Thermal equilibrium - Entropy driven

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Hey there. We are struggling with a problem from an old exam about statistical mechanics. I hope you can help us or give any clues. Here is the problem.

Homework Statement



A chamber is divided by a wall into two sections of equal volume. One section of the chamber is initially filled by an ideal gas at temperature T0, whereas the other section is empty. Then, a small hole is opened in the dividing wall such that the gas can flow through, until equilibrium is reached. Consider the chamber isolated towards the surrounding and no heat is exchanged with the walls.

a) What is the final Temperature of the system?


b)How does the result change if instead initially the chamber is filled with a Van der Waals gas?


Homework Equations



Any thermodynamical equation.

The Attempt at a Solution



We tried to solve the problem by using the entropy. Since thermal equilibrium is reached, wehn the entropy is maximized.
 

Answers and Replies

  • #2
fluidistic
Gold Member
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Hey there. We are struggling with a problem from an old exam about statistical mechanics. I hope you can help us or give any clues. Here is the problem.

Homework Statement



A chamber is divided by a wall into two sections of equal volume. One section of the chamber is initially filled by an ideal gas at temperature T0, whereas the other section is empty. Then, a small hole is opened in the dividing wall such that the gas can flow through, until equilibrium is reached. Consider the chamber isolated towards the surrounding and no heat is exchanged with the walls.

a) What is the final Temperature of the system?


b)How does the result change if instead initially the chamber is filled with a Van der Waals gas?


Homework Equations



Any thermodynamical equation.

The Attempt at a Solution



We tried to solve the problem by using the entropy. Since thermal equilibrium is reached, wehn the entropy is maximized.
The first equation that crosses my mind for part a) is the famous [itex]PV=NRT[/itex].
For part b), wikipedia tells us [itex]\left(p + \frac{n^2 a}{V^2}\right)\left(V-nb\right) = nRT[/itex]. So I suggest you to use these 2 equations rather than the entropy.
 
  • #3
Andrew Mason
Science Advisor
Homework Helper
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Hey there. We are struggling with a problem from an old exam about statistical mechanics. I hope you can help us or give any clues. Here is the problem.

Homework Statement



A chamber is divided by a wall into two sections of equal volume. One section of the chamber is initially filled by an ideal gas at temperature T0, whereas the other section is empty. Then, a small hole is opened in the dividing wall such that the gas can flow through, until equilibrium is reached. Consider the chamber isolated towards the surrounding and no heat is exchanged with the walls.

a) What is the final Temperature of the system?
Does the gas do any work? When you answer that question, apply the first law to determine the change in internal energy. What does that tell you about the change in temperature?

b)How does the result change if instead initially the chamber is filled with a Van der Waals gas?
Apply the same first law analysis. In this case, however, how is temperature related to volume and internal energy in a Van der Waals gas? (Hint: in an ideal gas, the temperature is a function of internal energy and independent of volume. Does the same apply to a Van der Waals gas?)

AM
 

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