Helmholtz Free Energy

What is Helmholtz free energy?

Helmholtz free energy is a thermodynamics concept in which, thermodynamic potential 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:

\(\large F=U-TS\)


  • F is the Helmholtz free energy in Joules
  • U is the internal energy of the system in Joules
  • T is the absolute temperature of the surroundings in Kelvin
  • S is the entropy of the system in joules per Kelvin

Formula development for laws of thermodynamics:

\(dU=\delta Q+\delta W\) which is from  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 auto-encoder:

Auto-encoder 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.

Related Articles:

What is Helmholtz Function?

Helmholtz function is a thermodynamic function which is defined as the decrease in the function and is equal to the maximum amount of work which is available during reversible isothermal process.

Difference Between Helmholtz free energy and Gibbs free energy

Helmholtz free energy Gibbs free energy
 It is defined as the useful work that is obtained from a particular system It is defined as the maximum reversible work that is obtained from a particular system
It is the energy required to create a system at constant temperature and volume It is the energy required to create a system at constant pressure and temperature
Helmholtz free energy finds lesser application as the volume of the system should be constant Gibbs free energy finds more application as the pressure of the system is constant


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