Helmholtz Free Energy Legendre Transformation

In summary, the conversation discusses the use of Legendre transformations to obtain the Helmholtz free energy A(T,V) from the internal energy and derive the general expression for its differential. The equations for internal energy (U) and Helmholtz free energy (A) are also mentioned, along with the variables (S, V, T) they are dependent on. The conversation ends with a request for verification of the derivation and a suggestion to post in a more advanced forum.
  • #1
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Homework Statement



Show how a Legendre transformation is used to obtain the Helmholtz free energy A(T,V) from the internal energy and derive the general expression for the differential of A.

Homework Equations


Internal Energy is a function of Entropy and Volume.

U Ξ (S, V)

A Ξ (T,V)

A = U - TS

where
U: Internal Energy
S: Entropy
V: Volume
T: Temperature
A: Helmholtz Free Energy

The Attempt at a Solution



A = U - (dU/dS)VS

dU = TdS - pdV

(dU/dS)V = T

A = U - TS (Helmholtz free energy)

dA - dU - TdS-SdT

dA = TdS - pdV - TdS - SdT

dA = -SdT - pdV

A Ξ (T,V)

I just wanted someone to check my derivation as I'm still a bit new to Legendre transformations. Cheers
 
  • #3
I would post this in the advanced physics forum. It's not introductory material.
 

FAQ: Helmholtz Free Energy Legendre Transformation

1. What is the Helmholtz Free Energy Legendre Transformation?

The Helmholtz Free Energy Legendre Transformation is a mathematical tool used in thermodynamics to convert the Helmholtz free energy function into a new function, called the Legendre transformed function. This transformation is used to simplify calculations and make it easier to analyze thermodynamic systems.

2. Why is the Helmholtz Free Energy Legendre Transformation important?

The Helmholtz Free Energy Legendre Transformation is important because it allows us to study thermodynamic systems in a more simplified way. By transforming the Helmholtz free energy function, we can easily calculate important properties such as temperature, pressure, and volume without having to directly measure them.

3. How is the Helmholtz Free Energy Legendre Transformation calculated?

The Helmholtz Free Energy Legendre Transformation is calculated by taking the derivative of the Helmholtz free energy function with respect to one of its independent variables, such as temperature or volume. The result is a new function that is easier to work with and contains important information about the thermodynamic system.

4. What are the applications of the Helmholtz Free Energy Legendre Transformation?

The Helmholtz Free Energy Legendre Transformation has various applications in thermodynamics, such as in the study of phase transitions, chemical reactions, and thermodynamic equilibrium. It is also used in other fields like statistical mechanics and quantum mechanics to analyze complex systems.

5. Are there any limitations to the Helmholtz Free Energy Legendre Transformation?

Although the Helmholtz Free Energy Legendre Transformation is a useful tool, it is not applicable to all thermodynamic systems. It can only be used for systems in equilibrium and where the Helmholtz free energy function is a well-defined and continuous function. It also cannot be used for systems with non-conservative forces or irreversible processes.

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