Topic 9 · page 7 of 15

Temperature and the curve

Raising the temperature flattens the Maxwell–Boltzmann curve and moves its peak to the right, but the area and the activation energy stay put. The share of molecules beyond the activation energy rises sharply, and that is why rate rises so fast.

12 min read9.49.1

Read first: The Maxwell–Boltzmann distribution

What temperature does to the curve 9.4

Temperature is a measure of the average kinetic energy of the particles. At a higher temperature the average energy is greater, so the whole distribution shifts towards higher energy.

Two Maxwell-Boltzmann curves on the same axes, for a lower and a higher temperature. The higher temperature curve has a lower peak at a higher energy and a broader tail, and the two curves enclose the same area. A dashed vertical line marks the activation energy, which is in the same place for both. The area to the right of that line is shaded for both curves and is much larger for the higher temperature. — At the higher temperature the peak is lower and further right, and a much larger share of the area lies beyond the activation energy.

Compare the two curves:

The peak moves to the right and is lower. The most probable energy is higher, but the molecules are spread over a wider range.
The curve is broader, with a longer, fatter tail.
The area under each curve is the same. The number of molecules has not changed.
The curves cross once. The cooler curve is higher at low energy, the hotter curve is higher at high energy.
$E_\text{a}$ does not move. It depends on the reaction, not on the temperature.
The area to the right of $E_\text{a}$ is much bigger at the higher temperature. A greater proportion of molecules, and so of collisions, have energy of at least $E_\text{a}$ .

In the model used to draw this figure, about 8% of molecules have energy of at least $E_\text{a}$ at the lower temperature and about 26% at the higher one. These percentages belong to the drawing and are not data for a real reaction. For real reactions at room temperature the share is much smaller, but the pattern is the same: a modest rise in temperature gives a large rise in the share.

Why a small rise in temperature has a big effect 9.1

Higher temperature does two things:

Particles move faster, so they collide more often. This is a small effect.
A much larger proportion of collisions have energy of at least $E_\text{a}$ . This is the large effect.

The second effect is why the rate of many reactions roughly doubles for every 10 °C rise near room temperature, as in the thiosulfate figures on page 1. That is a rule of thumb, not a law, and it fails for some reactions.

Worked exampleExplaining a rise in rate

Explain, using the Maxwell–Boltzmann distribution, why the rate of a reaction is greater at 310 K than at 300 K.

The curve at 310 K has a lower peak, moved to the right, and is broader. The area under it is the same.
The activation energy is in the same place, so the area to its right is larger: more molecules have energy greater than or equal to $E_\text{a}$ .
A greater proportion of collisions are successful. Collisions are also slightly more frequent. So the rate is higher.

Check yourself

A Maxwell–Boltzmann distribution is drawn at two temperatures. State what happens to (a) the area under the curve, (b) the height of the peak, (c) the position of $E_\text{a}$ when the temperature rises.
Explain why a 10 °C rise can roughly double the rate even though collisions become only slightly more frequent.
In the thiosulfate experiment the cross takes 80 s to disappear at 20 °C and 20 s at 40 °C. By what factor does the rate rise per 10 °C, on average?
State where the two curves cross, and what each curve shows on either side of that point.
A student's sketch shows the hot curve with a higher peak than the cold curve. Explain why this cannot be right.
Explain, in terms of the Maxwell–Boltzmann distribution, why increasing the temperature increases the rate of reaction.

Answers cover these until you have tried the questions

(a) Unchanged: the number of molecules has not changed. (b) The peak is lower (and moves to the right). (c) $E_\text{a}$ does not move.
The particles move faster so collisions are slightly more frequent, but the main effect is that many more molecules have energy of at least $E_\text{a}$ . The proportion of successful collisions rises by a large factor, which can roughly double the rate.
Rates: 1 ÷ 80 = 0.0125 s⁻¹ and 1 ÷ 20 = 0.050 s⁻¹, so the rate rises 4 times over 20 °C. That is two 10 °C steps, so each step gives a factor of √4 = 2.
The curves cross once, at an energy to the right of both peaks. At lower energy the cooler curve is higher (more cool molecules at low energy). At higher energy the hotter curve is higher (more hot molecules with high energy).
The area under each curve is the same (the same number of molecules). The hotter curve is broader, spreading the same area over a wider range, so its peak must be lower.
At a higher temperature the curve is flatter and moves to the right, with the same area. $E_\text{a}$ stays in the same place, so the area to the right of it is larger: more molecules have energy greater than or equal to $E_\text{a}$ , so a greater proportion of collisions are successful. (Collisions are also more frequent.) So the rate increases.