0:38

These questions are intimately related to the properties of adsorption.

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

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

Â between different kinds of adsorption, derive the isotherm for

Â localized and mobile adsorption,

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

Â 1:07

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

Â gas phase, they can bound to a solid substrate.

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

Â Now, why does this happen?

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

Â For example, van der Waals forces.

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

Â chemical, and biological processes, involving transformation at surfaces.

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

Â where there is negligible interactions between adsorbed molecules.

Â Now we will call this the non-interacting case.

Â 1:54

Now let's consider the following two scenarios.

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

Â where the molecule likes to adsorb.

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

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

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

Â any site on the surface.

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

Â this leads to the second case known as mobile adsorption.

Â Now in the case of localized adsorption, each adsolved molecule

Â is trapped in a potential well around the adsorbed site.

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

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

Â Now this system is akin to the one dimensional harmonic oscillator

Â that we have studied.

Â Now each direction can be assumed to be decoupled.

Â 2:59

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

Â partition function of an adsorbed molecule as simply the product

Â of the one dimensional harmonic oscillator in each direction.

Â Now each adsorbed molecule occupies a lattice site

Â with a distinct spatial location.

Â So are these particles distinguishable or indistinguishable?

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

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

Â What is the degeneracy of this configuration?

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

Â Now the canonical partition function for

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

Â times the single molecule partitition raised to the power N.

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

Â a partial derivative of the logarithm of the partition function

Â with respect to the number of adsorbed molecules.

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

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

Â the chemical potential of the adsorb state.

Â Remember for

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

Â One that depends only on temperature and

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

Â Now equality condition from the equilibrium

Â leads to the famous Langmuir Adsorption Isotherm.

Â 4:32

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

Â sites occupied by the molecules adsorbed versus the temperature and

Â the pressure of the gas phase adsorbent.

Â Now for localized adsorption, as the pressure increases,

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

Â Now how does mobile adsorption differ from this?

Â In the case of mobile adsorption,

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

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

Â Hence the single particle partition function

Â is composed of multiplying a one dimensional harmonic oscillator

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

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

Â Now comes the question of distinguishability.

Â What do you think?

Â 5:29

Are the mobile adsorbed molecules distinguishable or indistinguishable?

Â Well, the mobile adsorbed molecules are indistinguishable since

Â they're not restricted to a localized adsorption site.

Â Hence the canonical partition function needs to be corrected for

Â the indestinguishability factor that is N factorial.

Â Now we can evaluate the chemical potential similar to that

Â of the localized adsorption case.

Â For an adsorption from an ideal gas,

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

Â Now, all things being the same,

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

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

Â