How Do You Relate Entropies to Temperatures in Water Mixing?

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Discussion Overview

The discussion focuses on relating entropies to temperatures in the context of mixing water at different temperatures. Participants explore the principles of entropy balance, reversible processes, and the calculation of total entropy changes in a control volume involving mixed liquids.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about how to relate entropies S1 and S2 to temperatures in their entropy balance.
  • Another participant suggests using reversible processes to determine entropy differences, specifically by reducing the temperature of the hotter water to the final temperature of the mixed solutions.
  • A participant mentions conducting an entropy balance on a control volume but struggles to find absolute values for entropies S1 and S2.
  • It is noted that absolute entropies are not typically used in introductory physics; instead, differences in entropy between states are emphasized.
  • One participant claims to have solved the problem, highlighting the complexity of expressing the total entropy change when dealing with different masses of water.
  • Another participant agrees with the previous claim of having solved it and discusses the need for a single expression for total entropy, questioning how to combine mass-specific entropy changes.
  • A participant suggests that the total entropy change can be computed by independently calculating the entropy changes for each mass and then adding them together, while noting that the final temperature will not be the mean of the initial temperatures.
  • There is a clarification regarding the use of mass-specific heat in the calculations, indicating a potential misunderstanding in earlier posts.

Areas of Agreement / Disagreement

Participants express differing views on the approach to calculating entropy changes and the use of absolute versus relative entropies. There is no consensus on the final method or expression for total entropy change.

Contextual Notes

Some participants mention the importance of reversible processes and the specific heat of water, indicating that assumptions about these factors may affect the calculations. There are unresolved questions regarding the final temperature and the method of combining entropy changes.

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I don't know how to relate the entropies, S1 and S2, to the temperatures in my entropy balance
 

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Can't say I understand your attempt.

In order to determine entropy differences you have to come up with one or more reversible processes. Then you can compute the difference in entropies before & after.

Hint: take the hotter water solution and reversibly reduce its temperature from its original temperature to the final temperature of the two solutions. Compute ΔShotter.

Do the same for the cooler solution.

You wind up with two solutions of equal temperature, then you can mentally just pour them together without any further changes in any thermodynamic coordinate.

Answer will be ΔScooler - ΔShotter.
 
I am just doing an entropy balance on my control volume, being the tank where the two liquids are mixed. I just can't find an absolute value of entropy for S1 and S2
 
You never deal with absolute entropies in introductory physics. You deal in differences in entropy between two (or more) states.

So the hotter water's entropy changes by ΔSh and the colder by ΔSc. The total change in entropy is ΔSc + ΔSh.
 
I was able to solve it. The tricky thing at the end is to get it in the form they want, you have a squared term on top, so you have to take out a square from the top and bottom to make the bottom have the square root in it.

For the part with two masses being different, if I add them together, how can I get one expression for the total entropy? I can get the mass specific total entropy change, but not sure about the total entropy change.
 
Last edited:
Maylis said:
I was able to solve it. The tricky thing at the end is to get it in the form they want, you have a squared term on top, so you have to take out a square from the top and bottom to make the bottom have the square root in it.

yes, I noticed that too.

For the part with two masses being different, if I add them together, how can I get one expression for the total entropy? I can get the mass specific total entropy change, but not sure about the total entropy change.
I don't know how exactly you did the math. If you did it the way I suggested it's a straightforward extension of the method. You compute the two entropy changes independently and then add them. You don't need specific entropies. Of course, the final temperature will not be the mean of the two starting temperatures.

Post your math if you want to.

EDIT Sorry, I hadn't noticed mass-specific heat of water already used in part (a). Of course, use it again for part (b). This is usually written with a lower-case c which is what threw me off in the originally provided answer.
 
Last edited:

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