The Maxwell-Boltzmann distribution

The Maxwell-Boltzmann distribution is a graph of how the energies of the molecules in a gas are spread out at a given temperature. It shows the number of molecules with each energy. Most molecules have a moderate energy, a few have very little, and a few have a lot. Only molecules whose energy reaches the activation energy can react, which is why the curve explains how temperature and catalysts change the rate of a reaction.

What the Maxwell-Boltzmann distribution shows

The molecules in a gas are constantly colliding, so their energies are always changing. At any moment they don't all have the same energy. The Maxwell-Boltzmann distribution plots energy along the x-axis and the number of molecules with that energy up the y-axis.

Features of the curve

• It starts at the origin. No molecules have zero energy.
• The peak is the most probable energy: the energy more molecules have than any other.
• It is not symmetrical. It rises steeply, then falls more slowly to the right. The mean energy is to the right of the peak.
• It never touches the x-axis at high energy. There is no maximum energy; a very few molecules have a very large energy.
• The area under the curve is the total number of molecules.
• The area to the right of the activation energy, Eₐ, is the number of molecules with enough energy to react when they collide.

Eₐ is the smallest energy a collision must have for the particles to react. On most curves Eₐ is well to the right of the peak, so only a small fraction of collisions lead to a reaction.

The Maxwell-Boltzmann distribution and temperature

When the temperature rises, the molecules move faster on average, so the curve changes shape:

• The peak moves to the right: the most probable energy is higher.
• The peak gets lower and the curve gets wider.
• The total area does not change: there are just as many molecules as before.
• The area to the right of Eₐ gets much bigger.

At a lower temperature, the opposite happens: the peak is higher and further left, and fewer molecules have energy greater than Eₐ.

Why a small temperature rise makes a big difference

A higher temperature means slightly more frequent collisions, but that is a small effect. The big effect is that a much larger fraction of the molecules now has at least the activation energy, so a much larger proportion of collisions succeed. In the curves above, raising the temperature from T₁ to T₂ increases the fraction of molecules past Eₐ from about 3% to about 12%, more than four times as many.

Example: a 10 K rise

At A-level the fraction of molecules with at least the activation energy is close toe^{-E_a/RT}. For a reaction with Eₐ = 50 kJ mol⁻¹, compare 298 K with 308 K, using R = 8.31 J K⁻¹ mol⁻¹:

A rise of just 10 K, about 3% in absolute temperature, nearly doubles the number of molecules that can react, and so nearly doubles the rate. The Arrhenius equation guide takes this further.

Practise choosing the right curve in this question from the Reaction Kinetics course.

The Maxwell-Boltzmann distribution and catalysts

A catalyst provides an alternative reaction route with a lower activation energy. It does not change the curve at all, because the temperature and the molecules are the same. Instead, the Eₐ line moves to the left. Many more molecules now lie to the right of it, so more collisions succeed and the rate increases.

How to draw a Maxwell-Boltzmann distribution

1. Label the axes: energy along the bottom, number of molecules (or fraction of molecules) up the side.
2. Start the curve at the origin.
3. Rise to a peak, then fall more gently, getting closer to the x-axis without touching it.
4. Mark Eₐ on the x-axis to the right of the peak.
5. For a second, hotter curve: start at the origin, peak lower and to the right, and cross the first curve once, so it lies above the first curve at high energies. Keep the areas roughly equal.

The Maxwell-Boltzmann distribution in physics

Maxwell first worked out the distribution of molecular speeds in a gas, and physicists usually plot speed rather than energy. The shape is similar: zero at zero speed, a peak, and a long tail. The root mean square speed comes from the kinetic theory equation\frac{1}{2}m(c_{rms})^2 = \frac{3}{2}kT:

For nitrogen molecules (mass 4.65 × 10⁻²⁶ kg) at 300 K, with k = 1.38 × 10⁻²³ J K⁻¹,c_{rms}= √(3 × 1.38 × 10⁻²³ × 300 ÷ 4.65 × 10⁻²⁶) ≈ 517 m/s. The most probable speed, √(2kT/m) ≈ 422 m/s, is lower, because of the long tail of fast molecules. Double the kelvin temperature and every speed goes up by a factor of √2.

Common mistakes

• Starting the curve above the origin or ending it on the x-axis.
• Making the hotter curve taller. It is lower and wider, with the same area.
• Moving the curve for a catalyst. The curve stays the same; only Eₐ moves left.
• Saying the rate rises mainly because of more collisions. The main reason is that a much bigger share of collisions now reach Eₐ.
• Drawing a symmetrical, bell shaped curve. It is skewed, with a long tail to the right.
• Labelling the y-axis "energy". Energy goes along the x-axis.

Practise with a short timed paper

Kinetics, including the Maxwell-Boltzmann distribution, is in the first year of A-level Chemistry. The daily A-level Chemistry: Year 1 paper is a short timed paper of 8 questions, with a new one every day. You can also heat particles and watch collisions in the Reaction Rate Simulator.

Read next

Chemistry · Rates and equilibriumChemical reaction simulator: rates and collision theoryRun rate of reaction practicals as moving particles: change temperature, concentration, surface area and catalyst, and see why the rate changes.