How Should the Partition Function of Bound O2 Be Calculated?

[Kelly Lin]

Homework Statement

Homework Equations

I have question for (a) section.

The Attempt at a Solution

I have two answer for the question but I can't figure out which one is right.
(1)Since the partition function is to sum up all the state in the system, I write down the answer

(2)In other point of view, we can first find out the partition function of one O2 molecule.

(either bound or unbound)

And then, since a hemoglobin can only bind four O2 molecules, I can write down

In my opinion, I think the first solution is more accurate since I include all the situation of that system. However, I cannot persuade myself that the second solution is wrong.(Or the reason is that Z=[Z₁]^N can only apply to large system?)

Please help me out~ Thanks!

 

[mjc123]

In (1), you have neglected the multiplicity. How many ways are there of binding 1 of 4 molecules? How many for 2? 3? 4?

 

[Kelly Lin]

In (1), you have neglected the multiplicity. How many ways are there of binding 1 of 4 molecules? How many for 2? 3? 4?

What do you mean?
My perspective is that a hemoglobin can bind at most 4 O2 molecules.Therefore, first possibility is that a hemoglobin binds one O2 molecules and second possibility is that a hemoglobin binds two O2 and so on... Moreover, since O2 molecules are identical and indistinguishable (am I right?), I write down the first solution.

 

[mjc123]

OK, reading the question again, I don't think it's saying there are four O₂ molecules (bound or free) in the system, rather that the hemoglobin molecule has four oxygen binding sites, and the question is how many are filled (i.e. we are looking for the partition function of the hemoglobin, not the oxygen). We must assume there is an excess of oxygen, so that whether an oxygen molecule binds to a site is independent of whether the other sites are already filled. We than ask: how many ways are there of filling exactly one site? or two...?
(If we look for the PF of 4 oxygen molecules, with the equations you have written, the probabilities in (b) and (c) would be independent of λ.)

 

[Kelly Lin]

OK, reading the question again, I don't think it's saying there are four O₂ molecules (bound or free) in the system, rather that the hemoglobin molecule has four oxygen binding sites, and the question is how many are filled (i.e. we are looking for the partition function of the hemoglobin, not the oxygen). We must assume there is an excess of oxygen, so that whether an oxygen molecule binds to a site is independent of whether the other sites are already filled. We than ask: how many ways are there of filling exactly one site? or two...?
(If we look for the PF of 4 oxygen molecules, with the equations you have written, the probabilities in (b) and (c) would be independent of λ.)

Oh! I got your point!
So if we focus on how many sites are filled, then we can write down

Then my first and second solutions might be wrong? Am I correct?

 

[mjc123]

You need an extra +1 (for no sites filled). And you still need to work out the multiplicities. E.g. you have four sites. One of them is filled. How many ways can you do that?

 

[Kelly Lin]

You need an extra +1 (for no sites filled). And you still need to work out the multiplicities. E.g. you have four sites. One of them is filled. How many ways can you do that?

Oh! You mean that we should consider the position choice. Is that correct?

 

[mjc123]

Yes

 

[Kelly Lin]

Yes

Thanks for your help!