Chemistry 9701/12 — February/March 2019

Background Concept

The Boltzmann distribution graph shows the distribution of molecular energies in a sample of gas at a specific temperature. The y-axis represents the proportion of molecules (or the number of molecules) with a given energy $E$, and the x-axis represents the molecular energy $E$.

Key features of a Boltzmann distribution:

• Starts at the origin: At $E=0$, the proportion of molecules is zero (though technically some molecules can have near-zero energy, the distribution curve starts at 0,0 in these schematic representations).
• Asymptotic tail: The curve approaches the x-axis but never touches it, meaning there is always a small proportion of molecules with very high energy.
• Area under the curve: The total area under the curve represents the total number of molecules in the sample. This area is constant unless the amount of gas changes.
• Peak: The highest point on the curve corresponds to the most probable energy (the energy possessed by the largest proportion of molecules). The average kinetic energy is slightly to the right of the peak.

Understanding the Question

The question provides a Boltzmann distribution curve for a gas at temperature $T_1$. The gas is then heated to a higher temperature $T_2$ (so $T_2 > T_1$). We must identify which of the four graphs (A, B, C, D) correctly represents the distribution at $T_2$.

The command word is implicit in "What is the correct distribution...": we need to deduce the shape of the new curve based on the principles of kinetic molecular theory.

Approach

To solve this, we apply three rules for how the Boltzmann distribution changes when temperature increases (at constant volume/amount of gas):

1. Energy increases: Temperature is a measure of average kinetic energy. As $T$ increases, the average energy increases, so the peak must shift to the right (higher $E$).
2. Area is constant: The total number of molecules does not change. Since the curve spreads out to higher energies, the peak height must decrease to keep the total area the same.
3. More energetic molecules: A higher temperature means a significantly larger proportion of molecules have high energies (above the activation energy, $E_a$). Therefore, the right-hand tail of the curve must be higher at high energies.

Step-by-Step Reasoning

1. Analyze the shift in peak position: Since $T_2 > T_1$, the average energy is higher. The most probable energy (the peak) must move to the right along the x-axis. This eliminates graphs A and D, where the peak is to the left of the original $T_1$ peak (indicating a lower temperature).
2. Analyze the peak height: The total number of molecules is constant, so the area under the curve must be the same as the original curve. If the curve shifts to the right and broadens, the peak must become lower (flatter) to maintain the same area. Graph C has a peak height similar to the original but shifted right, which would imply a larger area (more molecules), so it is incorrect. Graph B has a lower peak.
3. Analyze the tail: At higher temperatures, more molecules have sufficient energy to react (energy > $E_a$). The curve at high energies (far right) should be higher than the original curve. Graph B shows a much higher tail extending to the right, consistent with more high-energy molecules.

• Graph A: Peak is higher and to the left. This is the distribution for a lower temperature.
• Graph B: Peak is lower and to the right. The tail is higher. Area is roughly conserved. This is the correct distribution for a higher temperature.
• Graph C: Peak is similar height but shifted right. The area under this curve is larger than the original, implying the number of molecules increased, which is not stated.
• Graph D: Peak is very high and to the left. This is the distribution for a much lower temperature.

Therefore, B is the correct answer.

Key Takeaways

• Temperature increase: Peak moves right (higher energy), peak height decreases (curve flattens), tail at high energy rises.
• Area constancy: The total area under a Boltzmann distribution curve is proportional to the total number of molecules. If amount of gas is constant, area is constant.
• Catalyst effect: A catalyst does not change the Boltzmann distribution; it lowers the activation energy threshold (a vertical line moving left), increasing the area under the curve to the right of the new $E_a$.

Common Mistakes

• Confusing temperature and catalyst effects: Students often think a catalyst changes the distribution curve (making the peak higher). A catalyst only changes the activation energy line, not the distribution of molecular energies.
• Ignoring area constancy: Choosing a graph where the peak is higher and to the right (like a mix of A and B features) without realizing the area would be huge, implying more molecules were added.
• Wrong direction of shift: Thinking that heating makes molecules slower or shifts the peak left. Heat = more energy = shift right.

Things to Be Careful About

• Starting point: The curve must always start at the origin $(0,0)$.
• Asymptote: The curve must approach the x-axis asymptotically on the right; it should not drop sharply to zero.
• Area: Always check that the "flattening" of the curve compensates for the "widening" to keep the area constant.
• Activation energy: While not drawn here, remember that the vertical line for $E_a$ stays in the same place; only the curve moves. The area to the right of $E_a$ increases significantly with temperature.