Sketch and interpret a Boltzmann distribution
Read a particle-energy distribution and identify the fraction with energy at or above activation energy.
AS Level; Cambridge International Chemistry 9701, 2025–2027 syllabus. Rate data and diagrams are illustrative unless a question supplies measurements. A diagram is qualitative unless labelled values are provided. Written practice is self-checked
Before you begin
- Rearrange simple ratios and use consistent units for time, volume, mass and concentration.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is shown on the horizontal axis of a Boltzmann distribution?
What you will learn
- Label a Boltzmann distribution
- Mark activation energy
- Interpret the high-energy fraction qualitatively
The curve distributes particles by energy
A Boltzmann distribution plots the number or fraction of particles against particle energy at a fixed temperature. The curve starts at the origin, rises to a maximum and then falls with a long tail towards high energy.
The total area under one distribution represents the total number of particles in the sample. At a fixed temperature and for the same sample, raising an energy threshold does not change the curve; it changes the area to the right of the threshold.
Activation energy selects the effective-collision fraction
Draw a vertical line at Eₐ. The area to the right represents particles with energy at least Eₐ, which is the fraction able to overcome the energy barrier in the simplified collision model.
The area is a qualitative proportion, not an exact particle count unless numerical data are supplied. The syllabus asks learners to sketch and use the distribution, not calculate the area by integration.
Pause and explain
What does the area to the right of Eₐ show?
Worked example
On a particle-energy distribution, what does the area to the right of the Eₐ line represent?
Show the worked solution
- The horizontal axis is particle energy.
- Particles to the right have energy at least Eₐ.
- They are the fraction that can overcome the barrier in the model.
Answer It represents the fraction of particles with energy ≥ Eₐ.
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Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Describe the shape and axes of a Boltzmann distribution.
Give me a hint
Name the energy quantity on the horizontal axis and the particle count/fraction on the vertical axis.
Compare my reasoning
- Horizontal axis: particle energy.
- Vertical axis: number or fraction of particles.
- The curve rises to a peak and tails off at high energy.
A Boltzmann distribution plots particle number or fraction against energy; it rises from zero to a peak and then falls into a high-energy tail.
Look for these in your work
- Axes named
- Shape described
- Tail mentioned
Two activation-energy lines are drawn on the same distribution, Eₐ₁ to the left of Eₐ₂. Which threshold has the larger area to its right? Explain.
Give me a hint
A lower threshold leaves more of the curve to its right.
Compare my reasoning
- Eₐ₁ is the lower energy threshold.
- More particles have energy at least Eₐ₁ than at least Eₐ₂.
- So the area to the right of Eₐ₁ is larger.
Eₐ₁ has the larger area to its right because more particles meet the lower energy threshold.
Look for these in your work
- Threshold order compared
- Right-hand areas interpreted
- Reason linked to energy
A learner says that drawing an Eₐ line removes particles from the distribution. Explain what the line actually does.
Give me a hint
The line is a reference threshold, not a physical change to the particles.
Compare my reasoning
- The curve represents the same particle population.
- The line divides the curve into energy ranges.
- The right-hand area identifies the fraction with energy at least Eₐ.
The Eₐ line does not remove particles or alter their energies; it marks the threshold used to read the proportion above it.
Look for these in your work
- Curve population preserved
- Threshold role explained
- Effective fraction identified
Common mistakes
- Putting particle energy on the vertical axis
- Treating Eₐ as the curve's peak
- Claiming the area to the right is the total number of particles
- Assuming the threshold line changes the distribution
What can you explain now?
What is shown on the horizontal axis of a Boltzmann distribution?
Compare with the explanation
Particle energy
The curve shows how particle numbers are distributed by energy.
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Make it stick
What do the axes, the total curve area and the right-hand area beyond Eₐ mean?
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