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

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SUMMARY

Atoms in an ideal gas exhibit quantum mechanical behavior at temperatures below T < (1/k)(h 2/3m) 3/5P 2/5. This relationship is derived from the ideal gas law P V = N kT, which helps deduce interatomic spacing. For helium at atmospheric pressure, the conditions for quantum behavior are met at low temperatures. The discussion also questions whether hydrogen in outer space, with an interatomic distance of approximately 1 cm and a temperature of about 3K, can be considered quantum mechanical.

PREREQUISITES
  • Understanding of the ideal gas law (P V = N kT)
  • Familiarity with quantum mechanics concepts
  • Knowledge of interatomic spacing and its implications
  • Basic thermodynamics principles
NEXT STEPS
  • Calculate interatomic spacing for various gases using the ideal gas law
  • Explore the implications of quantum mechanics on gas behavior at low temperatures
  • Research the properties of helium and hydrogen in quantum states
  • Study the effects of pressure on quantum mechanical behavior in gases
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Students studying thermodynamics, physicists interested in quantum mechanics, and researchers exploring the behavior of gases at low temperatures.

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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?
 
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