fraction
of sites that are occupied
by B, or this term is the fraction of vacant sites.
And the rate of desorption, the desorption rate constant
times the fraction of the sites occupied
by B instead of fraction.
Of course, it could be the number of sites
and that just changes a constant.
So if I just rearrange this equation,
I get that the fraction
of B is KB pressure B,
one plus KB pressure, B
and if instead of fraction,
I wanted to write it in terms of number of grams,
for example, this term, alpha would then be in the equation.
So you could look at theta, is this XB over alpha.