Kinetic theory of gases: formula for the most probable molecular speed Select the correct expression for the most probable speed of gas molecules in terms of absolute temperature T and molar mass M (or Boltzmann constant k and molecular mass m). Choose the correct formula.
Mechanical Engineering
Thermodynamics
Difficulty: Medium
Choose an option
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Av<sub>mp</sub> = √(2RT/M)
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Bv<sub>mp</sub> = √(3RT/M)
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Cv<sub>mp</sub> = √(8RT/(πM))
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Dv<sub>mp</sub> = √(kT/m) × √3
Answer
Correct Answer: v<sub>mp</sub> = √(2RT/M)
Explanation
ConceptFrom the Maxwell–Boltzmann distribution for molecular speeds, the characteristic speeds are:
- Most probable speed: vmp = √(2RT/M) = √(2kT/m)
- Mean (average) speed: v̄ = √(8RT/(πM))
- RMS speed: vrms = √(3RT/M)
Verification (units)R (J/mol·K), T (K), M (kg/mol) ⇒ RT/M has units m²/s²; square root gives m/s.
Common pitfallsDo not confuse vmp with vrms (factor 3) or with v̄ (factor 8/π).
Final Answervmp = √(2RT/M)