Need a formula for this scenario....

[Intle]

So I've been wondering if anyone knows a formula to relate these quantities in this scenario.
Say I have a container that originally has some volume, some amount of atoms at a constant velocity and is at a certain pressure and is airtight and then connect it to another container that has twice the volume and is at a very low pressure. Now some of these atoms will move from the first container to the second container due to the difference in pressure. Is there a formula I could use to calculate how many of these atoms move over to the second container to reach equilibrium?
What about a change in velocity of the atoms?
Is there a formula I can use?

 

[jedishrfu]

If you consider the two volumes reach equilibrium after some time then you could use volume ratios to determine the atoms in each one and knowing the atoms in the second container before they were connected you could determine how many moved, right?

 

[CWatters]

Looks like two thirds of the atoms will end up in the new container.

Thinks like final pressure and temperature depend if the containers are insulated or not. Google adiabatic, isothermal, and possibly isobaric.

 

[Intle]

Looks like two thirds of the atoms will end up in the new container.

Thinks like final pressure and temperature depend if the containers are insulated or not. Google adiabatic, isothermal, and possibly isobaric.

Ok, so could you show me how you got this value?
It would be an adiabatic scenario, and yes it is supposed to be insulated. It is an ideal scenario. I'm a bit more interested on how to find a change in velocity of these atoms. Ofcourse I am assuming that there will be a change in velocity, and I could be wrong in this assumption, but I thought that the change in pressure would cause the gas to move into the area of very low pressure. I would like to add that this area is supposed to be very close to a vacuum.

 

[CWatters]

γ

Ok, so could you show me how you got this value?

Say I have a container that originally has some volume, some amount of atoms at a constant velocity and is at a certain pressure and is airtight and then connect it to another container that has twice the volume and is at a very low pressure.

The volume increases from 1 unit to 3 units. That means the original container represents 1/3rd of the final volume and the new container 2/3rds. If the temperature and pressure ends up the same in both then the atoms will end up distributed uniformly. So there will be 1/3rd in the original container and 2/3rds in the new container.

It would be an adiabatic scenario, and yes it is supposed to be insulated. It is an ideal scenario.

If the container is insulated the gas will cool as it expands. I believe you need to know the adiabatic index γ of the gas to work out the final temperature and pressure. No matter.. If both containers end up at the same temperature and pressure then I would still expect the distribution of atoms to be uniform.

I'm a bit more interested on how to find a change in velocity of these atoms.

I believe you can calculate that...

The temperature will change as the gas expands and Temperature is related to the average (rms) velocity of the atoms...

If you had gas at temperature T₁ in an insulated syringe of volume V₁ and simply allowed the gas to expand to volume V₂ by pulling or releasing the syringe the gas would cool down to T₂..

Formulae for adiabatic process can be found here..
https://en.wikipedia.org/wiki/Adiabatic_process

I'm a bit rusty but I think..

T₁V₁^γ-1 = T₂V₂^γ-1

So if you know T₁, V₁, V₂ and γ you can work out the final temperature T₂

The before and after rms velocity of the atoms can be calculated from T₁ and T₂ as per...

https://en.wikipedia.org/wiki/Root-mean-square_speed

 

[CWatters]

Just to continue..

From https://en.wikipedia.org/wiki/Root-mean-square_speed

Vrms₁ = √ (3RT₁/M_m)

So the ratio of the rms velocities would be..

Vrms₂/Vrms₁ = √T₂/√T₁ = √(T₂/T₁) ........(1)

From earlier

T₁V₁^γ-1 = T₂V₂^γ-1

so T₂/T₁ = V₁^γ-1/V₂^γ-1 ...(2)

substitute into (1) gives

Vrms₂/Vrms₁ = √(V₁^γ-1/V₂^γ-1)

So quite a simple relationship between the rms velocity and change in volume.

As I said I might be rusty!

Edited to correct a few typos.

 

[Intle]

Just to continue..

From https://en.wikipedia.org/wiki/Root-mean-square_speed

Vrms₁ = √ (3RT₁/M_m)

So the ratio of the rms velocities would be..

Vrms₂/Vrms₁ = √T₂/√T₁ = √(T₂/T₁) ........(1)

From earlier

T₁V₁^γ-1 = T₂V₂^γ-1

so T₂/T₁ = V₁^γ-1/V₂^γ-1 ...(2)

substitute into (1) gives

Vrms₂/Vrms₁ = √(V₁^γ-1/V₂^γ-1)

So quite a simple relationship between the rms velocity and change in volume.

As I said I might be rusty!

Edited to correct a few typos.

Thanks, precisely what I was looking for.