Deriving Ideal Gas Law through partition function

[Mangoes]

Homework Statement

The pressure of a non-interacting, indistinguishable system of N particles can be derived from the canonical partition function

$$P = k_BT\frac{∂lnQ}{∂V}$$

Verify that this equation reduces to the ideal gas law.

The Attempt at a Solution

I have a very poor background in quantum mechanics and probability and this is the first course I'm taking which relies on any of the two. The course I'm taking teaches thermodynamics through a blend of the classical point of view and the microscopic statistical point of view.

From what I understand, for an indistinguishable system, the total partition function Q can be related to the individual molecular partition functions q by

$$Q = \frac{q^N}{N!}$$

Using properties of logarithms,

$$lnQ = Nlnq - lnN!$$

Since the partial with respect to V of lnN! is 0, I have so far:

$$P = k_BTN\frac{∂lnq}{∂V}$$

But when I look at the molecular partition function q, I have absolutely no idea how I'd be able to relate it to volume to be able to differentiate. The definition of the molecular partition function q I have is:

$$q = \sum{e^{-\frac{E_j}{k_BT}}}$$

The only thing that comes to my mind is that I know that the total energy is going to be the sum of translational, rotational, vibrational, and electronic energy levels in a molecule. It'd make sense to me for translational energy levels to have something to do with volume and pressure, but I'm not sure how to draw the connection or whether I'm even going down the right track. A lot of these concepts are completely new to me.

Where else can I go with this, assuming this isn't completely off? I'd appreciate any help.

 

[BvU]

You have a summation over something. If the number of these somethings is proportional to V then you are where you want to be ! As you say, so my response is a bit too quick. I think you are on the right track.
Total energy and N are fixed already...

 

[BvU]

The next step indeed has a link to quantum mechanics. Particle in a box. If unfamiliar, you have to look it up; very important and very useful later on as well. Energy levels are discrete and have something to do with n/L (length of one side). Three directions get you a factor L^3=V after summation (done by integrating) . http://pollux.chem.umn.edu/4501/Lectures/3501_lecture_E.pdf

 

[hjelmgart]

You are right, that it is related to tranlational and internal energy of the system, however, the evidence may be a bit long/complicated, but what it comes down to is a partition function for N particles, perhaps it has already been mentioned somewhere? You should be able to use that expression, I think.

What I mean is, you probably have a more generalised expression for q?

 

[Mangoes]

Thanks for the replies.

I looked through the link that BvU gave and it does mention the translational energy states equation that I was thinking of previously. My book goes through a derivation that shows that the translational partition function is given by

$$q_{trans} = (\frac{2πmk_BT}{h^2})^{3/2}V$$

If I'd assume particles were monatomic, I'd be able to disregard rotational and vibrational degrees of freedom. I'd still have to worry about electronic degrees of freedom, but my professor said we will mostly disregard it as the temperatures we're concerned with don't mess around with it.

It would be a stretch but the only thing that's coming to mind is by saying (assuming monatomic and low temp) that q will be equal to the translational partition function and then differentiating that to see if anything worthwhile comes out. Although that would come at the price of a loss of generality due to my assumptions.

Aside from that, I don't know where to go, probably because I don't really understand this part of the course very well. The most general expression I can think of for q would be of q defined as a product of the translational, rotational, vibrational, and electronic partition functions.

EDIT: Don't know why the equation isn't showing...

At below, fixed, thanks.

 

[BvU]

there is a } too many. TeX is picky.
Edit: add: It's safe to disregard the other DOF since they don't have much to do with the volume of the box, but more with interatomic stuff. Note: ideal gas law mentioned in the OP. non interacting point-like particles.

 

[Mangoes]

Well, going ahead and differentiating the log of the translational partition function pretty much leads to cancellation of the constants and a V term in the denominator, so it seems like a step in the right direction. I'd have

$$PV = k_BTN$$

And if I assume I have a mole N of particles, the product of N and boltzmann's constant is the gas constant, so out comes

$$PV = nRT$$

where n would be a scaling factor for the number of moles.

I still have the issue of the loss of generality though, since I'm disregarding rotational and vibrational degrees of freedom and the idea gas law works for polyatomic particles.

 

[BvU]

They enter in the partition function as a factor that has no dependence on V, so they don't come in when differentiating ln q

 

[Mangoes]

Alright. I have no idea what the rotational or vibrational partition function looks like but intuitively it makes sense that there isn't a dependence on volume. I'll just take what I have and call it a day.

Thanks for the help!