# Deriving Ideal Gas Law through partition function

1. Jan 28, 2014

### Mangoes

1. The problem statement, all variables and given/known data

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.

3. 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.

2. Jan 29, 2014

### Staff: Mentor

Moved the problem to advanced physics, it is more densely populated by right kind of people, so probability of getting an answer should be higher.

3. Jan 29, 2014

### Mangoes

Hey Borek, since this never got any replies I started a new thread in the introductory physics subforum and got the help I needed,

I have no idea how to lock threads on this site, so this one remained open. Sorry about that.

4. Jan 29, 2014

### Staff: Mentor

Never start a new thread - report the old one and ask mentors to move the thread.

Besides, you reposted just after 10 hours, that is worth a warning. You are lucky I have not noticed earlier.