Thermodynamics problem

Question

We solved this task in the class, but I don't understand it. I would be very happy if someone could explain it in general terms. I can write the basic equations for the adiabatic change of state, but then I am unable the progress with the task.

So the question is: Where are the pistons in their equilibrium position?

Condition: The movement of the pistons stops after quite a few oscillations (due to the internal friction of the gases). H-monatomic, O-diatomic(we take these as ideal gases)

And this is the task: We have a heat-insulating tank of height l. At half the height of the tank is a heat-conducting, thin piston and the tank is sealed from above by a heat-insulating, light piston. It initially contains gas at pressure p. The upper half of the tank contains helium, and the lower half contains oxygen. We place a heavy weight on the upper piston and then we release it.

Thanks in advance!

Answer 1

Hi Zac,

I can see that you understand the basic equations for the adiabatic process, but you might encounter difficulties breaking down the problem and/or applying the equations to it. I assume you have your lecture notes, so I will focus on how to approach this problem and the intuition behind the process, instead of overwhelming you with math (and typing math here is extremely messy, haha). It could seem too easy for you, but I'm trying to make sure I hit every small detail and the logic behind it.

Step 1: Break down the problem!

Tank Properties --> important to find the volume of each gas --> apply to Ideal gas law (if needed) --> can also be important in mathematically formulating with the position of the piston (see the end)

Height of the tank: l

At half the height (l/2), there is a heat-conducting, thin piston

allows heat transfer, but not mass transfer

The tank is sealed from above by a heat-insulating, light piston

no heat transfer --> isolate the system into an adiabatic system

Gases --> assumed ideal gas --> ideal gas law!

The upper half contains helium (He), a monatomic ideal gas

The lower half contains oxygen (O2), a diatomic ideal gas

Initial Conditions --> used for adiabatic process, remember that for those, we have PV^γ=constant, which means P₁V₁^γ=P₂V₂^γ

The entire tank initially contains gas at pressure p

γ_He = 5/3 ; γ_O2 = 7/3

Problem:

A heavy weight is placed on the upper piston and then released --> causing the pistons to move until they reach equilibrium --> find the equilibrium position

Step 2: "Visualize" the process and see what changed!

From the problem, we know that the light upper piston will move down due to the added weight until the pressures above and below it equalize (which means the pressure above and below are different initially, but are equalized afterwards), you can assign different variables to them, for example:

Let p1 be the pressure of helium in the upper section after equilibrium

Let p2 be the pressure of oxygen in the lower section after equilibrium

We also know that the middle heat-conducting piston allows for thermal equilibrium but no mass transfer, which implies that the temperature of the gases on both sides of this piston will be the same at equilibrium

Quick recap: for the two gasses, the pressure is different until equalizes, but the temperature is the same!

Now, let's try to "visualize" the process:

1. Initially, the gases are at the same pressure p
2. Adding a weight increases the pressure on the upper piston
3. The system undergoes adiabatic processes until thermal equilibrium is reached at a common temperature T
4. The equilibrium condition is determined by the balance of pressures and the adiabatic condition for each gas. Remember, the equilibrium condition is where...
5. The pressures in both gas sections are equal
6. The temperature is uniform due to the heat-conducting piston

Step 3: Math

If you follow through and have no trouble, and if you understand the equation you learned in class, you should be able to understand your teacher's solution. Other than those equations, one thing that could be pointed out is

Volume of helium at equilibrium: V_He = A (l - y)

Volume of V_O2 = A y

where y is the position of the middle piston from the bottom (remember y_0 = 1/2 l), and A is the cross section of the tank (no worries, I think you can cancel it out later)

I will simply explain what we could do with this information. To solve for y, we can use the adiabatic conditions and the fact that the final pressures are equal for helium and oxygen. So, express the final pressure of the gasses with the initial pressure, then equate them. After equating, you should have an equation that you can solve for y, which is what the question seeks.

I hope seeing this is helpful for you! Hope you have a great day :)

Let me know if I can help break down your teacher's solution for you, too!

Cheers,

Devany