Maxwell-Boltzmann Distribution Revision Notes | A Level AQA Chemistry

1. Why Molecules Do Not All Move at the Same Speed

Picture a gas in a sealed container. Billions of molecules are colliding with each other and with the walls every second. Each collision transfers energy between molecules — one molecule speeds up, the other slows down. Because these collisions are essentially random, no two molecules end up with exactly the same energy at any given moment.

This is the central insight behind the Maxwell-Boltzmann distribution: in any gas sample, molecular energies are spread across a wide range, not fixed at a single value. Some molecules happen to have very little energy after a series of collisions; a small number have gained a great deal. The distribution curve is the map of that spread.

Understanding this spread is not merely academic. The fraction of molecules that possess energy above a certain threshold — the activation energy — determines how fast a chemical reaction proceeds. That connection makes the Maxwell-Boltzmann distribution one of the most practically important ideas in A-Level kinetics.

2. The Shape of the Maxwell-Boltzmann Distribution Curve

The Maxwell-Boltzmann distribution is plotted with kinetic energy on the x-axis and number of molecules (or fraction of molecules) on the y-axis. The curve has a distinctive, asymmetric shape that you must be able to draw and interpret accurately in the exam.

• The curve starts at the origin (0, 0): no molecules have zero energy.
• It rises steeply to a peak — the most probable energy — then falls away more gradually.
• The right-hand tail extends towards high energies but never touches the x-axis: there is always a tiny fraction of molecules with very high energies.
• The area under the entire curve represents the total number of molecules in the sample.
• The mean energy lies to the right of the peak, because the long high-energy tail pulls the average upward.

The asymmetry is crucial. The curve is not a normal (bell-shaped) distribution. The left side is steeper than the right side, and the tail on the right never reaches zero. Examiners frequently award marks specifically for drawing this asymmetry correctly.

On the chart above, $E_{mp}$ marks the most probable energy (the peak), $E_{mean}$ marks the mean energy (slightly to the right of the peak), and $E_a$ marks the activation energy threshold. Only molecules to the right of $E_a$ have sufficient energy to react when they collide.

3. Activation Energy and the Reacting Fraction

Not every collision between molecules leads to a reaction. For a reaction to occur, the colliding molecules must together possess at least the activation energy, $E_a$. On the Maxwell-Boltzmann curve, $E_a$ is marked as a vertical line on the energy axis. The shaded area under the curve to the right of $E_a$ represents the fraction of molecules that have sufficient energy to react.

This shaded area is typically very small compared with the total area under the curve. That is why, even in a fast reaction, the vast majority of collisions are unsuccessful — most molecules do not carry enough energy at the moment of collision.

Here is the key link to reaction rate: the larger the shaded area, the greater the proportion of successful collisions, and the faster the reaction. Anything that shifts more of the distribution to the right of $E_a$ — or lowers $E_a$ itself — increases the reaction rate. This is the conceptual foundation for understanding both temperature effects and catalysis.

4. The Effect of Temperature on the Distribution Curve

When you heat a gas, you supply energy to the molecules. The total energy of the sample increases, so the distribution shifts. Comparing curves at two temperatures — say $T_1$ and a higher temperature $T_2$ — reveals a consistent and examinable pattern of changes.

• The peak of the curve shifts to the right: the most probable energy increases.
• The peak height decreases: fewer molecules are clustered at any single energy value.
• The curve becomes broader and flatter, spreading over a wider range of energies.
• The tail on the right extends further: a greater proportion of molecules now have very high energies.
• The total area under the curve remains the same, because the total number of molecules has not changed.

That last point is one of the most commonly tested. The area under the curve is proportional to the total number of molecules. If you are comparing two curves for the same sample at different temperatures, the areas must be equal. If you draw the higher-temperature curve with a larger area, you will lose marks.

Notice how the $T_2$ curve has a lower, broader peak shifted to the right, while the $T_1$ curve is taller and narrower. The area under both curves is the same. The activation energy $E_a$ is fixed — it does not change with temperature. However, the shaded area to the right of $E_a$ is noticeably larger for $T_2$, which is why higher temperatures produce faster reactions.

5. Interpreting the Curve to Explain Reaction Rate

The Maxwell-Boltzmann distribution gives you the language to explain, at a molecular level, why increasing temperature increases reaction rate. This explanation appears repeatedly in AQA mark schemes, so the phrasing matters.

At higher temperature, the distribution shifts to the right. The activation energy $E_a$ is fixed. Because the curve has shifted, a greater proportion of molecules now have energy equal to or greater than $E_a$. This means a greater proportion of collisions are successful, so the rate of reaction increases. The increase in the number of collisions per second also contributes, but the dominant effect at modest temperature increases is the shift in the energy distribution.

A catalyst provides an alternative reaction pathway with a lower activation energy. On the Maxwell-Boltzmann diagram, this is shown by moving the $E_a$ line to the left. The distribution curve itself does not change — the temperature is the same — but a much larger fraction of molecules now lies to the right of the new, lower $E_a$. This is why catalysts increase reaction rate without changing temperature.

6. Drawing the Curve: A Step-by-Step Walkthrough

Exam questions frequently ask you to sketch a Maxwell-Boltzmann curve, sometimes adding a second curve at a different temperature or marking $E_a$ on the diagram. Each feature carries marks, so a systematic approach prevents careless errors.

Worked example: Sketching a Maxwell-Boltzmann Distribution with a Second Temperature Curve

Sketch the Maxwell-Boltzmann distribution for a gas at temperature $T_1$. On the same axes, add a curve for the same gas at a higher temperature $T_2$. Mark the activation energy $E_a$ and shade the area representing molecules that can react at $T_2$.

1. x-axis: 'Kinetic energy / E' (or 'Energy'); y-axis: 'Number of molecules' (or 'Fraction of molecules'). Both axes start at the origin. — Correct axis labels are often worth a mark. 'Speed' or 'velocity' on the x-axis is incorrect for an energy distribution — the AQA specification refers to molecular energies.
2. Start at the origin. Rise steeply to a clear peak (the most probable energy at $T_1$), then fall away more gradually to the right. The right-hand tail must not touch the x-axis. — The asymmetry — steeper left side, longer right tail — is essential. A symmetric bell curve will lose marks. The curve must begin at (0, 0): no molecules have zero energy.
3. Draw a second curve that: (a) starts at the origin, (b) has a lower and broader peak shifted to the right of the $T_1$ peak, (c) has a longer right-hand tail, and (d) encloses the same total area as the $T_1$ curve. — Equal areas are critical. If the $T_2$ curve looks taller or the same height as $T_1$, the sketch is wrong. The two curves will cross somewhere to the right of the $T_1$ peak.
4. Draw a vertical dashed line on the x-axis at a position to the right of both peaks. Label it $E_a$. — $E_a$ is fixed — it does not move between the two curves. Place it in the tail region, where only a small fraction of molecules lie to its right for $T_1$, but a larger fraction for $T_2$.
5. Shade the area under the $T_2$ curve to the right of the $E_a$ line. — This shaded area is larger than the corresponding area under the $T_1$ curve, visually demonstrating why the reaction is faster at $T_2$. If the question asks for $T_1$, shade under the $T_1$ curve instead.
6. Label: the peak of each curve as '$E_{mp}(T_1)$' and '$E_{mp}(T_2)$'; the curves as '$T_1$' and '$T_2$'; and the shaded region as 'molecules with $E \geq E_a$'. — Clear labelling prevents ambiguity and secures method marks even if the sketch is not perfectly proportioned.

Answer: A correctly drawn sketch shows two asymmetric curves starting at the origin, with the $T_2$ curve lower, broader, and shifted right relative to $T_1$. Both curves enclose equal areas. $E_a$ is marked as a fixed vertical line in the tail region, with the shaded area under $T_2$ to the right of $E_a$ visibly larger than the corresponding area under $T_1$.

7. Exam Technique and Common Mistakes

Maxwell-Boltzmann questions appear in multiple formats: sketch questions, interpretation questions, and extended-response 'explain' questions. Each format has its own mark-winning requirements.

Extended 'explain' questions on this topic typically award marks for: (1) stating that the distribution shifts to higher energies at higher temperature, (2) stating that $E_a$ is unchanged, (3) stating that a greater proportion of molecules now have energy $\geq E_a$, and (4) concluding that the rate of successful collisions — and therefore the reaction rate — increases. Hitting all four points in a logical sequence is the mark-winning approach.