In a classroom at 290 K, there are typically 3 × 1027 air molecules each having five degrees of freedom and mass 4.8 × 10−26 kg. The room has dimensions 3 m by 6 m by 7 m. You are going to calculate how many quantum states are accessible to this system.
(a) What is the thermal energy of this system?
(b) The magnitude of the momentum of any molecule can range from 0 to p0. Estimating p0 to be roughly twice the root mean square momentum, what is the volume in momentum space that is available to any particle?
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(c) For reasons cited in the previous problem, we are going to calculate the number of accessible quantum states for translational motion and ignore the relatively fewer states for rotational motion. How many different quantum states would be accessible to any particle if it were all by itself?
(d) If all the particles were identical, what would be the corrected number of states per particle, ωc?
(e) How many different quantum states are accessible to the entire system?
(f) Answer parts (d) and (e) but now assuming that 79% are identical nitrogen molecules and 21% are identical oxygen molecules (of about the same mass).