Velocity Distributions: Videos & Practice Problems

Velocity Distributions Example

The distribution of molecular velocities can be analyzed through graphical representations, typically showing curves that indicate the number of molecules at various velocities. In this context, we have three curves: red, blue, and green, corresponding to different temperatures and molar masses of gas molecules.

Firstly, the temperature of a gas affects the breadth of its velocity distribution curve. As temperature increases, the curve becomes broader, indicating a wider range of molecular velocities. In this case, the red curve (curve A) is the narrowest, suggesting it corresponds to the lowest temperature. Conversely, the green curve (curve C) is the broadest, indicating it represents the highest temperature.

Next, the relationship between molar mass and the breadth of the velocity distribution is also significant. Gases with lower molar masses exhibit broader curves, while those with higher molar masses show narrower distributions. Since curve C is the broadest, it implies that it corresponds to gas molecules with the smallest molar mass.

Additionally, it is important to note that a narrower velocity distribution indicates a lower temperature. Therefore, the red curve (curve A), being the most narrow, confirms that it is associated with the lowest temperature.

In summary, analyzing the statements regarding the curves reveals that two of them are true: curve C represents gas molecules with the smallest molar mass, and the more narrow the velocity distribution, the lower the temperature. Thus, the correct choice is that two statements are true.

Here's what students ask on this topic:

How does temperature affect the Maxwell-Boltzmann velocity distribution of gas molecules?

The Maxwell-Boltzmann distribution curve shows how gas molecules' velocities vary with temperature. As temperature increases, the probable speed—the velocity at which most molecules move—also increases. For example, at 30°C, the probable speed might be around 400 m/s, but at 330°C, it rises to about 600 m/s. This means more molecules move at higher velocities at elevated temperatures. Additionally, the distribution curve broadens and lowers with increasing temperature, indicating a wider range of molecular speeds. This broadening reflects that more molecules achieve higher velocities, such as speeds above 800 m/s, which are rare at lower temperatures. Therefore, temperature directly influences both the average speed and the spread of molecular velocities in a gas.

What is the impact of molecular weight on the velocity distribution of gases according to the Maxwell-Boltzmann distribution?

Molecular weight significantly affects the velocity distribution of gases. According to the Maxwell-Boltzmann distribution, lighter gases have higher probable speeds compared to heavier gases. For instance, helium, with a molecular weight of about 4 g/mol, has a probable speed near 700 m/s, while xenon, weighing around 131 g/mol, has a probable speed close to 100 m/s. This occurs because heavier molecules require more energy to move at higher speeds. Consequently, the distribution curve for lighter gases is broader and lower, indicating a wider range of velocities, whereas heavier gases have narrower and taller curves, showing a more limited velocity range. Thus, as molecular weight increases, the average molecular speed decreases.

Why does the Maxwell-Boltzmann distribution curve broaden and lower with increasing temperature?

The Maxwell-Boltzmann distribution curve broadens and lowers with increasing temperature because higher temperatures provide more kinetic energy to gas molecules. This increased energy allows a greater number of molecules to reach higher velocities, spreading the distribution over a wider range of speeds. As a result, the peak of the curve, representing the most probable speed, decreases in height because the molecules are more evenly distributed across a broader velocity spectrum. This broadening indicates that the gas molecules exhibit more varied speeds at higher temperatures, reflecting increased molecular motion and energy diversity within the gas.

How can the Maxwell-Boltzmann distribution explain the difference in speeds between helium and xenon gas molecules?

The Maxwell-Boltzmann distribution explains that helium molecules move faster than xenon molecules due to their lower molecular weight. Helium has a molecular weight of about 4 g/mol, while xenon is much heavier at approximately 131 g/mol. Because kinetic energy is distributed among molecules, lighter molecules like helium achieve higher velocities to maintain the same average kinetic energy as heavier molecules. This results in helium having a probable speed around 700 m/s, whereas xenon’s probable speed is closer to 100 m/s. The distribution curve for helium is broader and lower, showing a wide range of speeds, while xenon's curve is narrower and taller, indicating slower and more uniform speeds.

What is the probable speed in the context of the Maxwell-Boltzmann velocity distribution?

The probable speed in the Maxwell-Boltzmann velocity distribution is the speed at which the largest number of gas molecules are moving. It corresponds to the peak of the distribution curve. This speed is significant because it represents the most common molecular velocity in a gas sample at a given temperature. The probable speed depends on temperature and molecular weight; it increases with temperature and decreases with molecular weight. Mathematically, the probable speed $v\_p$ can be expressed as $v\_p=\sqrt{\frac{2kT}{m}}$, where $k$ is Boltzmann's constant, $T$ is temperature, and $m$ is molecular mass.