Velocity Distributions: Videos & Practice Problems

The Maxwell-Boltzmann distribution curve demonstrates how gas molecules' velocities are influenced by temperature and molecular weight. As temperature rises, the probable speed increases, with molecules moving at higher velocities. For instance, at 330°C, a significant number of gas molecules travel at 800 m/s or more, compared to fewer at 30°C. Conversely, an increase in molecular weight results in a decrease in probable speed; heavier gases like Xenon move slower than lighter ones like Helium. The distribution curve broadens and lowers with higher temperatures, indicating a wider range of velocities, while it narrows and peaks higher for gases with greater molecular weights, reflecting a more limited range of velocities. Understanding these trends is crucial for comprehending the behavior of gaseous molecules under varying conditions.

Velocity Distributions

The Maxwell-Boltzmann distribution describes the distribution of speeds among gas molecules and is influenced by two primary factors: temperature and molecular weight. Understanding these factors is crucial for grasping how gas molecules behave under different conditions.

Firstly, temperature plays a significant role in determining the probable speed of gas molecules. For instance, at 30 degrees Celsius, the probable speed of gas molecules is approximately 400 meters per second, while at 330 degrees Celsius, this speed increases to around 600 meters per second. This trend indicates that as temperature rises, the average velocity of gas molecules also increases. The distribution curve becomes broader and lower at higher temperatures, signifying that a larger number of molecules can achieve higher speeds. Consequently, more molecules are found moving at speeds of 800 meters per second or greater when the temperature is elevated.

Secondly, molecular weight affects the speed of gas molecules inversely. For example, helium, with a molecular weight of about 4 grams per mole, exhibits a probable speed of around 700 meters per second, whereas xenon, with a much higher molecular weight of approximately 131 grams per mole, has a probable speed of only about 100 meters per second. This trend illustrates that as molecular weight increases, the average speed of the gas molecules decreases. Heavier gases, like xenon, find it more challenging to achieve high velocities compared to lighter gases like helium. Additionally, the distribution curve for lighter gases is broader and lower, indicating that more molecules can reach similar velocities compared to heavier gases.

In summary, the Maxwell-Boltzmann distribution reveals that increasing temperature leads to higher molecular speeds and a broader distribution of speeds, while increasing molecular weight results in lower speeds and a narrower distribution. These insights are essential for understanding the kinetic behavior of gases in various conditions.

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 distribution curve?

Temperature significantly impacts the Maxwell-Boltzmann distribution curve. As temperature increases, the probable speed of gas molecules also increases, meaning more molecules move at higher velocities. For example, at 330°C, a larger number of gas molecules travel at 800 m/s or more compared to 30°C. Additionally, the distribution curve becomes broader and lower with higher temperatures, indicating a wider range of velocities among the gas molecules. This broadening occurs because higher temperatures provide more energy to the molecules, allowing them to move faster and occupy a wider range of speeds.

What is the relationship between molecular weight and the Maxwell-Boltzmann distribution curve?

The molecular weight of a gas inversely affects the Maxwell-Boltzmann distribution curve. As the molecular weight increases, the probable speed of the gas molecules decreases. Heavier gases, such as Xenon, move slower compared to lighter gases like Helium. For instance, Helium, with a molecular weight of around 4 g/mol, has a probable speed of approximately 700 m/s, while Xenon, with a molecular weight of around 131 g/mol, has a probable speed of only about 100 m/s. Additionally, the distribution curve for heavier gases is narrower and peaks higher, indicating a more limited range of velocities, whereas lighter gases have broader and lower curves, reflecting a wider range of velocities.

Why do lighter gases have broader Maxwell-Boltzmann distribution curves?

Lighter gases have broader Maxwell-Boltzmann distribution curves because they can achieve a wider range of velocities. Since lighter molecules, such as Helium, have less mass, they require less energy to reach higher speeds. This results in a broader distribution of velocities, as more molecules can move at various speeds. In contrast, heavier gases, like Xenon, have more mass and require more energy to achieve the same speeds, leading to a narrower range of velocities and a more peaked distribution curve. Therefore, the lower molecular weight of lighter gases allows for a more extensive spread of velocities in the distribution curve.

How does the Maxwell-Boltzmann distribution curve change with increasing temperature?

With increasing temperature, the Maxwell-Boltzmann distribution curve becomes broader and lower. This change indicates that a wider range of velocities is present among the gas molecules. As temperature rises, the probable speed of the molecules increases, meaning more molecules move at higher velocities. For example, at 330°C, a significant number of gas molecules travel at 800 m/s or more, compared to fewer at 30°C. The broadening of the curve occurs because higher temperatures provide more energy to the molecules, allowing them to move faster and occupy a wider range of speeds. Consequently, the distribution curve flattens and spreads out as temperature increases.

What trends can be observed in the Maxwell-Boltzmann distribution curve when comparing gases of different molecular weights?

When comparing gases of different molecular weights, several trends can be observed in the Maxwell-Boltzmann distribution curve. Lighter gases, such as Helium, have higher probable speeds and broader distribution curves, indicating a wider range of velocities. For instance, Helium, with a molecular weight of around 4 g/mol, has a probable speed of approximately 700 m/s. In contrast, heavier gases, like Xenon, have lower probable speeds and narrower, more peaked distribution curves, reflecting a more limited range of velocities. Xenon, with a molecular weight of around 131 g/mol, has a probable speed of only about 100 m/s. These trends show that as molecular weight increases, the probable speed decreases, and the distribution curve becomes narrower and higher.