Minimum heat removed from gas to restore its state

[hnnhcmmngs]

Homework Statement

After a free expansion to quadruple its volume, a mole of ideal diatomic gas is compressed back to its original volume isobarically and then cooled down to its original temperature. What is the minimum heat removed from the gas in the final step to restoring its state?

Homework Equations

pV = nRT
W = pΔV
ΔEint = Q - W

The Attempt at a Solution

I think ΔEint would be zero since it returns to its original temperature, so
Q = W
However, I don't know where to go from here. Could somebody explain?

 

[TSny]

After the isobaric compression, I think the gas would need to be heated rather than cooled in order to get back to the initial state.

What happens to the temperature of the gas during the free (adiabatic) expansion?
What happens to the temperature of the gas during the isobaric compression?

Based on the answers to these questions, should the gas be heated or cooled after the isobaric compression in order to return to the initial state?

 

[hnnhcmmngs]

After the isobaric compression, I think the gas would need to be heated rather than cooled in order to get back to the initial state.

What happens to the temperature of the gas during the free (adiabatic) expansion?
What happens to the temperature of the gas during the isobaric compression?

Based on the answers to these questions, should the gas be heated or cooled after the isobaric compression in order to return to the initial state?

During the adiabatic expansion the temperature remains constant, correct?
And during the isobaric compression the temperature decreases since the pressure remains constant and the volume decreases, correct?
So the gas should be heated?

 

[TSny]

During the adiabatic expansion the temperature remains constant, correct?
And during the isobaric compression the temperature decreases since the pressure remains constant and the volume decreases, correct?
So the gas should be heated?

Yes, Yes, and Yes.

 

[hnnhcmmngs]

Yes, Yes, and Yes.

Then why does the question say "then cooled down to its original temperature"? How am I supposed to calculate the heat removed from the gas?

 

[TSny]

Then why does the question say "then cooled down to its original temperature"?

It appears that whoever wrote the problem was mistaken. The gas is heated in the last step.

How am I supposed to calculate the heat removed from the gas?

You can try to find the heat added during the final step.

Let T_o be the initial temperature of the gas before the free expansion. Can you find the temperature of the gas at the beginning and end of the final step in terms of T_o? Hint: What is the temperature at the beginning and end of the isobaric compression?

Knowing the initial and final temperatures for the final step, can you find the heat added during this last step (if you assume the last step takes place at constant volume)?