Derivation of the thermodynamic potentials using Legendre transformations

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Discussion Overview

The discussion revolves around the derivation of thermodynamic potentials using Legendre transformations, focusing on the rigorous demonstration of this process. Participants explore the connections between Legendre transformations and various thermodynamic potentials, including enthalpy, Helmholtz free energy, and Gibbs free energy, as well as related concepts like Maxwell transformations and Euler relations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses a lack of clarity on how Legendre transformations are rigorously applied to derive thermodynamic potentials.
  • Another participant provides a systematic approach to generating the potentials, starting from the differential form of internal energy and applying transformations to derive enthalpy, Helmholtz function, and Gibbs function.
  • A participant questions the necessity of Legendre transformations in the explanation provided, suggesting that the method described may not explicitly require them.
  • A link to a Wikipedia article on Legendre transformations in thermodynamics is shared for further reference.
  • One participant expresses uncertainty about whether their previous explanation involved true Legendre transformations, inviting further input from others.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity and application of Legendre transformations in the derivation of thermodynamic potentials, indicating that multiple views remain on this topic.

Contextual Notes

The discussion highlights potential limitations in understanding the rigorous application of Legendre transformations, as well as the dependence on definitions and interpretations of the transformations in the context of thermodynamics.

Andromon
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Hello guys, I'm studying Thermodynamics and I don't totally see how you introduce the potencials using Legendre transformations.

I have seen a non formal explanation showing how you can interpret them, but not a rigorous demonstration of how you get them via the Legendre transformations.

Do you know any site or book that covers it?

Also all the other issues, like the Maxwell transformations and the Euler equations and relations.

Ty.
 
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I expect I'm not giving you a full picture of what a Legendre transformation is, but here's a systematic way to generate potentials, H, F and G...

Start with dU = TdS- pdV.

From the product rule: dU = TdS- {d(pV) - Vdp}

We can write this as: d{U + pV} = TdS + Vdp

U + pV is usually designated as H. It is the enthalpy potential. Its 'natural variables' are S and P.

We can product-transform TdS instead of pdV, and obtain the Helmholtz function U - TS, with natural variables T and V.

Finally we can transform both TdS and pdV,obtaining the Gibbs function G = U + pVTS, with natural variables p and T.
 
Ok, I see it, is an add and subtract trick, but I don't see where there it's used the Legendre transformation, it's not needed at all?
 
I'd be surprised to be told that I wasn't actually doing Legendre transformations in my earlier post, but let wiser heads decide.
 

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