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Stat/Thermo Problem - Equilibrium Pressure in Permeable Sphere

1. Homework Statement

A sphere full of air at room temp and 1 atm is placed in a chamber filled with He gas at room temp and 1 atm. The sphere is permeable to He only. What will the equilibrium pressure in the sphere be?



2. Homework Equations

Variables:
Po=initial pressure
V=vol of sphere
V'=vol of chamber
Na=number of air particles
Nh=number of He particles

Ideal gas law: P=NRT/V

---------
I think that's all i need, but other eqs that might be relevant:

S=ln(Q) : entropy=ln(number of states)
dS/dE=1/T : derivative of entropy wrt Energy = 1/temp

3. The Attempt at a Solution

Here's what I think:

since He is a noble gas, we can assume it does not interact with the air particles. Therefore, at equilibrium, a single He molecule is equally likely to be found anywhere in the chamber (V'), which means that the mean number of He particles in the sphere (V) will be V/V'*Nh.

Then use the ideal gas law:
P=(V/V'*Nh+Na)RT/V

can also solve for Nh and Na initially in terms of Po, R, T and V or V':

Nh=PoV'/(RT), Na=PoV/(RT)

and then plug in, rearrange and reduce:

P=(Po/(RT))(V/V'*V'/V+V/V)*RT=Po(1+1)=2Po=2 atm

So, is there anything wrong with what I did? I don't know what the answer should be, but I'm not confident in my answer because my prof mentioned that we should start by maximizing entropy.... which I could do, but this way seemed so much simpler. Suggestions? Any errors in my assumptions or approach? I don't find this material intuitive, so I could very well have made mistakes.

Thanks!
 

Mapes

Science Advisor
Homework Helper
Gold Member
2,592
17
Your approach is somewhat limited because it can't easily accommodate complications such as a chemical reaction (for example). As long as you can also apply the entropy maximization approach, though, I don't see a problem. In fact, I think it's great that you're attacking problems from different angles.
 

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