Kinetic Theory of Gases Calculator — RMS Speed

Kinetic Theory Guide

What's the key thing to understand about Kinetic Theory Postulates?

Ideal gas assumptions: 1. Gas consists of many identical particles in random motion. 2. Particle volume is negligible compared to container. 3. Collisions between particles and walls are perfectly elastic. 4. No intermolecular forces (between collisions). 5. Particles obey Newton's laws of motion. Results: pV = (1/3)Nmv²_rms. Combined with ideal gas law pV = nRT = NkT: ½m·v²_rms = (3/2)kT. Where k = Boltzmann constant = 1.38×10⁻²³ J/K. Average kinetic energy per molecule = (3/2)kT — depends ONLY on absolute temperature, not on the identity, mass, or pressure of the gas — two different gases at the same temperature have the same average kinetic energy per molecule, even though their average speeds differ because of their different molar masses.

What do I need to know about RMS, Average, and Most Probable Speeds?

Three different speeds describe the Maxwell-Boltzmann distribution: v_rms = √(3kT/m) = √(3RT/M). v_average = √(8kT/πm) = √(8RT/πM). v_most_probable = √(2kT/m) = √(2RT/M). Ratios: v_mp : v_avg : v_rms = √2 : √(8/π) : √3 = 1 : 1.128 : 1.225. For nitrogen at 20°C (293K): v_rms = √(3 × 8.314 × 293 / 0.028) = √(261,000) = 511 m/s. Faster than the speed of sound in air (343 m/s) — actually, sound speed is related to gas particle speed. Speed of sound ≈ √(γRT/M) where γ = 1.4 for diatomic. Sound speed is somewhat lower than the RMS molecular speed because sound propagates via collisions transferring momentum through the gas rather than individual molecules travelling the full distance — but the two speeds are of the same order of magnitude and both scale with the square root of absolute temperature.

What should I know about Temperature and Energy?

Average translational KE per molecule = (3/2)kT. Internal energy of monatomic ideal gas: U = (3/2)nRT. Each translational degree of freedom contributes (1/2)kT per molecule. Diatomic gas (N₂, O₂) at room temperature: 5 degrees of freedom (3 translation + 2 rotation). U = (5/2)nRT. At very high temperatures: vibrational modes activate. Equipartition: at thermal equilibrium, each accessible mode of energy has (1/2)kT per particle on average. The fundamental link: temperature IS a measure of microscopic kinetic energy — there is no more fundamental definition. Zero kelvin represents the theoretical point at which all translational molecular motion stops (in practice unreachable, per the third law of thermodynamics), which is why the Kelvin scale, not Celsius, must be used in all kinetic theory calculations.

What do I need to know about Diffusion and Effusion?

Graham's law of effusion: rate of effusion ∝ 1/√M. Lighter gases effuse faster than heavier. Hydrogen effuses √(32/2) = 4× faster than oxygen. Application: uranium enrichment used effusion of UF₆ — the U-235 hexafluoride is slightly lighter than U-238 hexafluoride, enabling slow separation through thousands of stages. Diffusion: mixing of gases due to random motion. Mean free path (λ): average distance between collisions. λ = 1/(√2 × n × σ). For air at STP: λ ≈ 68 nm. Surprisingly small — but molecules travel at hundreds of metres per second between collisions, so a gas released in one corner of a room takes far longer to be smelled across the room than the raw molecular speed alone would suggest — the random, zig-zagging path between frequent collisions, not the speed itself, is what limits how fast diffusion appears macroscopically.