Measurable consequences of entropy of mixing

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Discussion Overview

The discussion revolves around the measurable consequences of the entropy of mixing, particularly in the context of distinguishable versus indistinguishable gases. Participants explore the implications of this entropy in macroscopic experiments, including potential methods to differentiate between a pure gas and a mixture of distinguishable particles.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions whether it is possible to design a macroscopic experiment to distinguish between a pure gas and a mixture of distinguishable gases without prior knowledge of the differences between the gas types.
  • Another participant suggests that distinguishing the gases would require a method such as a semipermeable membrane or differences in molecular weight.
  • A third participant references Gibbs' paradox, proposing that the solution involves assigning entropy to information regarding distinguishability.
  • One participant expresses appreciation for the insights shared and acknowledges the complexity of the topic, indicating a desire to further understand the referenced paper on Gibbs' paradox.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the feasibility of distinguishing the gases without additional information or methods. Multiple viewpoints regarding the implications of entropy and distinguishability are presented, indicating ongoing debate.

Contextual Notes

Participants highlight the need for specific experimental setups or conditions to explore the consequences of entropy of mixing, suggesting that assumptions about distinguishability and measurement methods are critical to the discussion.

crossword.bob
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Most textbooks include an example of entropy of mixing that involves removing a partition between two (in principle) distinguishable gases, and compare this to the case where the two gases are indistinguishable. What I’ve not yet been able to figure out is what the consequences of this additional entropy are for the distinguishable case.

Say you are given two cylinders; one filled with a pure gas, and the other a mixture of two distinguishable (in principle) particles. Is there a macroscopic experiment you could perform to determine which is which, without knowing how the distinguishable particle types actually differ from each other? Do (for example) heat capacities depend on total entropy, so that one could measure temperature versus heat input for each cylinder?
 
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No, you need some means to tell the two gasses apart, e.g. a semipermeable membrane, different molecular weight, so they can be separated.
 
crossword.bob said:
Most textbooks include an example of entropy of mixing that involves removing a partition between two (in principle) distinguishable gases, and compare this to the case where the two gases are indistinguishable. What I’ve not yet been able to figure out is what the consequences of this additional entropy are for the distinguishable case.

Say you are given two cylinders; one filled with a pure gas, and the other a mixture of two distinguishable (in principle) particles. Is there a macroscopic experiment you could perform to determine which is which, without knowing how the distinguishable particle types actually differ from each other? Do (for example) heat capacities depend on total entropy, so that one could measure temperature versus heat input for each cylinder?

The solution of Gibbs' paradox (which is the experiment you describe) consists of assigning entropy to *information*- information about distinguish-ability, for example.

http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf
 
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Thanks, that helps. I suspect it will take a little time for me to fully digest that paper, but there is some solace is knowing the problem that’s been bothering me has bothered better minds than mine in the past!
 

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