Kinetic Theory of Gases Calculator — RMS Speed

Kinetic Theory Guide

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

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

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 micros

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 mo