§ 12.3NCERT Class 11 · Physics · Chapter 12

Molecular Speeds: RMS, Average, Most Probable & Maxwell-Boltzmann Distribution

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 3 min read

      Statistical Character of Molecular Motion: Because gas molecules undergo billions of collisions per second, individual molecular speeds vary widely from zero to very high values. We describe molecular speeds using three key statistical parameters: Root-Mean-Square Speed (vrms), Average Speed (vavg), and Most Probable Speed (vmp).
    

1. The Three Statistical Speeds

Speed Type	Symbol & Definition	Formula (Molar Form)	Formula (Molecular Form)
Root-Mean-Square Speed	v_rms Square root of the mean of squares of speeds	v_rms = √(3RT / M₀)	v_rms = √(3k_BT / m)
Average Speed	v_avg (or ⟨v⟩) Arithmetic mean of all molecular speeds	v_avg = √(8RT / πM₀)	v_avg = √(8k_BT / πm)
Most Probable Speed	v_mp Speed possessed by the maximum fraction of molecules	v_mp = √(2RT / M₀)	v_mp = √(2k_BT / m)

2. Comparison & Ratio of the Three Speeds

Comparing the numerical coefficients:

v_mp : v_avg : v_rms = √2 : √(8 / π) : √3 ≈ 1.414 : 1.596 : 1.732 ≈ 1 : 1.128 : 1.224

NEET/JEE Memory Hook: "RAM in descending order" → v_rms > v_avg > v_mp.
v_rms is the largest, while v_mp is the smallest!

3. Dependence on Temperature and Molar Mass

All three speeds follow the exact same functional dependence:

v ∝ √T and v ∝ 1 / √M₀

Effect of Temperature: If absolute temperature T is quadrupled (4T), molecular speeds double (2v).
Effect of Molecular Mass: Lighter gas molecules move much faster than heavier molecules at the same temperature. (e.g., at room temperature, v_rms of H₂ is √(32/2) = 4 times that of O₂!).
Effect of Pressure at Constant Temperature: Since v_rms depends only on T and M₀, changing pressure at constant temperature has no effect on v_rms (because ρ changes in the same ratio: P / ρ = constant).

4. Maxwell-Boltzmann Speed Distribution Curve

James Clerk Maxwell and Ludwig Boltzmann derived the theoretical law for the distribution of speeds among molecules of a gas in thermal equilibrium:

dN = 4 π N · (m / 2 π k_B T)^3/2 · v² · e^{−mv² / (2 k_B T)} dv

Key Salient Features of the Distribution Curve:

Peak of the Curve: The peak corresponds to the most probable speed (v_mp), possessed by the maximum number of molecules.
Asymmetry: The curve is asymmetric, skewed toward higher speeds. As a result: v_mp < v_avg < v_rms.
Area Under the Curve: The total area under the distribution curve represents the total number of molecules N in the gas sample and remains strictly constant.
Zero and Infinite Speeds: The fraction of molecules with zero speed is zero. The curve approaches zero asymptotically as speed approaches infinity.
Effect of Increasing Temperature: As temperature increases:
1. The peak shifts toward the right (to higher speeds).
2. The peak height decreases (curve flattens and broadens) because the total area under the curve must remain constant.
3. The fraction of molecules with very high speeds increases significantly.

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