Maxwell-Boltzmann Distribution
Speed distribution f(v) — peaks shift with T.
Key Notes
Maxwell-Boltzmann distribution: probability density of molecular speeds in a gas at thermal equilibrium.
f(v) = 4π·n·(m/(2πkT))^(3/2)·v²·exp(−mv²/2kT). Peaks at v_mp.
Tail: very few molecules at extreme speeds, but always non-zero probability.
Distribution depends only on T and m — not on V, P, or n (those just rescale).
Higher T: distribution SHIFTS RIGHT (faster speeds), BROADENS (wider spread).
Heavier gases: distribution NARROWER, lower peak speed.
Maxwell-Boltzmann is correct for CLASSICAL gases at moderate T. Quantum gases (e.g., electrons in metals) need Fermi-Dirac instead.
Underlies effusion (Graham's law), reaction kinetics (Arrhenius), evaporation, atmospheric escape.
Formulas
Maxwell speed distribution
Probability density per unit speed.
Most probable speed (peak)
Where f(v) is maximum.
Mean speed
Average ⟨v⟩.
RMS speed
√⟨v²⟩.
Fraction with speed > v₀
High-speed tail; exponentially suppressed.
Important Points
Distribution is HEAVY-TAILED but exponentially DECAYING — few molecules at very high speeds.
Increase T: peak shifts right and broadens.
Increase m: peak shifts left and narrows.
Reaction rates often depend on the fast tail (high-energy molecules) — explains Arrhenius behavior.
Atmospheric escape: light gases (H, He) escape Earth's gravity because their high-speed tail exceeds escape velocity.
Three peaks (v_mp, v_mean, v_rms) all visible on the same plot in fixed ratios.
Maxwell-Boltzmann Distribution notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.