Kinetic Molecular Energy
High-Yield Summary
- Kinetic molecular theory (KMT): 5 postulates explaining macroscopic gas properties from molecular behavior — random motion, negligible molecular volume, elastic collisions, negligible attractions, KE depends only on T.
- Average kinetic energy per mole: KEavg = (3/2)RT — depends ONLY on temperature, not pressure/volume/identity.
- Root mean square velocity urms = √(3RT/M) — increases with T, decreases with molar mass M.
- Diffusion = molecules spreading high→low concentration; effusion = gas escaping through a tiny opening into a vacuum.
- Graham's Law: r₁/r₂ = √(M₂/M₁) — lighter gases effuse/diffuse faster than heavier ones.
Key Terms
- Kinetic molecular theory (KMT)
- Theory explaining a gas's macroscopic properties (P, V, T) from the behavior of its individual molecules.
- Root mean square velocity (urms)
- The speed of a particle carrying the average kinetic energy of a gas sample.
- Diffusion
- Spontaneous spreading of molecules from high to low concentration.
- Effusion
- Movement of gas particles through a tiny opening into an evacuated space (vacuum).
5 Postulates of Kinetic Molecular Theory
- 1Gas molecules are in constant, random motion, colliding with each other and container walls — pressure results from wall collisions.
- 2Gas molecules are extremely small compared to the distances between them — most gas volume is empty space.
- 3Collisions between gas molecules are elastic — no kinetic energy is lost.
- 4Attractions between gas molecules are extremely weak and negligible.
- 5Average kinetic energy depends only on temperature — not on pressure, volume, or gas identity.
Average Kinetic Energy
KEavg = (3/2)RT
- R = universal gas constant, 8.314 J/(mol·K)
- T = absolute temperature (K)
- Two gases at the same T have the same KEavg regardless of molar mass or identity.
Root Mean Square Velocity
urms = √(3RT/M)
- M = molar mass in kg/mol (to stay consistent with R in J/(mol·K))
- Increases with temperature; decreases as molar mass increases — lighter particles move faster on average.
- Worked example: He (M = 0.00400 kg/mol) at 298 K → urms ≈ 1,360 m/s.
Graham's Law of Effusion
r₁/r₂ = √(M₂/M₁)
- r₁, r₂ = effusion rates of gas 1 and gas 2
- M₁, M₂ = molar masses of gas 1 and gas 2
- Worked example: H₂ (2.02 g/mol) vs O₂ (32.00 g/mol) → r(H₂)/r(O₂) = √(32.00/2.02) ≈ 3.98 — H₂ effuses ~3.98× faster.
- Rate can also be expressed as ΔX/Δt (change in amount or volume over time) for quantitative problems.
Must-Know Points
- KEavg depends ONLY on temperature — never on pressure, volume, or gas identity. This is the single most-tested KMT fact.
- Because KE = ½mv², two gases with equal KEavg but different M must have different typical speeds — lighter gas moves faster.
- A smell spreading quickly through a room is a real-world diffusion example driven by lighter, faster-moving odor molecules diffusing ahead of heavier ones.
Common MCAT Trap
- Don't confuse diffusion (spreading through a medium/open space) with effusion (escape through a tiny opening into a vacuum) — Graham's Law technically describes effusion but the same inverse-sqrt-mass relationship is often tested for diffusion too.
- In urms = √(3RT/M), M must be in kg/mol (not g/mol) to match R = 8.314 J/(mol·K) — a common unit-conversion trap.
- Equal average KE does NOT mean equal average speed across different gases — only equal-mass gases at the same T have equal average speed.
Quick Recall
Two gases, He and Ar, are at the same temperature. Which has higher average kinetic energy?
Which gas effuses faster: CO₂ (44 g/mol) or Ne (20 g/mol)?
What happens to urms as temperature increases?