Average velocity of gas molecules in a container

In summary, the pressure exerted by an ideal gas on the top of a cube can be determined by the average force exerted by the gas molecules hitting the top of the cube in a given time interval, and the time interval is related to the average vertical component of the velocity of the gas molecules.
  • #1
steven george
11
1

Homework Statement


An ideal gas with molecules of mass m is contained in a cube with sides of area A. The pressure exerted by the gas on the top of the cube is P, and N molecules hit the top of the cube in a time
Δt. What is the average vertical component of the velocity of the gas molecules.

Homework Equations


F = ma P = f/A

The Attempt at a Solution



F = ma
F = mΔV /Δt
F = 2mV / Δt (ΔV = 2V since collisions are elastic)

Putting this into P = F/A leads to the correct answer but my problem is with the Δt. When used in the acceleration Δt should be the time that the molecules is in contact with the wall of the container and it seems to me to be unrelated to the Δt as specified in the question.

Thanks to anybody that can help clear this up.
 
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  • #2
steven george said:

Homework Statement


An ideal gas with molecules of mass m is contained in a cube with sides of area A. The pressure exerted by the gas on the top of the cube is P, and N molecules hit the top of the cube in a time
Δt. What is the average vertical component of the velocity of the gas molecules.

Homework Equations


F = ma P = f/A

The Attempt at a Solution



F = ma
F = mΔV /Δt
F = 2mV / Δt (ΔV = 2V since collisions are elastic)

Putting this into P = F/A leads to the correct answer but my problem is with the Δt. When used in the acceleration Δt should be the time that the molecules is in contact with the wall of the container and it seems to me to be unrelated to the Δt as specified in the question.

Thanks to anybody that can help clear this up.
The wall exerts bigger force on the molecule while in contact with it, but the pressure it exerts on the gas is also average quantity: the time average of the force exerted by unit area. It exerts some force f to one molecule during the collision , and zero between the collisions.
 
  • #3
ehild said:
The wall exerts bigger force on the molecule while in contact with it, but the pressure it exerts on the gas is also average quantity: the time average of the force exerted by unit area. It exerts some force f to one molecule during the collision , and zero between the collisions.
Thanks, Maybe I'm missing something that will seem obvious once it makes sense, but when considering the acceleration shouldn't we just consider the time that the molecule is actually in contact with the wall? It really doesn't seem right to me to use the same Δt.
 
  • #4
Instead of thinking in terms of acceleration, think of the force in terms of momentum delivered per unit of time. It's the momentum per collision divided by the time between collisions.
 

Related to Average velocity of gas molecules in a container

1. How is the average velocity of gas molecules in a container calculated?

The average velocity of gas molecules in a container is calculated by taking the square root of the average of the squared velocities of all the molecules. This is known as the root mean square (RMS) velocity.

2. Is the average velocity of gas molecules affected by the temperature and pressure of the container?

Yes, the average velocity of gas molecules is directly proportional to the temperature of the container and inversely proportional to the pressure. This means that as the temperature increases, the average velocity increases, and as the pressure increases, the average velocity decreases.

3. How does the mass of the gas molecules affect the average velocity?

The mass of the gas molecules does not directly affect the average velocity. However, it does affect the RMS velocity, as heavier molecules will have a lower RMS velocity compared to lighter molecules at the same temperature and pressure.

4. Can the average velocity of gas molecules be greater than the speed of light?

No, according to the theory of relativity, the speed of light is the maximum speed at which any object can travel. Therefore, the average velocity of gas molecules cannot be greater than the speed of light.

5. How does the average velocity of gas molecules relate to the kinetic theory of gases?

The average velocity of gas molecules is directly related to the kinetic theory of gases, which states that gas molecules are in constant motion and their average kinetic energy is directly proportional to the temperature of the gas. Therefore, as the temperature increases, the average velocity of gas molecules also increases.

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