Knudson Effusion between two gasses

[Robsta]

Homework Statement

Consider two chambers of equal volume separated by an insulating wall and containing an ideal gas, maintained at temperatures T₁ = 225K and T₂ = 400K. Initially the two chambers are connected by a long tube whose diameter is much larger than the mean free path in either chamber and equilibrium is established (while maintaining T₁ and T₂). Then the tube is removed, the chambers are sealed but a small hole is opened in the insulating wall, with diameter that is much less than the mean free path.

In what direction will the gas flow through the small hole? Why?

Homework Equations

I know that effusion rate (number of particles moving through a small hole of unit area per unit time) is p/sqrt(2πmkT)

The Attempt at a Solution

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So I figure that when the wide pipe is attached, the pressure in the two tanks is equal. But then when they're sealed and a small hole is punched, then the respective particle fluxes from each tank will be

Φ_1->2=p/sqrt(2πmkT₁) and Φ_2->1=p/sqrt(2πmkT₂)

So if the pressures are equal then the flux from the colder tank to the warmer tank will be higher? This doesn't make much sense to me. Can anybody explain to me why my calculations are wrong?

 

[lightgrav]

Your intuition is ignoring the equalized pressure:
which gas is denser?
which gas hits its wall with softer impact?
which gas hits the (or the opening in the wall) more often? (I'm not asking which hits faster ...)

 

[Robsta]

Thanks for your response. Just to clarify, am I right or wrong that the pipe equalises the pressure?

Once the pipe is disconnected, the higher temperature gas has a lower density. It has more energetic particles though so this balances it out and keeps the pressure equal. The higher temperature tank hits the sides harder per hit. I think the cooler gas hits the sides more often.

 

[lightgrav]

Yes. The first pipe is big Area, so will equalize pressure in very short time.

The hot ones hit harder, so they need to hit less often. => cold ones "hit" the small hole more often.
isn't that what your calculation showed?

 

[Robsta]

Yes, it is. So the colder ones hit more often, so the gas effuses from the colder container into the hotter container since the flux from cold to hot is more than the flux from hot to cold.