The Boltzmann Distribution

Learning Objectives (OCR Spec 3.2.2)

By the end of this lesson you should be able to:

• Sketch the Maxwell-Boltzmann distribution of molecular energies
• Interpret the main features of the curve: origin at (0,0), long tail at high energy, non-zero at Ea
• Show the effect of increasing temperature on the distribution
• Relate the area under the curve beyond Ea to the proportion of particles that can react
• Use the Boltzmann curve to explain why small temperature rises cause large rate increases

The Distribution of Molecular Energies

At any instant, the molecules in a sample of gas (or liquid) have a wide range of kinetic energies. The Maxwell-Boltzmann distribution shows how molecular energies are statistically distributed. The x-axis is the energy of a molecule, and the y-axis is the number of molecules (or fraction) with that energy.

graph LR
  A[Origin: few molecules with E = 0] --> B[Rises steeply to a peak]
  B --> C[Peak: most probable energy, Emp]
  C --> D[Falls gradually with a long tail]
  D --> E[Tail extends to very high E but does not touch axis]

Key Features to Label on a Boltzmann Curve

1. The curve starts at the origin (0, 0) - no molecules have zero kinetic energy (they are all in motion).
2. The curve rises to a peak at the most probable energy (Emp) - this is not the mean energy, which lies slightly higher.
3. The curve then falls gradually with a long tail at high energies.
4. The curve never touches the x-axis - theoretically, a small number of molecules can have arbitrarily high energies.
5. The total area under the curve equals the total number of molecules (a constant at fixed amount of gas).

Activation Energy on the Curve

Mark the activation energy Ea as a vertical line on the x-axis. The area under the curve to the right of Ea represents the number of molecules with enough energy to react in any collision. This is typically a small fraction of the total at room temperature - which is why most reactions need heat or a catalyst to proceed.

     molecules with E >= Ea
              (shaded tail)

At 298 K, for a typical Ea of ~60 kJ mol⁻¹, only about 2 × 10⁻¹⁰ of the molecules (one in ten billion) have enough energy to react - yet the collision frequency is so high (~10¹⁰ s⁻¹ per molecule) that reaction still occurs at a measurable rate.

Effect of Temperature

When temperature increases, the kinetic energies of all molecules increase on average. The Boltzmann distribution changes in three ways:

1. The peak shifts to the right (higher most probable energy)
2. The peak becomes lower (the curve flattens)
3. The area under the tail beyond Ea increases significantly

The total area stays the same because the total number of molecules does not change.

graph LR
  A[T1 curve: higher, narrower peak at lower Emp] --> B[T2 > T1: lower, broader peak at higher Emp]
  B --> C[Both curves cross; T2 has larger area beyond Ea]

Why Small Temperature Rises Cause Large Rate Increases

A common "rule of thumb" is that for many reactions the rate doubles for every 10 K rise in temperature near room temperature. This seems surprising: the collision frequency only rises by about 2% for a 10 K change, yet the rate doubles (100% increase).

The explanation is the Boltzmann distribution. The number of molecules with E ≥ Ea depends exponentially on temperature through the Boltzmann factor e⁻ᴱᵃᐟᴿᵀᵈ. For Ea = 50 kJ mol⁻¹:

T / K	e⁻ᴱᵃᐟᴿᵀᵈ	Fraction with E ≥ Ea
293	~1.0 × 10⁻⁹	tiny
303	~1.9 × 10⁻⁹	tiny but roughly doubled
313	~3.5 × 10⁻⁹	doubled again

A 10 K rise almost doubles the exponential factor, and that doubling of active molecules translates directly into a doubling of rate. The collision frequency increase (~2%) is a minor contributor compared with the exponential increase in the fraction above Ea.

Examiner phrasing: write "a greater proportion of molecules have energy greater than or equal to Ea" - not "more molecules have high energy" or "molecules move faster".

Common Drawing Mistakes