Maximum work obtained by mixing 2 gases

[superduper]

Homework Statement

2 boxes containing the same number of moles of 2 ideal identical gases with the same adiabatic index (this is given as gamma), at the same initial temperature T_i but with different volumes, V₁ and V₂ are brought together. Find the maximum mechanical work that can be obtained.

Homework Equations

The Attempt at a Solution

If the gases have all the parameters identical but the volumes, that means that they also have different pressures, so when we are mixing them, the gas with the higher pressure will do work on the gas with lower pressure. However, I have no idea how to calculate this work and the answer given is a big messy expression. It dosent say anything about the recipients being adibatically isolated, but I guess I have to assume that? The temperature will be constant? I think I should calculate the variation of entropy for the system and then relate this to the first principle to get the work done, but I have no idea how to do that.

 

[Chestermiller]

Hi again. Welcome to Physics Forums.

I don't think the problem intends for you to assume that the gases are intimately mixed. I think it intends for you to assume that there is an insulated partition separating the two gases, and to find the maximum amount of work that they can do on the partition. If I'm right, my solution to this problem should match the given answer. Do you want me to reveal my solution so that you can compare it with the given answer?

If my interpretation is correct, then in both chambers, an adiabatic reversible expansion occurs. So, in each chamber, PV^ν=constant. Let's focus on chamber 1. Let $P_0$ represent the initial pressure nRT/V₁, let P represent the final pressure when the pressures on both sides of the partition have equilibrated, and let $V_{1f}$ represent the final volume. In terms of $P_0$, V₁, and $V_{1f}$, what is the final pressure P in chamber 1?

Chet

 

[Chestermiller]

Maybe you can provide the expression for the given answer, and we can try to determine how it was derived.

Did the expression have a bunch of gamma's and volume ratios in it?

 

[Chestermiller]

Does it look anything like this:

$$W=nRT_i\left[2-\left(\frac{1+(V_2/V_1)^{\frac{\gamma-1}{\gamma}}}{1+(V2/V_1)}\right)^{\gamma-1}-\left(\frac{1+(V_1/V_2)^{\frac{\gamma-1}{\gamma}}}{1+(V1/V_2)}\right)^{\gamma-1}\right]$$

 

[superduper]

Sorry for the late response, but I'm away from uni and don't have the book at my disposal to see if that's the exact answer, but, yes, I remeber it has a bunch of gammas and volume ratios involved

 

[Chestermiller]

Sorry for the late response, but I'm away from uni and don't have the book at my disposal to see if that's the exact answer, but, yes, I remeber it has a bunch of gammas and volume ratios involved

So, where do you want to go from here? How would you like me to help you?