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These questions are intimately related to the properties of adsorption.

And adsorption is a phenomenon that engineers use in many designs.

Now, by the end of this lecture you should be able to distinguish

between different kinds of adsorption, derive the isotherm for

localized and mobile adsorption,

incorporate the effect of correlations, and finally relate it to the Ising model.

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Adsorption occurs when atoms or molecules, typically in the vapor or

gas phase, they can bound to a solid substrate.

The vapor or liquid phase is the adsorbate and the solid substrate is the adsorbent.

Now, why does this happen?

Well, typically, his is due to some sort of favorable interactions.

For example, van der Waals forces.

The phenomenon of adsorption is very important in a variety of physical,

chemical, and biological processes, involving transformation at surfaces.

Now let's first consider adsorption on a solid lattice,

where there is negligible interactions between adsorbed molecules.

Now we will call this the non-interacting case.

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Now let's consider the following two scenarios.

The first scenario is that there is a special spot on the surface

where the molecule likes to adsorb.

Now, this would mean that the adsorbed molecule is highly localized

on the surface, and we will call this localized adsorption.

Now, the second case is when the molecule does not have any real preference for

any site on the surface.

Now, in this case the molecule is highly mobile on the surface and

this leads to the second case known as mobile adsorption.

Now in the case of localized adsorption, each adsolved molecule

is trapped in a potential well around the adsorbed site.

Now to a first approximation what we would do is we could expand out that potential

in the x, y, and z direction as a quadratic function in displacement.

Now this system is akin to the one dimensional harmonic oscillator

that we have studied.

Now each direction can be assumed to be decoupled.

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And now what we can do is we can write down the single particle

partition function of an adsorbed molecule as simply the product

of the one dimensional harmonic oscillator in each direction.

Now each adsorbed molecule occupies a lattice site

with a distinct spatial location.

So are these particles distinguishable or indistinguishable?

Now because of their localized location, they are indeed distinguishable particles.

Let's take the case that the lattice has M sites and containing N adsorbed molecules.

What is the degeneracy of this configuration?

Well, this is simply given by the combinatorial factor M choose N.

Now the canonical partition function for

the localized adsorption is given by the product of the degeneracy factor

times the single molecule partitition raised to the power N.

Now the chemical potential of the adsorbate can be evaluated by taking

a partial derivative of the logarithm of the partition function

with respect to the number of adsorbed molecules.

Now, let's assume that the molecules are adsorbing from the gas phase.

Now this implies that there is an equilibrium between the gas phase and

the chemical potential of the adsorb state.

Remember for

an ideal gas, the chemical potential can be written as a sum of two parts.

One that depends only on temperature and

then a pressure dependent part that scales as the logarithm of the pressure.

Now equality condition from the equilibrium

leads to the famous Langmuir Adsorption Isotherm.

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This equation describes the surface coverage theta that is the fraction of

sites occupied by the molecules adsorbed versus the temperature and

the pressure of the gas phase adsorbent.

Now for localized adsorption, as the pressure increases,

then the coverage turns to unity, that is it covers the full total surface.

Now how does mobile adsorption differ from this?

In the case of mobile adsorption,

the molecule is free to vibrate in a potential well in the z direction.

However in the x-y plane, it has free mobility.

Hence the single particle partition function

is composed of multiplying a one dimensional harmonic oscillator

in the zed direction with a two dimensional ideal gas in the x-y plane.

Note that in this expression A is the surface area of the adsorbent.

Now comes the question of distinguishability.

What do you think?

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Are the mobile adsorbed molecules distinguishable or indistinguishable?

Well, the mobile adsorbed molecules are indistinguishable since

they're not restricted to a localized adsorption site.

Hence the canonical partition function needs to be corrected for

the indestinguishability factor that is N factorial.

Now we can evaluate the chemical potential similar to that

of the localized adsorption case.

For an adsorption from an ideal gas,

the surface coverage can be easily evaluated in an analogous way.

Now, all things being the same,

this leads to a number of adsorbed molecules in the mobile adsorption

case that is higher than that of localized adsorption at all pressures.