Adsorption of Molecules Onto a Surface
Many thanks to Bart Andrews for this contribution!
Question
Consider a gas in contact with a solid surface. The molecules of the gas can adsorb to specific sites on the surface. These sites are sparsely enough distributed over the surface that they do not directly interact. In total, there are N adsorption sites, and each can adsorb n = 0, n = 1, or n = 2 molecules. When an adsorption site is unoccupied, the energy of the site is zero.
When an adsorption site is occupied by a single molecule, the energy of the site is
The gas above the surface can be considered as a heat and particle reservoir with temperature T and chemical potential μ.
- Calculate the grand canonical partition sum
. - Calculate the grand canonical potential J.
- Calculate the mean number of adsorbed molecules on the surface directly from
. - Calculate the mean number of adsorbed molecules on the surface directly from J.
- Calculate the probability that an adsorption site is in the state with
and .
Solution
1. Calculate the Grand Canonical Partition Sum
In the question we are told that the binding sites are “sparsely enough distributed over the surface that they do not directly interact” and so we can use the relation
So the first term is for the
2. Calculate the Grand Canonical Potential
For this part you can use the definition of the grand canonical potential and simply substitute in the expression for the grand canonical partition sum.
3. Calculate the mean number of adsorbed molecules from
For this part we can use a similar approach as to what was shown in lectures.
So in the brackets we are calculating the mean number of adsorbed molecules per binding site and then we multiply this by
4. Calculate the mean number of adsorbed molecules from J
Here, we need to make use of the thermodynamic relation between the grand canonical potential, the chemical potential and the particle number.
This is the same expression as we derived in part three, as required.
5. Probability an adsorption site is in the state
This expression is simply the state divided by the sum of all possible states.