Chemistry 9701/12 — October/November 2014

Background Concept

The Maxwell-Boltzmann distribution describes the distribution of kinetic energies among molecules in a gas at a given temperature. The y-axis represents the number of molecules (or fraction of molecules) with a particular energy, and the x-axis represents the molecular kinetic energy. The area under the curve is proportional to the total number of molecules in the sample. Since the number of molecules is conserved when temperature changes (for a closed system), the area under the curve must remain constant.

Understanding the Question

The question asks to identify the correct Maxwell-Boltzmann distribution curve for a gas at 500 K (solid line), given the curve at 300 K (dotted line). We need to apply the effects of increasing temperature on the energy distribution of gas molecules and select the graph that correctly reflects these changes.

Approach

Recall the key features of a Maxwell-Boltzmann distribution curve and how they change with temperature:

1. Start at the origin: At zero kinetic energy, there are no molecules, so the curve begins at (0,0).
2. Peak position: The peak represents the most probable energy. As temperature increases, the average kinetic energy increases, so the peak shifts to the right (towards higher energy values).
3. Peak height: Since the total area under the curve is constant (total number of molecules is constant), a shift of the peak to the right must be accompanied by a decrease in peak height to keep the area the same. The curve becomes flatter and broader.
4. Intersection: The higher-temperature curve and lower-temperature curve must cross exactly once. At low energies, the lower-temperature curve has more molecules; at high energies, the higher-temperature curve has more molecules.

Step-by-Step Reasoning

• The gas is heated from 300 K (dotted line) to 500 K (solid line).
• Because the temperature is higher, the molecules have, on average, more kinetic energy. This means the peak of the distribution (the most probable energy) must be at a higher energy value. Therefore, the solid line's peak must be to the right of the dotted line's peak. This eliminates options B and D, where the solid line peak is to the left (which would happen if the gas cooled down).
• The total number of molecules in the sample does not change, so the area under both curves must be equal. If the solid line's peak is shifted to the right, the curve must spread out and become lower (flatter) to maintain the same total area. Option A shows the solid line peak being lower and to the right, with the curves crossing once, which is the correct shape.
• Option C shows the solid line peak lower and to the right, but the curves do not cross correctly (the solid line does not tail off to meet the x-axis properly, or the area is not conserved). In a correct Maxwell-Boltzmann distribution, the higher temperature curve must cross the lower temperature curve exactly once and both must approach the x-axis asymptotically. Option A is the standard, correctly drawn representation.

Key Takeaways

• Increasing temperature shifts the Maxwell-Boltzmann peak to the right (higher energy) and lowers it (flatter curve) to conserve the area under the curve.
• The curve always starts at the origin.
• A higher-temperature curve and a lower-temperature curve for the same gas will intersect exactly once.

Common Mistakes

• Thinking the peak gets higher when temperature increases: this would increase the area under the curve, implying more molecules, which is incorrect.
• Shifting the peak to the left: this represents a decrease in temperature (lower average kinetic energy).
• Forgetting that the curve must start at the origin or tail off to the x-axis.

Things to Be Careful About

• Ensure you can distinguish between the effect of temperature (which shifts the whole curve, changing the peak height and position but keeping the area constant) and the effect of a catalyst (which does not change the distribution curve at all, only the activation energy line on the graph).
• The y-axis is 'number of molecules' or 'fraction of molecules', not 'rate of reaction'. The shape of the curve remains the same; only the position and height of the peak change.