# Helmholtz Free Energy

## What is Helmholtz free energy?

Helmholtz free energy in thermodynamics is a thermodynamic potential which is used to measure the work of a closed system with constant temperature and volume. It can be defined in the form of the following equation:

$\large F=U-TS$

Where,

F: Helmholtz free energy in joules

U: internal energy of the system in joules

T: absolute temperature of the surroundings in kelvin

S: entropy of the system in joules per kelvin

Formula development for laws of thermodynamics:

$dU=\delta Q+\delta W$ (first law of thermodynamics for closed system)

$dU=TdS-pdV$ (𝜹Q = TdS and 𝜹W = pdV)

$dU=d(TS)-SdT-pdV$ (product rule ie; d(TS) = TdS+SdT)

$d(U-TS)= -SdT-pdV$ $dF=-SdT-pdV$ (from F=U-TS)

### Application of Helmholtz free energy

• In equation of state: Pure fluids with high accuracy (like industrial refrigerants) are represented using Helmholtz function as a sum of ideal gas and residual terms.
• In autoencoder: Autoencoder is an artificial neural network which is used to encode efficient data. Here Helmholtz energy is used to find the sum of code cost and reconstructed code.

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