Elementary Ideas About Entropy -- Is this textbook example correct?

[BrandonInFlorida]

Summary:: An elementary example calculation involving entropy in a textbook seems wrong

I was reading an elementary introduction to entropy and the second law of thermodynamics. The book gave the example of a gas in a chamber suddenly allowed to expand into an additional portion of the chamber so that it's volume us doubled, which is an irreversible process. The calculated the change in entropy. Entropy had been defined by:

dS = dQ/T

The chamber was completely insulated from the environment. They said that the quantities were undfined during the irreversible process, so calculated the integral of dQ/T by imagining a reversible process connecting the two states instead and got a non-zero integral. In the reversible process, the chamber was connected to an energy reservoir, which in the real process it is not. They said that since the entropy reflects only the state, and not how it was achieved, the entropy change in the real process would be the same.

Here's my question. In the actual, reversible process, the chamber is insulated so dQ will always be zero It doesn't matter if T is undefined or zero or whatever. dQ is rigorously zero at all times. How can the integral be positive, even if it's positive for an imaginary reversible process connecting the two states?

 

[kuruman]

If you know that the change in a function depends only on the end points but not how got there, what do you do in the case where you don't know how you got there? If it doesn't matter how you got from one point to the other, then any procedure that allows you to get there and that you can calculate mathematically will work.

Let me make an analogy. I watch a fly move from the floor to the top of a table following a path that has all sorts of twists and turns and changes in height as flies do. I have no idea what the path looks like mathematically and I want to calculate the work done by gravity on the fly. I know that the expression for that work is $$W_{grav.}=\int_{\text{floor}}^{\text{table}}\vec F_{grav.}\cdot \vec s~,$$but I don't know the path. What do I do?

Do I throw up my hands in despair and say I cannot do it because the exact path is unknown? Of course not. I know that gravity is a conservative force and, therefore, the work done by it on the fly depends only on the end points and not on how the fly got there. Therefore, I invent another path in two segments: horizontally across from where the fly started to directly below where it landed and then straight up. The work integral is zero for the horizontal segment because the force is perpendicular to the path, so all that counts is the vertical segment. This integral I can easily do to find $$W_{grav.}=\int_{\text{floor}}^{\text{table}}\vec F_{grav.}\cdot \vec s=-mg\Delta y.$$Do you see the reasoning behind the strategy?

The fact that $dQ$ is "rigorously zero" in the first scenario is irrelevant. If you were shown the system in its initial state, left the room and when you came back you found the system in its final state, you wouldn't know if it got there following the first scenario or the second scenario used by the textbook to calculate the entropy change. Same thing with the fly. If the path doesn't matter, the path doesn't matter.

 

[BrandonInFlorida]

If you know that the change in a function depends only on the end points but not how got there, what do you do in the case where you don't know how you got there? If it doesn't matter how you got from one point to the other, then any procedure that allows you to get there and that you can calculate mathematically will work.

Let me make an analogy. I watch a fly move from the floor to the top of a table following a path that has all sorts of twists and turns and changes in height as flies do. I have no idea what the path looks like mathematically and I want to calculate the work done by gravity on the fly. I know that the expression for that work is $$W_{grav.}=\int_{\text{floor}}^{\text{table}}\vec F_{grav.}\cdot \vec s~,$$but I don't know the path. What do I do?

Do I throw up my hands in despair and say I cannot do it because the exact path is unknown? Of course not. I know that gravity is a conservative force and, therefore, the work done by it on the fly depends only on the end points and not on how the fly got there. Therefore, I invent another path in two segments: horizontally across from where the fly started to directly below where it landed and then straight up. The work integral is zero for the horizontal segment because the force is perpendicular to the path, so all that counts is the vertical segment. This integral I can easily do to find $$W_{grav.}=\int_{\text{floor}}^{\text{table}}\vec F_{grav.}\cdot \vec s=-mg\Delta y.$$Do you see the reasoning behind the strategy?

The fact that $dQ$ is "rigorously zero" in the first scenario is irrelevant. If you were shown the system in its initial state, left the room and when you came back you found the system in its final state, you wouldn't know if it got there following the first scenario or the second scenario used by the textbook to calculate the entropy change. Same thing with the fly. If the path doesn't matter, the path doesn't matter.

The answer to the question is supposed to be the integral of dQ/T. dQ is zero at all times because of the insulating walls. An integral whose integrand has a zero numerator will be zero, irrespective of anything else happening with the integral. If the correct answer is the integral of zero divided by T, the integral must be zero, regardless of any other considerations.

 

[Chestermiller]

Summary:: An elementary example calculation involving entropy in a textbook seems wrong

I was reading an elementary introduction to entropy and the second law of thermodynamics. The book gave the example of a gas in a chamber suddenly allowed to expand into an additional portion of the chamber so that it's volume us doubled, which is an irreversible process. The calculated the change in entropy. Entropy had been defined by:

dS = dQ/T

The chamber was completely insulated from the environment. They said that the quantities were undfined during the irreversible process, so calculated the integral of dQ/T by imagining a reversible process connecting the two states instead and got a non-zero integral. In the reversible process, the chamber was connected to an energy reservoir, which in the real process it is not. They said that since the entropy reflects only the state, and not how it was achieved, the entropy change in the real process would be the same.

Here's my question. In the actual, reversible process, the chamber is insulated so dQ will always be zero It doesn't matter if T is undefined or zero or whatever. dQ is rigorously zero at all times. How can the integral be positive, even if it's positive for an imaginary reversible process connecting the two states?

Your source does not say that the change in entropy is the integral of dQ/T for all paths between the initial and final states, irrespective of whether the path is reversible or irreversible. It only says that the change in entropy between the two ends states is equal to the integral of dQ/T for a reversible path. For an adiabatic irreversible process, there is no reversible path that goes between the same two end states. So you need to devise (dream up) an alternate reversible path between the same two end states, and calculate the integral of dQ/T for that path. This is also the change in entropy for the irreversible path, since the change in entropy depends only on the two end points.

For a better understanding of how to determine the change in entropy for a closed system that experiences an irreversible process, see my Physics Forums Insights article: https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe/

 

[BrandonInFlorida]

Your source does not say that the change in entropy is the integral of dQ/T for all paths between the initial and final states, irrespective of whether the path is reversible or irreversible. It only says that the change in entropy between the two ends states is equal to the integral of dQ/T for a reversible path. For an adiabatic irreversible process, there is no reversible path that goes between the same two end states. So you need to devise (dream up) an alternate reversible path between the same two end states, and calculate the integral of dQ/T for that path. This is also the change in entropy for the irreversible path, since the change in entropy depends only on the two end points.

For a better understanding of how to determine the change in entropy for a closed system that experiences an irreversible process, see my Physics Forums Insights article: https://www.physicsforums.com/insights/grandpa-chets-entropy-recipe/

I think you've given me the key concept. The change in entropy for the actual situation isn't equal to the integral applied to those circumstances. That is what I needed.

I think you're giving my textbook more credit than it deserves, because it kind of does say that the change in entropy in the actual situation is the integral. Resnick and Halliday are usually pretty clear, but not in this case.

 

[Chestermiller]

Try a decent thermo book like Moran et al. Fundamentals of Engineering Thermodynamics. I believe this is available online.

 

[BrandonInFlorida]

Try a decent thermo book like Moran et al. Fundamentals of Engineering Thermodynamics. I believe this is available online.

I recently requested, and was given for a birthday, an introductory book by Schroeder, but I'll make a note of Moran. I am attempting to review every area of physics, although I suspect I may not live long enough to complete that, since my program involves reading every word of each book and solving every problem in every chapter. It's good to have a goal, though.

 

[bob012345]

...my program involves reading every word of each book and solving every problem in every chapter. It's good to have a goal, though.

I used to feel like I had to do it that way and it usually paralyzed me into inaction. Like when I wanted to study Gravitation by Misner, Thorne and Wheeler. Now I cut to the chase and only hunt for the answer I need to the question at hand. Of course I probably am missing things but the key point is, I no longer have to take tests.

 

[BrandonInFlorida]

I am eager to get to modern physics, but my overall goal is the same as it was when I was 15 years old (53 years ago) - to learn as much as possible about physics. I did get a BS and MS, but then went into engineering and drifted away from pure physics. Now that I'm retired, I can do anything I want and studying physics is what I want.

 

[bob012345]

I am eager to get to modern physics, but my overall goal is the same as it was when I was 15 years old (53 years ago) - to learn as much as possible about physics. I did get a BS and MS, but then went into engineering and drifted away from pure physics. Now that I'm retired, I can do anything I want and studying physics is what I want.

Any thought of getting a doctorate in physics?

 

[BrandonInFlorida]

No, it's too late, but my little pipe dream would be to write a paper accepted by a peer reviewed journal.

 

[bob012345]

No, it's too late, but my little pipe dream would be to write a paper accepted by a peer reviewed journal.

My English cousin was a computer analyst in the 60's. When he retired he became among other things an Egyptologist. Then he got a doctorate in German Literature at the age of 94, the oldest man in the U.K. to receive a doctorate. He just loved learning new things.

 

[BrandonInFlorida]

My feeling is that now I'm learning every day and the only person giving me tests is me (I actually do). All of these other people would just add inefficiency and irritation. If I were younger, I'd put up with it, but I'm not. I also cannot read a distant blackboard as well as I once could and it's unlikely to be correctable.

The only thing I miss is people to talk about physics with. In two years of going to the book store every day to study, I've only once met someone in a similar enough field to chat about physics with. I can talk to an old college buddy who's a physics professor, but only occasionally. That's the only major drawback to not being in school.

 

[Chestermiller]

I am eager to get to modern physics, but my overall goal is the same as it was when I was 15 years old (53 years ago) - to learn as much as possible about physics. I did get a BS and MS, but then went into engineering and drifted away from pure physics. Now that I'm retired, I can do anything I want and studying physics is what I want.

What kind of engineering work did you do? I was a ChE.

 

[bob012345]

My feeling is that now I'm learning every day and the only person giving me tests is me (I actually do). All of these other people would just add inefficiency and irritation. If I were younger, I'd put up with it, but I'm not. I also cannot read a distant blackboard as well as I once could and it's unlikely to be correctable.

The only thing I miss is people to talk about physics with. In two years of going to the book store every day to study, I've only once met someone in a similar enough field to chat about physics with. I can talk to an old college buddy who's a physics professor, but only occasionally. That's the only major drawback to not being in school.

If you live near a university you might consider attending physics department seminars which generally are open to the public. You can hobnob with the physics crowd. Of course that's why I joined PF.

 

[BrandonInFlorida]

What kind of engineering work did you do? I was a ChE.

So, I had the BS and MS in physics. I went to an employment agent and said, "Anything but teaching," so she got me a job as a teacher in a boy's Catholic high school. In less than a year, I left that and got a job with the Westinghouse Steam Turbine Generator Division. They made electrical generators for power plants. I'd still be there except that their business was going downhill very severely and there were a lot of layoffs which eventually affected me. Shortly thereafter they were bought by a German company. I then got a job with McDonnell Douglas Missile Systems. I'd still be there except that they lost a bid-off with another company for cruise missiles and then closed. But during my time there, I had migrated into software. I spent the rest of my career as a software engineer.

 

[BrandonInFlorida]

If you live near a university you might consider attending physics department seminars which generally are open to the public. You can hobnob with the physics crowd. Of course that's why I joined PF.

I like PF, but sometimes you want to speak to people face to face.