Derivation of the thermodynamic potentials using Legendre transformations

In summary, The conversation discusses the concept of Legendre transformations in Thermodynamics and the generation of potentials, including the enthalpy potential, Helmholtz function, and Gibbs function, through these transformations. The conversation also mentions the use of Legendre transformations in the process.
  • #1
Andromon
3
0
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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  • #2
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.
 
  • #3
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?
 
  • #5
I'd be surprised to be told that I wasn't actually doing Legendre transformations in my earlier post, but let wiser heads decide.
 

1. What is the purpose of using Legendre transformations in deriving thermodynamic potentials?

The purpose of using Legendre transformations is to simplify the mathematical expressions of thermodynamic variables and make them more convenient for calculations. This allows for easier analysis and interpretation of thermodynamic systems.

2. How are Legendre transformations used in the derivation of thermodynamic potentials?

Legendre transformations are used to eliminate one variable in a mathematical expression and replace it with its conjugate variable. This results in a new expression with a different independent variable, which is often more useful for thermodynamic calculations.

3. Can you explain the concept of conjugate variables in thermodynamics?

In thermodynamics, conjugate variables refer to pairs of variables that are related by a Legendre transformation. Examples include temperature and entropy, pressure and volume, and chemical potential and particle number. These variables are useful in describing the state of a thermodynamic system and how it changes under different conditions.

4. Is it possible to derive all thermodynamic potentials using Legendre transformations?

Yes, it is possible to derive all thermodynamic potentials using Legendre transformations. Each potential is derived by performing a Legendre transformation on a different pair of conjugate variables.

5. Are there any limitations or assumptions when using Legendre transformations in thermodynamics?

One limitation of using Legendre transformations is that they only apply to systems in equilibrium. Additionally, the transformations assume that the thermodynamic variables are continuous and differentiable, and that the thermodynamic system is in a state of internal equilibrium.

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