Pressure/Temperature mechanical equilibrium

[AKJ1]

1. Homework Statement

Three thermally insulated containers of equal volume are connected together by thin tubes of negligible volume so that gas can flow from one container to the next when a valve between them is open. Initially, the pressure and temperature in each container is at the same value Po and To, respectively.

The valves between the vessels are opened, and thermal insulation is removed from two of the vessels. Then two vessels are surrounded by heat reservoirs so that their temperatures are maintained at 2To and 2.3To while the third remains at To. What is the final pressure of the system when mechanical equilibrium is reached?

Homework Equations

P₀/T₀ = P₁/T₁ = P₂/T₂ ...[/B]

The Attempt at a Solution

Very stuck. Sorry for bad english

This problem was labeled in my text as a quick easy review, however I cannot seem to make any progress as to solve it. The answer given is 1.55Po.

My initial attempt was to work with PV=nRT, and try to relate the individual "new" pressures to their old pressures, then proceed to sum them up.

Meaning, I was told they each originally had P₀/T₀ , so for vessel one, I set this equal to
P₁/2T₀ . I did this for the remaining cases and solved the new pressures in terms of Po, however I soon got stuck and it didn't lead me anywhere.

 

[haruspex]

Meaning, I was told they each originally had P0/T0 , so for vessel one, I set this equal to
P1/2T0 . I did this for the remaining cases and solved the new pressures in terms of Po

You're overlooking that gas flows from one to another, so n doesn't stay constant for each.
But n for the whole system is constant. Assign separate variables for the final molecular quantities in the three containers.

 

[AKJ1]

If I understand correctly, I need to also examine the change in n.

P₀/(T₀n₀) =P₁/(2T₀n_f1) =P₂/(2.3T₀n_f2) =P₃/(T₀n_f3)

Is this the necessary relationship? If so, I suppose I raise another question, is the sum of the individual pressures equal to the total pressure of the system once mechanical equilibrium is achieved?

I apologize for the many questions, I am trying to understand some of this material before the topics are introduced.

 

[haruspex]

I need to also examine the change in n.

Yes.

is the sum of the individual pressures equal to the total pressure of the system once mechanical equilibrium is achieved?

These pressures don't add together, they're not partial pressures of a gas mixture. What will happen if the pressures are different? Is that equlibrium?

 

[Chestermiller]

If V is the volume of each tank, what is the number of moles in each of the tanks to start with?

What is the total number of moles in the tanks to start with?

If P is the pressure in the three tanks in the final equilibrium state, what is the number of moles in each of the tanks at the end (in terms of P)?

What do the final number of in the three tanks have to add up to?

Chet

 

[AKJ1]

If V is the volume of each tank, what is the number of moles in each of the tanks to start with?

What is the total number of moles in the tanks to start with?

If P is the pressure in the three tanks in the final equilibrium state, what is the number of moles in each of the tanks at the end (in terms of P)?

What do the final number of in the three tanks have to add up to?

Chet

Each tank starts with n₀, therefore, the total is 3n₀. Also since the pressure in one tank is P₀, this must also mean the pressure of the system (all tanks) is P₀ ; I should be able to say the same for the temperature.

So, before anything happens, I have P₀/(3n₀T₀) , assuming n₀ represents the number of moles in each tank. Is what I have stated true?

If so, the other side of the expression will be straight forward for me.

 

[Chestermiller]

Each tank starts with n₀, therefore, the total is 3n₀. Also since the pressure in one tank is P₀, this must also mean the pressure of the system (all tanks) is P₀ ; I should be able to say the same for the temperature.

So, before anything happens, I have P₀/(3n₀T₀) , assuming n₀ represents the number of moles in each tank. Is what I have stated true?

No. Not even close. The number of moles of gas initially in each tank is, using your notation, $n_0=\frac{P_0V}{RT_0}$. The total number of moles in the 3 tanks is
$$3n_0=3\frac{P_0V}{RT_0}$$

Chet