Molecular Speeds

The Maxwell-Boltzmann distribution is a probability density function f(u) that gives the fraction of molecules in a gas with speeds between u and u + du. It is expressed as f(u) = 4π (M/2πRT)^(3/2) · u² · exp(−Mu²/2RT), where M is the molar mass (kg/mol), R is the universal gas constant (8.314 J mol⁻¹ K⁻¹), and T is the absolute temperature (K). The curve is asymmetric: it rises from zero, reaches a peak (the most probable speed), and then falls off gradually toward higher speeds with a long tail.

The most probable speed is the speed at the peak of the Maxwell-Boltzmann curve — the speed possessed by the largest fraction of molecules. Mathematically, it is found by taking the derivative of f(u) with respect to u and setting it equal to zero. The result is u_mp = √(2RT/M). For nitrogen gas (M = 0.02802 kg/mol) at 298 K: u_mp = √(2 × 8.314 × 298 / 0.02802) ≈ 421 m/s. This is always the smallest of the three characteristic speeds.

The average speed is the arithmetic mean of the speeds of all molecules in the sample. It is computed by integrating u·f(u) over all speeds from zero to infinity: u_avg = √(8RT/πM). Because of the asymmetric tail at high speeds, u_avg is slightly larger than u_mp. For N₂ at 298 K: u_avg = √(8 × 8.314 × 298 / (π × 0.02802)) ≈ 475 m/s. The average speed is particularly important in effusion and diffusion calculations.

The root-mean-square speed is the square root of the average of the squared speeds: u_rms = √(3RT/M). It is the most directly linked to kinetic energy, since the average kinetic energy per molecule is KE_avg = ½mu_rms² = (3/2)k_BT. For N₂ at 298 K: u_rms = √(3 × 8.314 × 298 / 0.02802) ≈ 515 m/s. The ordering u_mp < u_avg < u_rms always holds for any gas at any temperature, and the three speeds are in the ratio 1 : 1.128 : 1.225.

All three speeds increase with temperature (proportional to √T) and decrease with molar mass (proportional to 1/√M). Doubling the absolute temperature increases each speed by a factor of √2 ≈ 1.41. A heavier gas such as CO₂ (M = 0.04401 kg/mol) has lower molecular speeds than a lighter gas such as He (M = 0.00400 kg/mol) at the same temperature. This is the molecular basis of Graham's Law of Effusion: the rate of effusion is inversely proportional to √M, because lighter molecules travel faster on average.

• u_{\text{rms}} — Root-mean-square speed (m s⁻¹): Square root of the mean of the squared molecular speeds; directly related to average kinetic energy
• u_{\text{avg}} — Average speed (m s⁻¹): Arithmetic mean of all molecular speeds in the sample
• u_{\text{mp}} — Most probable speed (m s⁻¹): Speed at the peak of the Maxwell-Boltzmann distribution curve
• R — Universal gas constant (J mol⁻¹ K⁻¹): R = 8.314 J mol⁻¹ K⁻¹
• T — Absolute temperature (K): Temperature measured on the Kelvin scale; must not be in Celsius
• M — Molar mass (kg mol⁻¹): Mass of one mole of the gas in kilograms (e.g., for O₂: M = 0.03200 kg mol⁻¹)

Problem. Calculate the root-mean-square speed, average speed, and most probable speed of oxygen gas (O₂) molecules at 300 K. Molar mass of O₂ = 32.00 g/mol = 0.03200 kg/mol, R = 8.314 J mol⁻¹ K⁻¹.

Answer. For O₂ at 300 K: u_mp ≈ 395 m/s, u_avg ≈ 446 m/s, u_rms ≈ 483 m/s

• Kinetic Theory of Gases — Molecular speeds arise directly from the postulates of kinetic theory, which models gas molecules as point masses in constant random motion.
• Ideal Gas Law — The ideal gas law (PV = nRT) is derived from kinetic theory using the concept of molecular speeds; pressure is a direct consequence of molecular collisions.
• Graham's Law of Effusion and Diffusion — Graham's Law (rate ∝ 1/√M) is a direct application of average molecular speeds — lighter gases effuse faster because their molecules move faster at the same temperature.
• Kinetic Energy and Temperature — The average kinetic energy of gas molecules (½mu_rms²= 3/2 k_BT) directly links molecular speed to the macroscopic property of temperature.
• Activation Energy and Reaction Rates — Only molecules in the high-speed tail of the Maxwell-Boltzmann distribution have enough kinetic energy to overcome the activation energy barrier, connecting molecular speed distributions to reaction kinetics.