27. The Maxwell-Boltzmann distributions of molecular speeds in samples of two different gases at the — States of Matter Chemistry Question

STEPS:

1. Identify the constant variable:
The problem specifies that both gas samples are kept at the same temperature.
2. Recall the relationship between temperature and kinetic energy:
According to the kinetic molecular theory, the average kinetic energy of gas molecules is directly proportional to their absolute temperature. Because the two gases are at the same temperature, they must have the same average kinetic energy.
3. Relate molecular speed to mass using the kinetic energy equation:
The standard formula printed on the AP equation sheet relates the kinetic energy of a molecule to its mass ($m$) and velocity ($v$):
$$2\text{KE per molecule} = mv^2 \quad$$
This can be rearranged to express velocity:
$$v = \sqrt{\frac{2\text{KE}}{m}}$$
Because both gases have the same average kinetic energy, the molecular speed is inversely proportional to the square root of the mass of the gas particles. Consequently, at any given temperature, heavier gas molecules (greater molar mass) will have a slower average speed, whereas lighter gas molecules (smaller molar mass) will have a faster average speed.
4. Analyze the Maxwell^-Boltzmann distribution curves:
* The x-axis of the graph represents molecular speed, and the y-axis represents the fraction of molecules.
* The peak of each curve represents the most probable speed of the molecules in that sample.
* Looking at the graph, the peak of Gas A is located significantly further to the left (closer to the origin), showing that its molecules have a lower average speed.
* The peak of Gas B is shifted further to the right (with a broader distribution), showing that its molecules have a higher average speed.
5. Conclude:
Since the molecules of Gas A are moving significantly slower on average than those of Gas B at the same temperature, Gas A must have heavier particles. Therefore, Gas A has the greater molar mass, making Option A the correct answer.

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WHY_OTHERS_WRONG:

• Option B is incorrect: Gas B's velocity distribution peak is shifted further to the right. Because its molecules have a higher average speed at the same temperature, Gas B must be composed of lighter molecules with a smaller molar mass.
• Option C is incorrect: If both gases had the same molar mass, their molecules would have the same average speed at the same temperature, resulting in identical Maxwell^-Boltzmann distribution curves.
• Option D is incorrect: The distribution of molecular speeds in a gas depends solely on absolute temperature and molar mass. Knowing the pressure of the samples is completely unnecessary because pressure does not influence the kinetic energy or speed distribution of the gas particles.