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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)

_{V}S

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