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LearnYPH11WPH15 · Unit 5: Thermodynamics, Radiation, Oscillations and CosmologyThermodynamicsKinetic theory of gases
Thermodynamics · Subtopic 4 of 5

Kinetic Theory of Gases

Gas pressure comes from molecules colliding with the walls, each collision changing momentum. Under the kinetic theory assumptions, pV=13Nm⟨c2⟩pV=\tfrac13Nm\langle c^2\rangle. Comparing with pV=NkTpV=NkT gives the mean translational kinetic energy of a molecule: 12m⟨c2⟩=32kT\tfrac12m\langle c^2\rangle=\tfrac32kT. Temperature is a direct measure of average molecular kinetic energy; the r.m.s. speed is ⟨c2⟩\sqrt{\langle c^2\rangle}.

Cambridge 9702 Paper 4Edexcel YPH11 WPH15AQA 7408 Paper 2≈ 40 minRevision 5 · updated 2026-10-27
Section 1 of 14

Learning objectives

  • State the assumptions of the kinetic theory of gases.
  • Explain pressure in terms of molecular momentum change.
  • Derive and use pV=13Nm⟨c2⟩pV=\tfrac13Nm\langle c^2\rangle.
  • Use 12m⟨c2⟩=32kT\tfrac12m\langle c^2\rangle=\tfrac32kT and find r.m.s. speeds.
Pause and think

At the same temperature, which molecules move faster on average in air: oxygen or nitrogen?

Section 2 of 14

Core idea

Start herePressure from collisions

The air pushing on your skin is billions of molecules hitting it every nanosecond. Model them as tiny bouncing balls and Newton's laws predict the pressure, and reveal what temperature really is: average molecular kinetic energy.

The big idea

pV=13Nm⟨c2⟩pV=\frac13Nm\langle c^2\rangle. With pV=NkTpV=NkT: 12m⟨c2⟩=32kT\frac12m\langle c^2\rangle=\frac32kT.

1. Assumptions

  • Very many molecules, moving randomly.
  • Molecular volume negligible.
  • Elastic collisions; no forces except during collisions.
  • Collision time negligible.
Molecules in a box, moving randomly
Each wall collision reverses a molecule's momentum perpendicular to the wall. Billions of such impacts each second give a steady pressure.

2. Derivation

pV=13Nm⟨c2⟩pV=\frac13Nm\langle c^2\rangle
  1. Δp=mcx−(−mcx)=2mcx\Delta p=mc_x-(-mc_x)=2mc_x
    One molecule hits a wall of a cube of side LL and rebounds.
  2. Δt=2Lcx\Delta t=\frac{2L}{c_x}
    Time to return to the same wall.
  3. F=2mcx2L/cx=mcx2LF=\frac{2mc_x}{2L/c_x}=\frac{mc_x^2}L
    Rate of change of momentum.
  4. p=FL2=mcx2Vp=\frac F{L^2}=\frac{mc_x^2}V
    Pressure on that wall.
  5. pV=Nm⟨cx2⟩pV=Nm\langle c_x^2\rangle
    Sum over NN molecules.
  6. ⟨cx2⟩=13⟨c2⟩⇒pV=13Nm⟨c2⟩\langle c_x^2\rangle=\tfrac13\langle c^2\rangle\Rightarrow pV=\tfrac13Nm\langle c^2\rangle
    Random motion: equal in xx, yy, zz.
  7. 13Nm⟨c2⟩=NkT⇒12m⟨c2⟩=32kT\tfrac13Nm\langle c^2\rangle=NkT\Rightarrow\tfrac12m\langle c^2\rangle=\tfrac32kT
    Compare with the ideal gas equation.
Molecular speeds spread out and rise with temperature
20040060080010001200140016000.20.40.60.810speed (m/s)number of molecules300 K800 K
A range of speeds at any temperature. Hotter gas: higher average speed, wider spread. cr.m.s.=3kTmc_{\text{r.m.s.}}=\sqrt{\dfrac{3kT}{m}}.
Worked exampleNitrogen at 300 K

m=4.65×10−26m=4.65\times10^{-26} kg.

Goal
Mean KE; r.m.s. speed.
Strategy
32kT\frac32kT; crms=3kTmc_{\text{rms}}=\sqrt{\frac{3kT}m}.
  1. 32(1.38×10−23)(300)=6.21×10−21 J\tfrac32(1.38\times10^{-23})(300)=6.21\times10^{-21}\text{ J}
    Mean KE.
  2. 3(1.38×10−23)(300)4.65×10−26=517 m s−1\sqrt{\frac{3(1.38\times10^{-23})(300)}{4.65\times10^{-26}}}=517\text{ m s}^{-1}
    r.m.s. speed.
Answer6.21×10−216.21\times10^{-21} J; 517517 m s−1^{-1}.
✓ CheckFaster than sound in air (sound travels via molecules) ✓.
Common misconceptionMean speed² ⟨c2⟩\langle c^2\rangle is the mean of the squares, not the square of the mean.
Pause and think

300 K → 1200 K: effect on r.m.s. speed?

Summary

  • pV=13Nm⟨c2⟩pV=\frac13Nm\langle c^2\rangle.
  • ⟨Ek⟩=32kT\langle E_k\rangle=\frac32kT.
Key words
r.m.s. speed
Square root of the mean of squared speeds.
HistoryMaxwell and Boltzmann. James Clerk Maxwell worked out the distribution of molecular speeds in 18601860, and Ludwig Boltzmann extended it into statistical mechanics. Boltzmann's ideas were fiercely opposed by scientists who doubted atoms existed. His equation S=klog⁡WS=k\log W is engraved on his tombstone in Vienna.
Fun factAir molecules are supersonic. At room temperature, nitrogen molecules have an r.m.s. speed of about 510510 m s⁻¹, faster than the speed of sound (340340 m s⁻¹). That's no coincidence: sound is carried by molecular collisions, so it can't travel faster than the molecules themselves.
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