Maxwell–Boltzmann Distribution

01Maxwell–Boltzmann Distribution

Distribution of energies

This section covers the shape of a Maxwell–Boltzmann distribution and what its main features represent.

Molecular energies in a gas

At one temperature, the molecules in a gas sample possess a range of energies. A Maxwell–Boltzmann distribution represents how the molecules are spread across those different energies.

Relatively few molecules occur at either extreme of the distribution. Most have energies between the very low- and very high-energy regions.

Features of the curve

  • At zero energy there are no molecules to count, so the graph passes through the origin.
  • The curve climbs to a single maximum before decreasing.
  • The right-hand tail continues towards increasingly high energies, getting progressively closer to the horizontal axis without ending on it.
  • The curve is asymmetric.

The peak identifies the most probable energy, EmpE_\mathrm{mp}, because this is the individual energy possessed by the greatest number of molecules.

The mean energy is positioned at a higher energy than EmpE_\mathrm{mp}, so the mean does not coincide with the peak.

Number of moleculesMolecular energyEmpmeanpeakhigh-energy tail

Area under the curve

The whole region beneath the curve accounts for all of the molecules in the sample. If the same sample is examined at another temperature, this total area stays constant because the total number of molecules is unchanged.

Exam Tip: In a Maxwell–Boltzmann sketch, start the curve at the origin, include one peak and draw a long right-hand tail that approaches the energy axis but never meets it.

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