Helmholtz Free Energy Legendre Transformation

AI Thread Summary
The discussion focuses on deriving the Helmholtz free energy A(T,V) from internal energy U(S,V) using a Legendre transformation. The key equation A = U - TS is established, where T is temperature and S is entropy. The differential form of A is derived, resulting in dA = -SdT - pdV, indicating how changes in temperature and volume affect Helmholtz free energy. The user seeks validation of their derivation, emphasizing their novice status with Legendre transformations. This topic is positioned as advanced physics material, suitable for deeper exploration in thermodynamics.
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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
 
I would post this in the advanced physics forum. It's not introductory material.
 
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