When Do Atoms in an Ideal Gas Exhibit Quantum Mechanical Behavior?

[Bashyboy]

Homework Statement

For what temperatures are the atoms in an ideal gas at pressure P quantum mechanical ? Hint: use the ideal gas law P V = N kT to deduce the interatomic spacing (the answer is T < (1/k)(h 2/3m) 3/5P 2/5). Obviously we want m to be as small as possible and P as large as possible for the gas to show quantum behavior. Put in numbers for helium at atmospheric pressure. Is hydrogen in outer space (interatomic distance ≈ 1 cm and temperature ≈ 3K) quantum mechanical ?

Homework Equations

The Attempt at a Solution

According to the answer key, to find the interatomic spacing, we need to find the size of a single gas particle. One gas particle corresponds to N=1, and the volume is V = d^3. This leads to

$d = \left( \frac{kT}{P} \right)^{1/3}$.

I have two objections to this, for which I hope you provide correction. Firstly, assigning the volume $V = d^3$ implies that we are assuming that the atoms are square? Secondly, how does finding the size of a single gas particle provide us with the interatomic spacing. It would seem that the most we could deduce from such information is, that closest two gas particles could get. Are we to assume that the gas particles are this closely packed? Wouldn't the gas solidify at this point?

 

[Bashyboy]

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