Internal energy + entropy for molecule

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Discussion Overview

The discussion revolves around the definitions and relationships between internal energy, entropy, and temperature, particularly in the context of statistical mechanics. Participants explore whether these concepts can be applied to individual molecules or if they are inherently macroscopic in nature.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant suggests that internal energy can be defined for a single molecule using the equation U = 1/2 Kb T, and questions if entropy can similarly be defined for one molecule.
  • Another participant challenges the initial claim by asking if the equation for internal energy describes the kinetic energy of each molecule or just the average kinetic energy consistent with the Boltzmann distribution.
  • A response argues that if internal energy can describe one molecule, then temperature can also describe one molecule, leading to the question of whether entropy can be defined for a single molecule.
  • A participant introduces a metaphor about the average number of children per family to illustrate the potential issues with applying average concepts to individual cases.
  • Some participants assert that in statistical mechanics, it is acceptable to define entropy for a single molecule.
  • A later reply seeks to clarify how the temperature of a single molecule relates to the macroscopic temperature of a larger system containing many molecules.

Areas of Agreement / Disagreement

Participants express differing views on whether internal energy, entropy, and temperature can be meaningfully defined for individual molecules. The discussion remains unresolved, with multiple competing perspectives presented.

Contextual Notes

There are unresolved questions regarding the definitions and applicability of thermodynamic concepts at the molecular level versus the macroscopic level, as well as the implications of statistical mechanics on these definitions.

tonyjk
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Hello,
Internal energy can be defined theoretically for one molecule (U = 1/2 Kb T) for example but entropy is defined for a system thus for many molecules. Then we define temperature equal to δU / δS but here U can be defined for one molecule, so S can also be defined for one molecule? How?

Thank you
 
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tonyjk said:
Hello,
Internal energy can be defined theoretically for one molecule (U = 1/2 Kb T) for example but entropy is defined for a system thus for many molecules. Then we define temperature equal to δU / δS but here U can be defined for one molecule, so S can also be defined for one molecule? How?

Thank you
Do you really feel that the equation you presented for U describes the kinetic energy of each and every molecule of an ideal gas, or is it just the average kinetic energy over all the molecules, consistent with the Boltzmann distribution?
 
Chestermiller said:
Do you really feel that the equation you presented for U describes the kinetic energy of each and every molecule of an ideal gas, or is it just the average kinetic energy over all the molecules, consistent with the Boltzmann distribution?
No like I said U can theoretically describe both. For example if U can describe one molecule thus the temperature can describe one molecule. So S can describe one molecule? Then if S cannot describe one molecule, a temperature cannot describe one molecule, so there is a contradiction in the U definition for one molecule
 
Last edited:
The average number of children per family in the US was 2.7 in 1961.
Can you say that any specific family had 2.7 children?
 
In statistical mechanics, there is nothing wrong with defining entropy for a single molecule.
 
DrDu said:
In statistical mechanics, there is nothing wrong with defining entropy for a single molecule.
Great. So how the temperature of one molecule in statistical mechanics is related to macroscopic temperature (T= dU/dS) of a volume containing this molecule and many others?
 

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