Questions: Maxwell-Boltzmann Distribution and Molecular Speeds

Question 1

At the same temperature, which gas has the higher most probable speed, and what determines the difference?

• A. O₂ (M=32), because more massive molecules carry more momentum and move faster
• B. H₂ (M=2), because lighter molecules achieve higher speed from the same thermal energy — v_p = √(2k_BT/m) scales inversely with mass
• C. Both have the same most probable speed, because temperature sets average kinetic energy equally for all ideal gases
• D. H₂, but only marginally — mass differences have little effect on speed distributions at typical laboratory temperatures

Answer: (B)

v_p = √(2k_BT/m): at fixed temperature, most probable speed is inversely proportional to the square root of molecular mass. H₂ is 16 times lighter than O₂, so v_p(H₂) = 4 × v_p(O₂). While it is true that average kinetic energy is the same for both gases (½m⟨v²⟩ = 3/2 k_BT), equal kinetic energy means lower speed for heavier molecules — mass and speed trade off to give the same energy product.

Question 2

The Maxwell-Boltzmann speed distribution rises from zero at v=0, reaches a peak, then tails off at high speeds. What two competing factors produce this shape?

• A. Molecular collisions preferentially create intermediate speeds; very slow and very fast molecules are destroyed by collisions
• B. The v² factor (more directions in velocity space correspond to higher speeds) competes with the Boltzmann factor exp(−mv²/2k_BT) (higher-energy states are exponentially less probable)
• C. The ideal gas approximation breaks down at very low and very high speeds, artificially suppressing the distribution at both extremes
• D. Intermolecular forces slow very fast molecules and accelerate very slow ones, creating the bell-shaped distribution

Answer: (B)

The v² term in f(v) = 4π(m/2πkT)^(3/2) v² exp(−mv²/2kT) reflects the spherical shell in velocity space: more directions correspond to speed v as v increases, so more molecules have speed v at the low end purely from geometry. The Boltzmann factor exp(−mv²/2kT) suppresses high-energy states exponentially. Their product creates a peak at v_p where the marginal increase from v² is exactly balanced by the exponential decay.

Question 3

For any ideal gas, the root-mean-square speed v_rms is always greater than the most probable speed v_p, regardless of temperature or molecular mass.

• T. True
• F. False

Answer: True

The ordering v_p < ⟨v⟩ < v_rms always holds for the Maxwell-Boltzmann distribution. This is a mathematical consequence of the distribution's asymmetric shape: the long high-speed tail (a few molecules moving very fast) pulls the mean and especially the RMS above the peak. The RMS is most sensitive to high-speed molecules because squaring amplifies fast outliers. This ordering persists at all temperatures and for all molecular masses.

Question 4

Heavier gas molecules at a given temperature have lower average kinetic energy than lighter molecules, which is why they have a lower most probable speed.

• T. True
• F. False

Answer: False

All ideal gas molecules at the same temperature have exactly the same average kinetic energy: ½m⟨v²⟩ = 3/2 k_BT, independent of mass. Heavier molecules move more slowly not because they have less energy, but because more mass requires less velocity to carry the same energy (KE = ½mv²). This is a common but critical misconception: the equipartition theorem guarantees equal average kinetic energy per translational degree of freedom at the same temperature, regardless of mass.

Question 5

The Arrhenius equation for reaction rates contains the factor exp(−E_a/k_BT). Explain how the Maxwell-Boltzmann distribution connects molecular speed to this exponential factor and why raising temperature dramatically increases reaction rates.

Model answer: Chemical reactions require that colliding molecules have kinetic energy above a threshold E_a along the reaction coordinate. The Maxwell-Boltzmann distribution gives the fraction of molecules with kinetic energy exceeding E_a: this fraction is proportional to exp(−E_a/k_BT) — the Boltzmann factor evaluated at the activation energy. Raising temperature broadens and flattens the distribution, shifting the high-speed tail to higher energies. Even a modest temperature increase substantially increases the fraction of molecules in the tail above E_a, because the exponential function amplifies small changes in T. This is why a 10°C temperature rise often doubles or triples a reaction rate.

The connection between the Maxwell-Boltzmann distribution and Arrhenius kinetics is fundamental: the Arrhenius exponential is not an empirical fitting parameter — it arises directly from the fraction of gas molecules exceeding the activation energy barrier, as determined by the Boltzmann distribution. This links thermodynamics, statistical mechanics, and kinetics in a single framework.