Question

A mixture of a light gas and heavy gas is in a closed container, the light gas will (1) have a lower average speed per molecule than the heavy gas (2) have a higher average speed per molecule than the heavy gas (3) rise to the top of the container (4) all are wrong

Short answer

The light gas will have a higher average speed per molecule than the heavy gas.

Step-by-step solution

Understand the Problem

A mixture of a light gas and heavy gas is in a closed container. The task is to determine the behavior of the light gas in terms of its average speed compared to the heavy gas.

Recall Kinetic Theory of Gases

According to the kinetic theory of gases, the average kinetic energy of gas molecules is proportional to the temperature of the gas. Mathematically, it can be expressed as: \[ E_{k} = \frac{1}{2}mv^2 \] where \( m \) is the mass of a gas molecule and \( v \) is its speed.

Relate Average Kinetic Energy to Temperature

At a given temperature, the average kinetic energy of gas molecules remains the same for different gases. Therefore, \[ \frac{1}{2} m_{light} v_{light}^2 = \frac{1}{2} m_{heavy} v_{heavy}^2 \] where \( m_{light} \) and \( m_{heavy} \) are the masses of light and heavy gas molecules respectively, and \( v_{light} \) and \( v_{heavy} \) are their respective speeds.

Solve for Speed

Rearranging the equation for speed, we get: \[ v_{light} = \frac{v_{heavy}}{\text{sqrt}(m_{heavy}/m_{light})} \] Since \( m_{heavy} > m_{light} \), \( \text{sqrt}(m_{heavy}/m_{light}) > 1 \), which implies that \( v_{light} > v_{heavy} \). Therefore, the light gas molecules will have a higher average speed than the heavy gas molecules.

Evaluate Other Options

Option (3) suggests that the light gas will rise to the top of the container. This behavior is unrelated to the context of average molecular speed in a closed container. Hence, option (3) is incorrect given the current information.

Conclusion

Comparing all given options and based on the analysis, the correct answer is that the light gas will have a higher average speed per molecule than the heavy gas.

Key concepts

Average Kinetic Energy

The average kinetic energy of gas molecules plays a significant role in understanding gas behavior. According to the kinetic theory of gases, the temperature of a gas is directly proportional to its average kinetic energy. This relationship can be described by the equation:
Rewriting this using molecular masses and speeds:
Here, we see that at the same temperature, different gases have the same average kinetic energy, but their speeds will differ based on their molecular masses. Light gas molecules, having less mass, need to move faster to maintain the same kinetic energy as heavier molecules.

Molecular Speed

Molecular speed is another crucial concept when analyzing gases. The speed of a molecule is influenced by its mass and the temperature of the gas. For our given problem, we compare light and heavy gas molecules in a closed container to determine their average speeds.
Given that lighter gas molecules have lower masses, they exhibit higher speeds at the same temperature to balance their kinetic energy. Using the relationship:
we understand that the speed of the light gas ( is greater than that of the heavy gas (...).

Gas Behavior

Understanding the comprehensive behavior of gases helps in applying the kinetic theory effectively. In a closed container, both light and heavy gases will show different behaviors due to their molecular speeds. The light gas molecules, due to their higher speeds, will tend to move more rapidly compared to the heavy gas molecules.
Note that gas behavior also includes other aspects, such as diffusion rates and how gases mix. However, in our current context, we focus on average speed and kinetic energy variations due to mass differences. This helps in predicting the movement and interactions within a mixture of gases accurately.