Maxwell-Boltzmann distribution and temperatureAQA A-Level Chemistry: Revision notes
Section 1
The Maxwell–Boltzmann distribution
In a gas, molecules have a wide range of energies and are constantly colliding and exchanging energy. The Maxwell–Boltzmann distribution shows the number of molecules with each energy, with energy on the x-axis and number of molecules on the y-axis.
Features of the curve:
- it starts at the origin, as no molecules have zero energy
- it has a peak, the most probable energy, and is asymmetrical with a long tail to high energy
- it never touches the energy axis at high energy
- the area under the curve equals the total number of molecules, so is the same at every temperature
Letting the curve touch the x-axis at high energy, or drawing a curve that does not start at the origin.
Section 2
Effect of temperature on the distribution
When the temperature increases:
- the peak moves to a higher energy and becomes lower
- the curve becomes flatter and broader, with a longer high-energy tail
- the area under the curve stays the same, since the number of molecules does not change
When drawing curves for two temperatures, the higher-temperature curve is below the lower-temperature one at the left, crosses it once, and is above it at high energy. The two curves should not touch the axis.
Draw the lower-temperature curve with the higher, narrower peak. The two curves cross once, to the right of the peaks.
Section 3
Activation energy and the fraction that can react
The activation energy (Ea) is marked on the energy axis, to the right of the peak. Only molecules with energy equal to or greater than Ea can react on collision. The area under the curve to the right of Ea is the number (proportion) of molecules that can react.
The activation energy does not change with temperature. At a higher temperature, the area to the right of Ea is larger, so a greater proportion of molecules have enough energy, and a greater proportion of collisions is successful.
Section 4
Rate of reaction and temperature
The rate of reaction is the change in concentration of a reactant or product per unit time (units mol dm⁻³ s⁻¹).
Raising the temperature increases the rate because:
- molecules move faster, so collisions are slightly more frequent
- much more importantly, a greater proportion of molecules has energy ≥ Ea, so more collisions are successful
Why a small rise gives a large increase in rate: Ea lies in the high-energy tail of the distribution, where the curve is low. A small shift of the distribution therefore increases the small area to the right of Ea by a large proportion. For many reactions near room temperature, a rise of about 10 K roughly doubles the rate. The effect becomes smaller at higher temperatures.
Explaining the effect of temperature only by saying that molecules move faster and collide more. The main reason is the larger proportion with energy ≥ Ea.
Section 5
Required practical 3: temperature and rate
Mix a fixed volume and concentration of sodium thiosulfate solution with hydrochloric acid at different temperatures, and time how long a cross takes to disappear as sulfur forms. The rate is proportional to 1/time.
To make the test fair:
- keep the volumes and concentrations of both solutions constant
- warm each solution to the required temperature in a water bath before mixing
- use the same cross and the same observer, or a light sensor
- repeat and take a mean
Results: rate against temperature is a curve that rises more steeply as temperature increases. For example, the time falling from 64 s at 20 °C to 31 s at 30 °C means the rate is 64 ÷ 31 = 2.1 times as great.
Must know
- Maxwell–Boltzmann curve: starts at origin, asymmetrical, area = number of molecules
- Higher temperature: peak lower and to the right, same area
- Area beyond Ea gives the proportion of molecules that can react
- A small temperature rise greatly increases the proportion with energy ≥ Ea, so the rate rises substantially
- Rate ∝ 1/t in the thiosulfate and acid experiment
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Maxwell-Boltzmann distribution and temperature
- The distribution of molecular energies in a sample of a gas at 300 K is described by a Maxwell–Boltzmann distribution curve, in which the number of molecules is plotted against their energy. The same sample is then heated to 310 K. A reaction of this gas has an activation energy that is much greater than the most probable molecular energy.Explain the significance of the area under the curve to the right of the activation energy.2 marks
- A student investigates how temperature affects the rate of the reaction between sodium thiosulfate solution and dilute hydrochloric acid, which forms a precipitate of sulfur. She mixes 50.0 cm³ of 0.10 mol dm⁻³ sodium thiosulfate solution with 5.0 cm³ of 1.0 mol dm⁻³ hydrochloric acid and measures the time taken for a cross viewed through the mixture to disappear. At 20 °C the time is 64 s and at 30 °C the time is 31 s.Suggest two ways in which the reliability of the investigation could be improved.2 marks
- A gas-phase reaction between two gases is carried out at constant volume at 300 K and at 310 K. At 300 K, 1.0 × 10⁻⁸ of the molecules in the mixture have energy equal to or greater than the activation energy. At 310 K the proportion is 1.8 × 10⁻⁸.Calculate the factor by which the proportion of molecules with energy equal to or greater than the activation energy increases, and use it to estimate the effect on the rate of reaction.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).