I am having a hard time applying the grand canonical theory to a simple example. I expose my understanding of the matter, the problem, my attempt of solution, the solution and my question on this solutions; I apologize for the lengthy question and will be very grateful to whoever feels like going through it!
I also add an answer including some ideas and a second solution.
I follow Kubo, Statistical Mechanics, but having a look around the notation should be standard. An open system is in contact with a reservoir fixing temperature and chemical potential . A microstate of the open system is denoted by ; the grand canonical partition function is
where denotes each available microstate of the system, the number of particles in that microstate, and the energy of that microstate. This can be related to the canonical partition function : let denote a microstate for fixed number of particles, then
This is useful: is relatively hard to compute due to the fixed number of particles condition, but allows to get rid of this condition. We consider the single particle properties:
- runs over the single particle possible microstates
- denotes the energy of the state , that is the energy that a single particle has when happens to be in the state
- is the occupation number of the state , that is the number of particles that happen to be in the state . For fermions ; for bosons .
A microstate of the whole system is then specified by the sequence of occupation numbers , and
We define the single state grand canonical partition function
We consider a gas in contact with a solid surface (e.g. argon on graphene or molecular nitrogen on iron, as in the Haber-Bosch synthesis). The gas molecules can be adsorbed at specific adsorption sites while one site can only bind one molecule. The energies of the bound and unbound state are and 0, respectively. The gas acts as a reservoir fixing and .
The grand canonical partition function should then read ()
What is wrong with this approach?
With no further explanation beyond the fact that the sites are non-interacting the lecturer, this page and this page claim
- Is the used here the same as the single state grand canonical partition function defined above?
- Where is from?
The similar canonical relation for non interacting systems of identical particles goes like this: we start with N distinguishable particles labeled by ; is the -energy level of the -particle. Then
The subscript can not be dropped in the relation , as is an object strictly related to a state , so again, how is obtained?
I'm not sure I will be able to clarify all your doubts, but this is the right approach to tackle this problem.
We consider available adsorption sites, an energy for each bound state, chemical potential and temperature .
The grandpartition function is always expressed as
EDIT: One can also reason directly using the grandpartition function as follows. Using the occupation number representation for noninteracting particles, with labelling single-particle states with energy and -fold degeneracy ,
The questions took literally hours to be written and during the writing I may have gained a partial understanding of the problem, which I'll try to expose here.
What is wrong with this approach?
The choice of the system: a fixed number of sites does not make a good gran canonical ensamble and does not apply.
As given in the solution, . Formally this is precisely
This expression resembles the one of the full grand canonical partition function and suggests the following interpretation: it describes a system
This system is one adsorption site, and the elements in the system are the captured particles. This number is allowed to change (between zero and one), so this is a good gran canonical ensamble. This picture may clarify the situation. On the left the array of adsorption sites is the system and one site is an element; on the right one site is the system and the captured (or not) particle is the element.
It then makes sense to write, for the system on the right side
Kubo, pag. 92. He denotes by the number of full sites, i.e. of captured particles, and calculates the canonical partition function. What should not be done in two wrong albeit attempting ways giving the same result:
Way one
Way two
This time we consider as the system the occupied sites: the number of occupied sites can vary between and and the energy of the system in the microstate is , so (using Newton's formula)
What should be done is calculating first the canonical partition function for a given value of between and . For this fixed value the energy of the system is always with degeneracy :
This could be taken as the sought proof that
I've found this way to understand (please someone notify me if this is considered a wrong general formula!):
We know that:
ZG= ZG1 * ZG2 * ...
Lets consider the case with 1 single-particle energy, but with a degeneracy of g1=2. Then the ZG can be written as:
ZG= ZG1 * ZG1 ( since we have to take into consideration every singe particle state).
So it's actually ZG= (ZG1)^g1 .
You can easily expand to r singe-particle states with degeneracy gr each, to :
ZG= ((ZG1)^g1 )* ((ZG2)^g2) * * * ((ZGr)^gr)
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