Maxwell-Boltzmann Distribution — AQA A-level Chemistry

SPEC 3.1.5.2–3.1.5.3 • KINETICS

Maxwell-Boltzmann Distribution & Effect of Temperature

Understanding molecular energy distribution and why temperature has such a dramatic effect on reaction rates

Why Do Particles Have Different Energies?

In Lesson 5.01, we learned that particles must have energy ≥ Eₐ to react. But not all particles in a sample have the same energy — there's actually a distribution of energies.

Key Concept

In any sample of gas at a given temperature, the particles are in constant random motion. They collide with each other billions of times per second, exchanging kinetic energy with each collision.

As a result of these countless random collisions:

Some particles gain energy and move faster

Some particles lose energy and move slower

Most particles have energies somewhere in the middle

This creates a characteristic distribution of molecular energies — the Maxwell-Boltzmann distribution — which is crucial for understanding reaction rates.

Why This Matters

Only particles with energy ≥ Eₐ can react. The Maxwell-Boltzmann distribution tells us what fraction of particles have enough energy to react at any given temperature.

The Maxwell-Boltzmann Distribution

Definition

The Maxwell-Boltzmann distribution is a graph showing how kinetic energies are distributed among the molecules in a gas at a particular temperature.

The x-axis shows kinetic energy, and the y-axis shows the number (or proportion) of molecules with that energy. The curve has a very characteristic shape:

Notice the curve starts at the origin — this is because no molecules have zero kinetic energy. Gas molecules are always moving, so every molecule has at least some kinetic energy.

Exam Tip

E(mp) = most probable energy — the energy that the largest number of molecules have. This is at the peak of the curve.

Key Features of the M-B Curve

You must be able to describe and explain each feature of the Maxwell-Boltzmann distribution. These are frequently tested in exams.

FeatureDescriptionExplanation

Starts at origin (0,0)Curve begins at zeroNo molecules have zero kinetic energy — they're always moving

Peak at E(mp)Maximum point of curveMost probable energy — the energy most molecules have

Long tail to the rightCurve extends but never touches x-axisSome molecules have very high energies; there's no upper limit

Asymmetric shapeSkewed to the rightMean energy is higher than most probable energy

Area under curveTotal area = total number of moleculesThis stays constant at a given temperature

Common Mistake

The curve never touches the x-axis on the right side. There is always a small number of molecules with very high energies. If you draw the curve touching the axis, you will lose marks.

Worked Example01

Sketch a Maxwell-Boltzmann distribution curve and label: (a) the axes, (b) the most probable energy E(mp), (c) the mean energy.

Key:

The mean energy is always slightly higher than E(mp) because the long tail on the right pulls the average up.

Activation Energy on the M-B Curve

The real power of the Maxwell-Boltzmann distribution is that it shows us what proportion of molecules can react. We mark the activation energy (Eₐ) on the curve:

Molecules with E < Eₐ

These particles will cause ineffective collisions only — they don't have enough energy to break bonds and cause a chemical reaction, no matter how they collide.

Molecules with E ≥ Eₐ

These molecules have sufficient energy to cause an effective collision which will result in a chemical change (if the orientation is correct).

The shaded area to the right of Eₐ represents the proportion of molecules that can potentially undergo successful collisions. The larger this area, the faster the reaction rate.

Worked Example02

A reaction has a high activation energy. On a Maxwell-Boltzmann diagram, explain why this reaction would be slow at room temperature.

1

A high activation energy means Eₐ is positioned far to the right on the energy axis.

2

At room temperature, the M-B curve shows most molecules have relatively low energies (clustered around the peak).

3

The shaded area to the right of Eₐ is very small — only a tiny fraction of molecules have energy ≥ Eₐ.

4

Therefore, very few collisions are successful, and the reaction rate is slow.

Effect of Temperature on the M-B Distribution

When we increase the temperature of a gas, the molecules gain kinetic energy. An increase in temperature results in an increased mean energy of the molecules, so there is a wider spread of values:

Key Changes at Higher Temperature

Peak shifts RIGHT (molecules have higher average energy)

Peak gets LOWER (curve flattens and broadens)

Curve spreads out more (wider range of energies)

Area under curve stays THE SAME (same number of molecules)

Why the Peak Lowers

The total number of molecules doesn't change, so the area under the curve must remain constant. When the curve spreads out (gets wider), it must also get shorter to keep the same total area. This is why the peak lowers as temperature increases.

Why Temperature Has Such a Large Effect

Increasing the temperature of a reaction mixture increases the rate of reaction in two ways. Understanding which effect dominates is crucial for exam success.

Effect 1: Minor

More frequent collisions

At higher temperatures, particles move faster, so they collide more frequently. A higher number of collisions in total means a higher number of successful collisions.

Typically increases by only ~2% for every 10 K rise

Effect 2: Major

Greater proportion with E ≥ Eₐ

At higher temperatures, a higher proportion of molecules have the activation energy or more. This means a higher proportion of collisions are successful.

Increases exponentially with temperature

Examiner Tip

The increase in proportion of molecules having kinetic energy greater than the activation energy has a greater effect on the rate of reaction than the increase in collision frequency. Always emphasise this in your answers.

The "10 K Rule" Explained

A common observation is that reaction rates approximately double for every 10 K (or 10°C) rise in temperature. But why such a large effect from such a small temperature change?

The Mathematics (Extension)

The fraction of molecules with energy ≥ Eₐ is given approximately by:

Fraction \propto e^{- E_{a} / RT}

Because this is an exponential relationship, small changes in T cause large changes in the fraction that can react.

Note: This equation comes from the Arrhenius equation (spec 3.1.9, Year 2). You don't need to memorise this formula for AS — just understand the qualitative effect.

Temperature ChangeEffect on Collision FrequencyEffect on Proportion with E ≥ Eₐ

300 K → 310 K (+10 K)Increases by ~2%May double (increase by ~100%)

310 K → 320 K (+10 K)Increases by ~2%May double again

Why This Happens

Look at the M-B curve near Eₐ: the curve rises steeply as energy increases. Even a small shift of the curve to the right (from increased T) brings a large additional area into the "can react" region. This is the exponential effect in action.

Worked Example03

Sketch Maxwell-Boltzmann distributions at temperatures T₁ and T₂ (where T₂ > T₁) on the same axes. Mark the activation energy Eₐ and explain why the reaction is faster at T₂.

1

Draw the T₁ curve: Start at origin, rise to peak, long tail to right (never touching x-axis).

2

Draw the T₂ curve: Lower peak, shifted right, but with the same area under the curve.

3

Mark Eₐ: Vertical line on the x-axis. Shade the areas to the right of Eₐ for both curves.

4

Explanation:

•

At T₂, the curve has shifted to the right

•

A greater proportion of molecules have energy ≥ Eₐ (larger shaded area)

•

More collisions are successful

•

Therefore the rate of reaction is faster

Key point: The total area under both curves is the same (same number of molecules), but the distribution of energies has changed.

Worked Example04

A student claims: "Increasing temperature doubles the rate because molecules move twice as fast and collide twice as often." Evaluate this statement using your understanding of the Maxwell-Boltzmann distribution.

1

The statement is incorrect — it oversimplifies the effect of temperature on rate.

2

Collision frequency: Increasing temperature by 10 K only increases collision frequency by approximately 2%, not 100% (doubling).

3

The main effect: Temperature affects the proportion of molecules with energy ≥ Eₐ. This is shown on the Maxwell-Boltzmann distribution as a shift of the curve to the right.

4

Exponential relationship: The proportion of molecules exceeding Eₐ increases exponentially with temperature, which is why a small temperature rise (10 K) can double the rate.

5

Correct explanation: "Increasing temperature shifts the Maxwell-Boltzmann distribution to the right, significantly increasing the proportion of molecules with energy greater than or equal to Eₐ, resulting in more successful collisions."

Common Exam Questions

Maxwell-Boltzmann questions appear frequently in AQA exams. Here are the most common types:

Question TypeWhat They AskKey Points to Include

Sketch the curveDraw and label a M-B distributionOrigin, peak at E(mp), long tail, never touches axis

Two temperaturesDraw curves at T₁ and T₂Higher T: lower peak, shifted right, same area

Explain rate increaseWhy is reaction faster at higher T?"Greater proportion have E ≥ Eₐ"

ShadingShade molecules that can reactArea to the RIGHT of Eₐ line

Area under curveWhat does the area represent?Total number of molecules (stays constant)

Model Answer Phrase

Always include this phrase when explaining temperature effects: "At higher temperature, a greater proportion of molecules have kinetic energy greater than or equal to the activation energy, so a greater proportion of collisions are successful."

Practice

Question 1: Reaction A has an activation energy of 50 kJ mol⁻¹. Reaction B has an activation energy of 150 kJ mol⁻¹. Using your knowledge of the Maxwell-Boltzmann distribution, explain which reaction will have a faster rate at room temperature.

Reaction A will be faster.

1

On a Maxwell-Boltzmann distribution, the activation energy is marked as a vertical line on the x-axis.

2

Reaction A has a lower Eₐ (50 kJ mol⁻¹), so its Eₐ line is positioned further to the left on the energy axis.

3

The area to the right of Eₐ represents molecules with sufficient energy to react.

4

With Eₐ further left, a greater proportion of molecules have energy ≥ Eₐ for Reaction A.

5

Therefore, more collisions are successful and the rate of Reaction A is faster.

Practice

Question 2: A student draws two Maxwell-Boltzmann curves at temperatures T₁ and T₂ (where T₂ > T₁). The student makes the curve at T₂ have a larger total area than the curve at T₁. Identify the error and explain what should be true about the areas of the two curves.

The error:

1

The student has drawn curves with different areas, implying a different number of molecules at each temperature.

What should be true:

2

The area under both curves should be exactly the same.

3

The area under a M-B curve represents the total number of molecules in the sample.

4

Changing temperature does not change the number of molecules — it only changes how their energies are distributed.

5

At higher T, the curve should be lower and broader (flattened), but with the same total area as at lower T.

Practice

Question 3: Explain why increasing the temperature by just 10 K can approximately double the rate of a chemical reaction, even though this represents only a 3% increase in absolute temperature (from 300 K to 310 K).

1

The increase in collision frequency is small (only ~2%) for a 10 K rise.

2

However, the proportion of molecules with energy ≥ Eₐ increases exponentially with temperature.

3

On the Maxwell-Boltzmann distribution, the curve shifts right at higher temperature.

4

Because the curve rises steeply near Eₐ, even a small shift brings a large additional number of molecules above the Eₐ threshold.

5

This exponential increase in successful collisions (not the small increase in collision frequency) is why a 10 K rise can double the rate.

Answer

Summary

The Maxwell-Boltzmann Distribution

FeatureDescription

What it showsDistribution of molecular kinetic energies at a given temperature

ShapeStarts at origin, peak at E(mp), long tail to right (never touches axis)

Area under curveTotal number of molecules (constant at given conditions)

Eₐ on diagramVertical line; area to the right = molecules that can react

Effect of Increasing Temperature

ChangeEffect

Peak positionShifts RIGHT and gets LOWER

Curve shapeFlattens and broadens

Area under curveSTAYS THE SAME

Proportion with E ≥ EₐINCREASES significantly (exponentially)

Key Exam Phrase

"At higher temperature, a greater proportion of molecules have kinetic energy greater than or equal to the activation energy, so a greater proportion of collisions are successful and the rate of reaction increases."

What's Next?

In this lesson, you learned how the Maxwell-Boltzmann distribution explains why temperature has such a dramatic effect on reaction rates.

In Lesson 5.03

You will learn about:

Effect of concentration on reaction rate (collision frequency)

Effect of pressure on gaseous reactions

Catalysts — how they provide an alternative pathway with lower Eₐ

How to show catalyst effect on a M-B diagram (Eₐ shifts left, curve unchanged)

Study Tip

Practice sketching M-B curves until you can do it quickly and accurately. Pay special attention to:

Starting at the origin

The long tail never touching the x-axis

Keeping the area constant when comparing temperatures

Correctly shading the region above Eₐ

Now complete the practice questions and end of lesson test to consolidate your understanding.