Calculating Molecules in a punctured tire

[J.Welder12]

If a tire is punctured (or if any container full of air is holed) the air starts to leak out. Consider a small area A of the wall of the container. Show that the number of molecules striking this surface in a time interval Δt is

pAΔt/2m<Vx>

p is the pressure
m is the average mass of the air molecule
<Vx> is the average x-velocity of the molecules
Assume collisions with the wall are elastic

By having elastic collisions, the KE is conserved and none of the KE is transferred in any other form
In class, we showed that p= -ΔP/Δt
where P is the average molecular momentum

pressure (p)=F/A
therefore -ΔP/Δt=F/A
F=ma => -ΔP/Δt=(ma)/A

that is where I have gotten so far. Not sure if I am doing this right or where to go from here
Please help!

 

[Delphi51]

I don't follow your formula p= -ΔP/Δt. Which p is momentum, which is pressure? The units don't seem to work out either way for me.

It seems to me you need to start with Force = ma = m*Δv/Δt.
From force, you can get pressure easily. m is related to the number of molecules and the molar mass. Δv would be double the average speed of the molecules. Solve for number of molecules and see if it looks anything like the desired formula!

 

[J.Welder12]

does m=M(molar mass)*N(number of particles)
and F=pA?

 

[Delphi51]

Looks good! I think you are almost there.

 

[asteriatic]

Your thought process so far is on the right track. To calculate the number of molecules striking a small area A in a time interval Δt, we need to consider the number of molecules that pass through that area in that time. This can be calculated by considering the average velocity of the molecules and the number of collisions they make with the wall in that time interval.

We know from the kinetic theory of gases that the average kinetic energy of a gas is directly proportional to its temperature. This means that the average kinetic energy of the molecules in the tire is constant, regardless of the size of the tire or the number of molecules in it. This also means that the average velocity of the molecules is constant.

Now, let's consider the molecules that are in the tire and are moving towards the punctured area. The molecules that are moving towards the punctured area will have a positive x-velocity, while the molecules moving away from the punctured area will have a negative x-velocity. Therefore, the average x-velocity of the molecules, <Vx>, will be the average of these velocities.

Using the ideal gas law, we can also relate the pressure (p) to the number of molecules (N), the temperature (T), and the volume (V) of the tire: pV=NkT, where k is the Boltzmann constant. This means that, for a given temperature and volume, the pressure is directly proportional to the number of molecules.

Now, let's consider the number of collisions that occur in the time interval Δt. We know that the molecules are moving in random directions, so only a fraction of them will collide with the punctured area in a given time interval. This fraction is equal to the ratio of the area of the punctured area (A) to the total surface area of the tire (4πr^2, where r is the radius of the tire). Therefore, the number of collisions in a time interval Δt is given by:

Number of collisions = (A/4πr^2) * (N/2)

Since we are only considering the molecules that are moving towards the punctured area, we can divide the number of collisions by 2. This is because only half of the molecules will be moving towards the punctured area at any given time.

Putting all of this together, we can calculate the number of molecules striking the punctured area in a time interval Δt as:

Number of molecules = (