- #1
TroyElliott
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- 3
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!
