I Focus Problem for Entropy Change in Irreversible Adiabatic Process

[Chestermiller]

I am afraid I don't follow your insight here...
Secondly, this is not obvious to me. why you compare two reversible processes C and D instead of compare processes E, C (or D) and irreversible...

What I learned by comparing reversible processes C and D is that, if all the heat transfer is carried out at constant volume (no work in such steps) and all the work is carried out at constant temperature, then I can make the amount of work done in the overall process as low as I desire by carrying out the expansion at as low a temperature as I choose. In process E, all the heat transfer is carried out in steps 1 and 3 at constant volume, and I chose a low enough temperature for step 2 (To/4) so that the overall reversible work is very low (even lower than for the irreversible process between the same two end states).

 

[guhag]

What I learned by comparing reversible processes C and D is that, if all the heat transfer is carried out at constant volume (no work in such steps) and all the work is carried out at constant temperature, then I can make the amount of work done in the overall process as low as I desire by carrying out the expansion at as low a temperature as I choose. In process E, all the heat transfer is carried out in steps 1 and 3 at constant volume, and I chose a low enough temperature for step 2 (To/4) so that the overall reversible work is very low (even lower than for the irreversible process between the same two end states).

This is a BEAUTIFUL explanation. Wow ! Thank you, Chet. I've learned a LOT already from you in this thread and there is still more to go :)

Is there a mathematical generalization behind this ? Or, it depends upon a particular situation, like this case?

 

[Chestermiller]

 

[Chestermiller]

This is a BEAUTIFUL explanation. Wow ! Thank you, Chet. I've learned a LOT already from you in this thread and there is still more to go :)

Is there a mathematical generalization behind this ? Or, it depends upon a particular situation, like this case?

I do't know what you mean.

 

[guhag]

View attachment 329674

Thank you. Mine looks like this....

 

[Chestermiller]

Thank you. Mine looks like this....

View attachment 329676

Is nobody willing to attempt part (c), the entropy change for each of the 5 reversible paths that we have identified?

 

[guhag]

Is nobody willing to attempt part (c), the entropy change for each of the 5 reversible paths that we have identified?

I haven't checked this thread in a while. Company layoffs looming ahead. I'll try part c) this week, for sure.

 

[Lollo]

@ChetMiller I have not found any explicit treatment, in books, about irreversible adiabatic transformations. 1.1) Very often it is said that, in an adiabatic expansion, given the same initial conditions and final pressure (or final volume) the work done by an irreversible one is less than that done by a reversible one, but I have not yet found an explicit demonstration of this.
1.2) Also, it is said that in an adiabatic compression, the work done by an irreversible is greater than that done by a reversible, given the same initial conditions and final pressure (or final volume).
In short, it should hold $$|W_irr < W_rev|$$. How to prove it?
3) Moreover, it was not understood what inequalities should exist between pressure, volume and temperature between irreversibles and reversibles. Given the same initial conditions and final pressure, is the final temperature in an irreversible adiabatic transformation greater or less than the final temperature in a reversible adiabatic transformation? If yes, why?

 

[Chestermiller]

@ChetMiller I have not found any explicit treatment, in books, about irreversible adiabatic transformations. 1.1) Very often it is said that, in an adiabatic expansion, given the same initial conditions and final pressure (or final volume) the work done by an irreversible one is less than that done by a reversible one, but I have not yet found an explicit demonstration of this.
1.2) Also, it is said that in an adiabatic compression, the work done by an irreversible is greater than that done by a reversible, given the same initial conditions and final pressure (or final volume).
In short, it should hold $$|W_irr < W_rev|$$. How to prove it?
3) Moreover, it was not understood what inequalities should exist between pressure, volume and temperature between irreversibles and reversibles. Given the same initial conditions and final pressure, is the final temperature in an irreversible adiabatic transformation greater or less than the final temperature in a reversible adiabatic transformation? If yes, why?

Have you read over the posts in this thread. If you had, you would know that there are an infinite number of reversible paths between the initial and final states of the irreversible adiabatic process analyzed here and that the work along some of these reversible paths is greater than for the irreversible path while, along others, the work is less than for the irreversible path; however the entropy change is the same for all the reversible paths as for the irreversible path.

If you have any comments or questions about the analysis presented here, let's hear them.

 

[Lollo]

@Chestermiller I am very surprised that there are cases where, in reversible adiabatic processes, the work is less than in the corresponding irreversible path. I always thought that the work carried out on the system in irreversible adiabatic compression is greater than the work carried out on the system in reversible adiabatic compression, and that the work carried out on the system in irreversible adiabatic expansion is always less than the work carried out on the system in reversible adiabatic compression. Is this not so? I have seen several answers on StackExchange saying this, and they have never been refuted. See for example https://chemistry.stackexchange.com...rreversible-adiabatic-compression-greater-tha. "the entropy change is the same for all the reversible paths as for the irreversible path." How to prove that? It seems to me that point (c) has been left open. Can you post a solution for the benefit of the forum?

 

[Chestermiller]

@Chestermiller I am very surprised that there are cases where, in reversible adiabatic processes, the work is less than in the corresponding irreversible path.

There exists no reversible adiabatic process between the same two end states as for an irreversible adiabatic process. We know this because the entropy change for an adiabatic irreversible process is positive and the entropy change for an adiabatic reversible process is zero. So if we have an adiabatic irreversible process, all reversible processes between the same two end states must be non-adiabatic.

I always thought that the work carried out on the system in irreversible adiabatic compression is greater than the work carried out on the system in reversible adiabatic compression, and that the work carried out on the system in irreversible adiabatic expansion is always less than the work carried out on the system in reversible adiabatic compression. Is this not so? I have seen several answers on StackExchange saying this, and they have never been refuted. See for example https://chemistry.stackexchange.com...rreversible-adiabatic-compression-greater-tha.

As I showed in the examples in the present thread, this is not correct.

"the entropy change is the same for all the reversible paths as for the irreversible path." How to prove that? It seems to me that point (c) has been left open. Can you post a solution for the benefit of the forum?

If the two end states are the same for all the paths and, if entropy is a state function, what other possibility is there?

If you think you have found any errors in the analysis that I have done in this thread, please articulate them.