Two gases separated by a piston

[GandalfTheGrey]

Homework Statement

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Two ideal gases are contained adiabatically and separated by an insulating, fixed piston that blocks the molecules of gas 2 but allows the molecules of gas 1 through(in both directions). The initial pressures, volumes, temperatures and number of molecules on each side is given. What is the equilibrium state?

Homework Equations

I don't think they're explicitly necessary for the question.

The Attempt at a Solution

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The volumes remain the same, so we basically have 5 variables: The final pressures, temperatures and the net number of molecules that went through the piston. At equilibrium, the partial pressure of gas 1 must be equal on both sides, that gives 1 equation. The ideal gas law, applied to both sides gives 2 more, and the fact that the whole system's internal energy is constant gives 1 more. So I have 5 variables and only 4 equations. What am I missing?

 

[Chestermiller]

The portion of gas 1 that has remained in its chamber at final equilibrium has experienced and adiabatic reversible expansion in driving the other portion of this gas through the piston into the other chamber.

 

[Chestermiller]

Now that I've given you the above hint, have you figured out how to solve this problem?

 

[GandalfTheGrey]

Now that I've given you the above hint, have you figured out how to solve this problem?

I think this means that $$P_{1f}V_1^{\gamma} = P_{1i}(\frac{n_{1rf}}{n_1}V_1)^\gamma$$ Where ##P_1## is the pressure of gas 1 on the right, ##V_1## is the volume of the right chamber, ##n_1## is the total number of moles of gas 1, ##n_{1rf}## is the final number of moles of gas 1 remaining in the right chamber at equilibrium. (I'm assuming that gas 1 starts on the right side and gas 2 on the left side)

 

[Chestermiller]

I think this means that $$P_{1f}V_1^{\gamma} = P_{1i}(\frac{n_{1rf}}{n_1}V_1)^\gamma$$ Where ##P_1## is the pressure of gas 1 on the right, ##V_1## is the volume of the right chamber, ##n_1## is the total number of moles of gas 1, ##n_{1rf}## is the final number of moles of gas 1 remaining in the right chamber at equilibrium. (I'm assuming that gas 1 starts on the right side and gas 2 on the left side)

Very nice.

 

[Chestermiller]

Have you been able to complete the solution now? If not, I can help further.

 

[Chestermiller]

Have you been able to complete the solution now? If not, I can help further.

I've completed the solution to this problem, but would like to compare notes. Any chance? What about someone else besides the OP?