Can the Entropy of a System Always Be Expressed as dU/dT?

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Discussion Overview

The discussion centers around whether the entropy of a system can always be expressed as the partial derivative of internal energy with respect to temperature, ##\partial U / \partial T##. Participants explore this concept in various contexts, including large-scale cosmological discussions and specific systems like gas containers.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the validity of expressing entropy as ##\partial U / \partial T## and seeks clarification on its applicability in different scenarios.
  • Another participant suggests that entropy is often defined in terms of multiplicity, indicating a potential alternative perspective on the definition of entropy.
  • There is a challenge regarding the expression for entropy, with participants discussing the relationship ##dU = TdS + pdV## and whether it is universally applicable.
  • Participants express uncertainty about the expression for entropy and its dependence on specific conditions or definitions.
  • Clarifications are made regarding the correct formulation of the relationship between internal energy, entropy, and volume, with acknowledgment of previous mistakes in notation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the expression for entropy as ##\partial U / \partial T## is universally applicable. Multiple competing views regarding the definition and applicability of entropy are present.

Contextual Notes

The discussion reflects limitations in the assumptions made about the relationship between internal energy and entropy, as well as the contexts in which these definitions may or may not hold true.

Tio Barnabe
Can we always express the entropy of a given system as ##\partial U / \partial T##, i.e. as the variation of the internal energy of the system w.r.t. its temperature?

By always I really mean, in every discussion we are eventually engaged in. Like, when I want to talk about the evolution of the universe (in large scale), or when I want to talk about a container of gas.
 
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As opposed to what? A definition involving multiplicity?
 
Are you sure about the expression you gave for entropy?
 
mishima said:
As opposed to what? A definition involving multiplicity?
Well, the definition I know for entropy does involve multiplicity. "The entropy is the logarithm of the multiplicity". So I'm not aware of an other definition which is opposed to the one I mentioned in the opening post.
Chestermiller said:
Are you sure about the expression you gave for entropy?
I thought that would be the expression for the entropy, since we have ##dU = S dT \ + \ ... \ ##.
 
Tio Barnabe said:
Well, the definition I know for entropy does involve multiplicity. "The entropy is the logarithm of the multiplicity". So I'm not aware of an other definition which is opposed to the one I mentioned in the opening post.

I thought that would be the expression for the entropy, since we have ##dU = S dT \ + \ ... \ ##.
I think you mean TdS
 
Tio Barnabe said:
Well, the definition I know for entropy does involve multiplicity. "The entropy is the logarithm of the multiplicity". So I'm not aware of an other definition which is opposed to the one I mentioned in the opening post.

Ok. So you're more asking if dU = TdS + pdV ever fails to be applicable?
 
Chestermiller said:
I think you mean TdS
yes, I'm sorry for the mistake
mishima said:
Ok. So you're more asking if dU = TdS + pdV ever fails to be applicable?
yes, exactly
 
Tio Barnabe said:
yes, I'm sorry for the mistake

yes, exactly
The equation ##dU=TdS-PdV## is really a physical property relationship that connects the changes in internal energy, entropy, and volume for a single-phase material between two closely neighboring thermodynamic equilibrium states at (U,S,V) and (U+dU, S+dS, and V+dV).
 

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