Chemistry 9701/12 — May/June 2016

Background Concept

The Boltzmann distribution curve shows the distribution of molecular energies in a gas at a given temperature. The y-axis represents the proportion (or number) of molecules with a given energy, and the x-axis represents the molecular energy. The area under the curve represents the total number of molecules in the sample, which is constant regardless of temperature. The peak of the curve (P) corresponds to the most probable energy (the energy possessed by the largest fraction of molecules). The activation energy ($E_a$) is the minimum energy required for a successful collision. The area under the curve to the right of $E_a$ represents the fraction of molecules with energy $\ge E_a$, which determines the reaction rate.

Understanding the Question

The question provides a Boltzmann distribution curve at an initial temperature and asks for the effect of increasing the temperature. We need to determine how the peak P changes and how the number of molecules with energy $> E_a$ changes.

Approach

We must consider two constraints: (1) the total number of molecules is constant (area under curve is constant), and (2) increasing temperature increases the average kinetic energy of the molecules (curve shifts right). By combining these, we can deduce the new shape of the curve.

Step-by-Step Reasoning

1. Average energy increases: When temperature increases, the molecules gain kinetic energy. The average energy increases, so the entire distribution shifts to the right along the x-axis.
2. Peak P lowers: The total area under the curve represents the total number of molecules (1 mole in this case), which does not change. As the curve shifts to the right and becomes broader (flatter) to accommodate the higher energies, the peak must decrease in height to keep the total area constant. Therefore, peak P will be lower.
3. More molecules have energy $> E_a$: The activation energy $E_a$ is a fixed value for the reaction (temperature does not change $E_a$, only a catalyst does). Because the curve has shifted to the right and flattened, the area under the curve to the right of the vertical line at $E_a$ has increased. This area represents the proportion of molecules with energy greater than the activation energy. Therefore, more molecules now have energy $> E_a$, which explains why the reaction rate increases with temperature.

Key Takeaways

• Increasing temperature shifts the Boltzmann curve to the right and lowers the peak.
• The area under the curve remains constant (total number of molecules is unchanged).
• The area to the right of $E_a$ increases, meaning a greater proportion of molecules have sufficient energy to react.

Common Mistakes

• Thinking the peak gets higher: This is incorrect because the total area must remain constant. If the curve shifts right and flattens, the peak must lower.
• Thinking fewer molecules have energy $> E_a$: This contradicts the observation that reaction rates increase with temperature. The shift to the right increases the tail area beyond $E_a$.
• Confusing the effect of temperature with a catalyst: A catalyst lowers $E_a$ (shifting the vertical line left), which also increases the area to the right, but does not change the shape of the Boltzmann curve itself.

Things to Be Careful About

• The y-axis is 'proportion of molecules' or 'number of molecules', not 'rate of reaction'.
• The area under the curve is constant; this is the key to understanding why the peak lowers.
• $E_a$ is fixed unless a catalyst is added; temperature only changes the distribution of energies, not the activation energy threshold itself.