# Average velocity of gas molecules in a container

• steven george
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.

## 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.

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.

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.

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.

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.

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.