3.5 Kinetic Theory

Maxwell-Boltzmann Distribution

A particulate model sits above the live curve on fixed axes, so heating visibly flattens the peak without changing the area. Pin a second curve to compare, then switch the axis to kinetic energy and watch every gas land on one curve.

KE = ½mv²Speed vs EnergyPinned Comparison6 Gases
Topic 3.5

Kinetic Molecular Theory

Explain the relationship between the motion of particles and the macroscopic properties of gases with (i) the kinetic molecular theory, (ii) a particulate model, and (iii) a graphical representation.

Kinetic molecular theory (KMT) is the particulate model that explains why PV = nRT works. Its assumptions:

  • Gas particles are in continuous, random motion.
  • Particle volume is negligible compared to the container volume.
  • There are no significant attractive or repulsive forces between particles.
  • Collisions are perfectly elastic — kinetic energy is conserved.
  • Average kinetic energy is proportional to absolute temperature.

Two quantitative relationships anchor the topic. First, KE = ½mv² relates a particle's kinetic energy to its speed. Second, and central: the kelvin temperature of a sample is proportional to the average kinetic energy of its particles.

Put those together and you get the result students most often miss: at the same temperature, all gases have the same average kinetic energy. Since KE = ½mv², a heavier gas must therefore move more slowly. Helium and xenon at 300 K have identical average kinetic energies; helium's average speed is far greater.

The Maxwell–Boltzmann distribution shows this graphically: the fraction of particles at each energy or speed. Its features:

  • The curve is asymmetric with a long high-energy tail.
  • Raising the temperature broadens and flattens the curve and shifts the peak to higher energy.
  • The total area under the curve is always 1 (every particle has some energy).
  • At a fixed temperature, a heavier gas gives a narrower curve peaked at a lower speed.

That high-energy tail matters enormously in Unit 5: it is the fraction of particles with enough energy to react.

Key points

  • Same temperature ⇒ same average kinetic energy, regardless of the gas.
  • Heavier gas at the same T ⇒ slower average speed (and narrower, lower-peaked distribution).
  • Raising T broadens, flattens, and right-shifts the Maxwell–Boltzmann curve; area stays 1.
  • Pressure arises from particle collisions with the container walls — more frequent or more forceful collisions mean higher pressure.

Equations

  • on the exam sheetPer-particle kinetic energy. On the sheet under Gases, Liquids, and Solutions.
    • particle mass
    • particle speed
  • not on the sheetThe proportionality is to absolute temperature. This is the conceptual heart of KMT.
    • temperature in kelvin

Common mistakes

  • Same temperature does not mean same speed. It means same average kinetic energy.
  • Do not say the curve “gets taller” when T rises. It gets shorter and wider — the area is fixed.
  • Average speed is not the peak of the speed distribution. The distribution is skewed, so the average sits slightly right of the most probable speed.
  • KMT assumptions are idealizations. Real gases have volume and attractions (see 3.6).

Worked example

A container holds a mixture of He and Ar at 400 K. Compare (a) the average kinetic energies, (b) the average speeds, and (c) the shapes of the Maxwell–Boltzmann curves for the two gases.

(a) Average kinetic energy: identical. Average kinetic energy depends only on absolute temperature, and both gases are at 400 K.

(b) Average speed: helium is much faster. Since KE = ½mv² and the two KE values are equal, the gas with the smaller mass must have the larger speed. Argon's molar mass (39.9) is about ten times helium's (4.00), so helium's average speed is roughly √10 ≈ 3.2 times greater.

(c) Curve shapes: plotted against speed, helium gives a broad curve peaked at high speed while argon gives a narrower curve peaked at low speed. Plotted against kinetic energy, the two curves are essentially the same, since both gases share the same energy distribution at a given temperature.

Full notes for topic 3.5 →