The shape of the lesson
By the end of the lesson, learners can
- The shape of the curve: Use the simulator to raise and lower the temperature and watch the distribution change shape.
- Why temperature matters so much: Raising the temperature by 10 K increases the average kinetic energy by only about 3%, yet many reactions go roughly twice as fast.
- Axes: number of molecules against energy.
- Origin start, tail never touches the axis.
- Higher T: peak lower and right, same area.
- Ea unchanged by temperature.
One picture every idea returns to
The Boltzmann distribution shows the number of molecules with each energy. It starts at the origin (no molecule has zero energy), rises to a peak (the most probable energy) and then falls, approaching but never touching the energy axis, because there is no maximum energy. The area under the curve equals the total number of molecules. Only molecules to the right of Ea can react. At a higher temperature the curve is lower and broader, with its peak moved to the right and the same area; the area beyond Ea increases greatly, so a much larger proportion of collisions succeed.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Not every molecule is equal.. In any gas some molecules crawl and a few race. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 17 min | Teach and check: The shape of the curve | Use the simulator to raise and lower the temperature and watch the distribution change shape. Project tab 2 of the lab (Reaction kinetics lab, Boltzmann distribution) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Why temperature matters so much | Raising the temperature by 10 K increases the average kinetic energy by only about 3%, yet many reactions go roughly twice as fast. | Mix-Pair-Share |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 13 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Why temperature matters so much, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| On the higher-temperature curve the peak is lower and further right, and the area is the same, because the number of molecules has not changed. | And temperature does not change E a; it changes how many molecules have enough energy to overcome it. The main reason for the faster rate is the larger proportion of molecules with E ≥ E a, not the increase in collision frequency. |
The Boltzmann distribution and activation energy: worksheet
6 questions, 13 marks. Written in the style of the papers; no past paper question is reproduced.
Sketch the Boltzmann distribution for a gas. Label the axes and mark the activation energy.
On the same axes, add the curve for the same gas at a higher temperature and describe two differences.
Use the Boltzmann distribution to explain why increasing the temperature increases the rate of reaction.
Explain why the area under the curve does not change when the temperature is raised.
Define activation energy.
A student says: “Heating a reaction lowers the activation energy.” Explain why this is wrong.
Answer key and marking notes
- Axes: number (or proportion) of molecules on y, energy on x.
- Curve from the origin, rising to a peak, then a long tail approaching but not touching the x-axis.
- A vertical line for Ea well to the right of the peak.
- The peak is lower and shifted to the right.
- The curve is broader, with more molecules at high energy; the area under both curves is the same.
- At higher temperature the distribution shifts to higher energies.
- A greater proportion of molecules have energy equal to or greater than Ea.
- So there are more successful collisions per unit time (collision frequency also rises slightly).
- The area represents the total number of molecules.
- Heating does not change the number of molecules, only how their energies are spread.
The minimum energy that colliding particles must have for a reaction to occur.
- The activation energy is fixed for a given reaction pathway; temperature does not change it.
- Heating increases the number of molecules with energy above Ea; only a catalyst lowers Ea.
8 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Which statement describes the shape of the Boltzmann distribution curve for the molecules in a gas?
- Ait starts at the origin, rises to a peak and then approaches the energy axis at high energy without ever meeting it
- Bit is symmetrical about the peak, like a bell-shaped curve
- Cit falls to zero at a maximum energy where it meets the energy axis
- Dit starts at a non-zero value, because many molecules have zero energy
What does the total area under a Boltzmann distribution curve represent?
- Athe total number of molecules in the sample
- Bthe total energy of all the molecules
- Cthe rate of the reaction
- Dthe number of molecules with energy greater than or equal to the activation energy
The graph shows Boltzmann distributions for the same sample of gas at two temperatures. Which statement about the curve at the higher temperature is correct?
- Athe peak is lower and moves to higher energy, and the area under the curve is unchanged
- Bthe peak is higher and moves to higher energy, and the area under the curve is unchanged
- Cthe peak is lower and moves to lower energy, and the area under the curve is unchanged
- Dthe peak is lower and moves to higher energy, and the area under the curve is larger
In the graph, what does the shaded area represent?
- Athe number of molecules at the higher temperature T2 with energy greater than or equal to Ea
- Bthe number of molecules at the higher temperature T2 with energy less than Ea
- Cthe total number of molecules at the higher temperature T2
- Dthe extra energy given to the molecules when the temperature is raised from T1 to T2
Which statement best explains why the rate of a reaction increases when the temperature is raised?
- Aa larger proportion of molecules have energy greater than or equal to Ea, so a larger proportion of collisions are successful
- Bthe activation energy decreases, so more molecules have enough energy to react
- Cthe molecules collide more often; this is the only reason the rate increases
- Dthe total number of molecules increases, so there are more collisions
For many reactions near room temperature, a 10 K rise roughly doubles the rate, yet the collision frequency increases by less than 2%. What accounts for most of the increase in rate?
- Athe number of molecules with energy greater than or equal to Ea roughly doubles, because this lies in the high-energy tail of the distribution
- Bthe collision frequency roughly doubles, because the molecules move twice as fast
- Cthe activation energy roughly halves for every 10 K rise
- Dthe area under the Boltzmann curve roughly doubles, so there are twice as many molecules
A reaction takes 160 s to produce a fixed amount of product at 25 °C. Assuming the rate doubles for every 10 °C rise in temperature, how long will the same reaction take at 45 °C?
- A40.0 s
- B80.0 s
- C640 s
- D20.0 s
More gas is added to a sealed container at constant temperature, so that there are twice as many molecules. How does the Boltzmann distribution change?
- Aevery point on the curve doubles in height: the area doubles, the peak stays at the same energy and the proportion of molecules with energy at least Ea is unchanged
- Bthe curve becomes flatter and moves to higher energy, with the same area
- Cthe curve is unchanged, but the Ea line moves to lower energy
- Dthe curve is unchanged, because only temperature affects the Boltzmann distribution
Quiz key
No molecules have zero energy, so the curve starts at the origin. It rises to a peak (the most probable energy) and then tails off, becoming asymptotic to the energy axis because there is no maximum energy: a very few molecules always have very high energies. The curve is not symmetrical; the high-energy tail is long and the rise from zero is steep. It never meets the axis at a maximum energy, which is the key reason a few molecules can always exceed Ea. Starting at a non-zero value would mean some molecules are stationary with zero energy, which is not the case.
The curve plots the number of molecules with each energy, so the area under the whole curve counts every molecule: it is the total number of molecules. That is why the area stays the same when the temperature changes. The total energy would need each number to be multiplied by its energy, which the area does not do. The rate depends on how many molecules exceed Ea and on collision frequency; it is not the whole area. Only the area to the right of the Ea line gives the number of molecules with at least the activation energy.
At the higher temperature (T2) the mean energy rises, so the distribution spreads to the right: the peak moves to higher energy and, because the same number of molecules is spread over a wider range, the peak is lower. The area is unchanged because the number of molecules is the same. A higher peak would require more molecules or a narrower spread; neither happens on heating. Moving to lower energy is what happens on cooling. A larger area would mean more molecules have appeared, which heating cannot do.
The shading lies under the T2 curve and to the right of the Ea line, so it counts the molecules at T2 that have at least the activation energy and so can react on collision. Molecules with energy less than Ea are to the left of the line and are not shaded. The total number of molecules is the whole area under the curve, not just the tail. The area under a Boltzmann curve counts molecules, not energy, so it cannot represent the energy supplied on heating.
Heating spreads the Boltzmann distribution to higher energies, so the area beyond the fixed Ea line becomes much larger: a greater proportion of collisions have enough energy to succeed. Ea is a property of the reaction pathway and does not change with temperature; only a catalyst provides a route with a lower Ea. Molecules do collide slightly more often at higher temperature, but that effect is small; the main reason is the energy effect. Heating does not create extra molecules; the area under the curve stays the same.
Ea lies well out in the high-energy tail, where the curve is small. A small shift of the distribution to the right makes a large relative change in the area beyond Ea, so the number of molecules able to react roughly doubles and so does the rate. The collision frequency rises only slightly because average speed depends on the square root of the absolute temperature: 310 K is only about 3% hotter than 300 K. Ea does not change with temperature. The area under the curve is the total number of molecules, which is constant.
From 25 to 45 °C is two 10 °C steps, so the rate increases by 2 × 2 = 4 and the time falls by a factor of 4: 160 / 4 = 40.0 s. 80.0 s allows for only one doubling. 640 s multiplies the time by 4, as though a faster rate took longer; time and rate are inversely related. 20.0 s divides by 8, counting three steps (25, 35, 45) instead of two intervals.
At the same temperature the shape of the distribution is the same, but there are twice as many molecules at every energy, so the curve is stretched vertically and its area doubles. The peak stays at the same energy and the same fraction of molecules exceeds Ea; the rate rises because there are more collisions per unit time, not because a bigger proportion succeed. Flattening and moving to higher energy is the effect of a temperature rise, not of adding molecules. Adding gas does not change Ea. The shape depends only on temperature, but the height depends on the number of molecules, so the curve does change.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.