The Maxwell-Boltzmann distribution of molecular speeds in an ideal gas.

Question

BlackTom AI · Accepted Answer

The question concerns the Maxwell-Boltzmann distribution of molecular speeds in an ideal gas and asks us to evaluate statements about characteristic speeds.

First, recall the key speeds for a molecule of mass m at temperature T: 
- Most probable speed vp = sqrt(2kT/m)
- Average (mean) speed v̄ = sqrt(8kT/πm)
- Root-mean-square speed vrms = sqrt(3kT/m)
Here, k is Boltzmann's constant.

Now compare these speeds numerically in terms of the square roots of constant factors in front of kT/m:
- vp is proportional to sqrt(2) ≈ 1.414
- v̄ is proportional to sqrt(8/π) ≈ sqrt(2.546) ≈ 1.595
- vrms is proportional to sqrt(3) ≈ 1.732
From these, we obtain the ordering vp < v̄ < vrms.

Option evaluation without provided alternatives: since the single listed option is 'The most likely speed vp is less than the rms speed vrms.' we can assess its truth value based on the ordering above. vp is indeed less than vrms because 1.414 < 1.732. This aligns with the standard MB distribution properties, where the peak (vp) occurs at a speed smaller than the rms speed, reflecting that the distribution has a long tail toward higher speeds contributing to a larger vrms.

If there were additional options, we would compare each one against the MB facts: whether vp < v̄ or vp > vrms or any other misstatements about the relationships among vp, v̄, and vrms. Common misconceptions include assuming all three speeds are equal or misplacing vp relative to v̄ and vrms; the correct hierarchy is vp < v̄ < vrms.

Note: In this input, there are no answer choices provided beyond the single statement listed in the answer field, and the 'answer_options' array is empty. This means the analysis above is focused on evaluating the truth of the given statement in light of MB theory, rather than selecting among multiple provided options.

SPHY001 Practice Quiz - Unit 13 | LA053

The Maxwell-Boltzmann distribution of molecular speeds in an ideal gas.

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