Maxwell Speed Measures and Ordering
For a classical ideal gas in Maxwell-Boltzmann equilibrium, v_p = sqrt(2k_BT/m), v_avg = sqrt(8k_BT/(pi m)), and v_rms = sqrt(3k_BT/m).
Why this shows up in the exam
Comparing Maxwell speed definitions · Reading distribution questions · Deriving ratios among characteristic speeds
Learn the idea
Most-probable, mean, and rms speeds are distinct summaries of the same Maxwell distribution. A molecular sample contains a spread of speeds; squaring before averaging weights the fast tail more strongly, making rms speed the largest of the three measures.
🧠 Memory hook: Probable uses 2, average uses 8 over pi, rms uses 3.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v_p = sqrt(2k_B T/m) — most-probable speed for a Maxwell distribution
- v_avg = sqrt(8k_B T/(pi m)) — arithmetic mean molecular speed
- v_rms = sqrt(3k_B T/m) — root-mean-square molecular speed
- v_p < v_avg < v_rms — ordering for the same gas at the same temperature
How to approach it
- 1Identify which average the question names
- 2Use one mass convention consistently
- 3Check the standard ordering
Common slip-ups that cost marks
- •Treating the three speeds as equal
- •Using molar mass with k_B without conversion
- •Assuming a mixture changes a species' equilibrium speed at fixed T
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A gas has rms molecular speed 300 m/s at 300 K. What is its rms speed at 1200 K, assuming ideal behavior?
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