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

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The discussion centers on whether entropy can always be expressed as the variation of internal energy with respect to temperature, specifically through the equation dU = TdS - PdV. Participants clarify that entropy is defined as the logarithm of multiplicity, which is a standard definition in thermodynamics. The conversation highlights the importance of the equation dU = TdS - PdV as a physical relationship applicable to single-phase materials in thermodynamic equilibrium. There is an acknowledgment of the need for precision in discussing these thermodynamic relationships. Overall, the dialogue emphasizes the foundational concepts of entropy and internal energy in thermodynamics.
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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