💨 Kinetic Theory of Gases

Molecular speeds, pressure from collisions, kinetic energy and temperature, mean free path & diffusion

P = (1/3) ρ v_rms² K.E_avg = (3/2) kT v_rms = √(3RT/M) λ = 1/(√2 π d² n)
← Physics Tools
💡 Molecules in random motion

Kinetic theory models a gas as a large number of tiny molecules in constant random motion making perfectly elastic collisions with each other and the container walls. From these microscopic assumptions, we can derive macroscopic quantities like pressure P = (1/3) (N/V) m v_rms² and show that average kinetic energy per molecule K.E_avg = (3/2) kT is proportional to absolute temperature. The theory also explains molecular speed distributions, mean free path, diffusion/effusion and the specific heats of gases via degrees of freedom.

Kinetic theory calculators

Speeds, pressure, kinetic energy, degrees of freedom, mean free path & diffusion

Temperature and molar mass
v_mp = —, v_avg = —, v_rms = —
Uses v_rms = √(3RT/M), v_avg = √(8RT/(πM)), v_mp = √(2RT/M) and K.E_avg = (3/2)kT.
Moles, volume and temperature
Molar mass (for v_rms and ρ)
P_kinetic = —, P_ideal = —
Uses P = (1/3) (N/V) m v_rms² and compares with ideal gas law P = nRT/V.
Gas type and amount
U = —, C_v = —, C_p = —, γ = —
Uses equipartition: U = (f/2) nRT, C_v = (f/2)R, C_p = C_v + R, γ = C_p/C_v.
Temperature, pressure and molecular diameter
λ = —, Z = —
Uses λ = (kT)/(√2 π d² P) and Z = v_avg/λ with v_avg from temperature.
Molar masses for diffusion/effusion
r₁/r₂ = —, t₁/t₂ = —
Uses r₁/r₂ = √(M₂/M₁) and t₁/t₂ = √(M₁/M₂) at same T and P.

📈 Maxwell speed distribution

🧮 Step-by-Step Solution ▼ Show

Pressure, kinetic energy and speeds

Concept Formula Notes
Pressure of ideal gas P = (1/3) (N/V) m v_rms² From molecular collisions
Density form P = (1/3) ρ v_rms² ρ = Nm/V
Average KE per molecule (1/2) m v_rms² = (3/2) kT Kinetic interpretation of T
Total KE K_total = (3/2) N kT = (3/2) nRT Translational KE
Root mean square speed v_rms = √(3RT/M) Also √(3kT/m)
Average speed v_avg = √(8RT/(πM)) v_avg ≈ 0.921 v_rms
Most probable speed v_mp = √(2RT/M) v_mp : v_avg : v_rms ≈ 1 : 1.128 : 1.224

Degrees of freedom, mean free path & diffusion

Concept Formula / Values Notes
Equipartition Each f gives (1/2)kT per molecule Or (1/2)RT per mole
Internal energy U = (f/2) nRT f = degrees of freedom
Mean free path λ = (kT)/(√2 π d² P) Air at STP: ~10⁻⁷ m
Collision frequency Z = v_avg/λ Collisions per second
Graham's law r₁/r₂ = √(M₂/M₁) Diffusion/effusion rates
Effusion times t₁/t₂ = √(M₁/M₂) Same T and P
Ideal gas law (kinetic) PV = (1/3) N m v_rms² = nRT Bridges micro and macro

About this kinetic theory tool

This tool focuses on the kinetic theory part of thermodynamics: how random molecular motion leads to gas pressure, temperature, energy and transport properties. Use it to quickly compute rms/average/most probable speeds, relate kinetic energy to temperature, compare kinetic and ideal-gas pressure, find internal energy from degrees of freedom, estimate mean free path and collision frequency, and apply Graham's law for diffusion or effusion questions.