# Confused on statistical mechanics problem

## Homework Statement

A dilute gas of N non-interacting atoms of mass m is contained in a volume V and in equilibrium with the surroundings at a temperature T. Each atom has two (active) intrinsic states of energies ε = 0 and ∆, respectively. Find the total partition function of the gas.

## Homework Equations

Partition function.

## The Attempt at a Solution

For discrete energy levels we normally write $Z = \sum_{i}e^{-\beta \epsilon_{i}},$ and the volume of the system appears in the energy spectrum e.g. particle in a box energy spectrum. For a classical system we write $$Z=\frac{1}{h^{3}}\int e^{H(q,p)}d^{3}qd^{3}p.$$ We can write

$$Z = \frac{V}{h^{3}}\int e^{-\beta p^{2}/2m}d^{3}p,$$
$\frac{p^{2}}{2m} = \epsilon$ implies $d\epsilon = \frac{p}{m}dp = \frac{\sqrt{2m\epsilon}}{m}dp$ which means we can write $dp = \sqrt{\frac{m}{2\epsilon}}d\epsilon$ and the partition function becomes

$$Z = (\frac{m}{2})^{3/2}\frac{V}{h^{3}}\int e^{-\epsilon}\epsilon^{-3/2}d^{3}\epsilon.$$

Should I be including delta functions in order to pick off the two discrete energy levels? or is there an easier way in going about this problem?

Thank you!

Related Advanced Physics Homework Help News on Phys.org
You said yourself; “for discrete energy levels we use the discrete sum over energy levels,” but then you use the continuous energy formulation??

Finding a discrete sum by integrating over Dirac-delta spikes seems pretty silly!

(Sorry if I’m missing something; I’m also learning the subject.)

I am just confused on how to incorporate the volume $V$ into the picture by evaluating $Z = \sum_{i} e^{-\beta \epsilon_{i}} = (1+e^{-\beta \Delta})^{N}.$ Maybe it is as simple as multiplying this by $\frac{V}{h^{3}}.$

Just because they give the volume V doesn’t mean we have to use it!

I don’t know quantum mechanics, but I would think V would appear inside Δ. But since some authority tells us what Δ is, then I don’t think we need to use V.

Anyway I’ll shut up now. Let’s wait for someone who actually knows what they’re talking about to chime in!