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Helmholtz Equation

Last updated date: 17th Apr 2024
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Helmholtz Free Energy

The Helmholtz equation is named after a German physicist and physician named Hermann von Helmholtz, the original name Hermann Ludwig Ferdinand Helmholtz.This equation corresponds to the linear partial differential equation: where ⛛2 is the Laplacian, is the eigenvalue, and A is the eigenfunction.In mathematics, the eigenvalue problem for the Laplace operator is called the Helmholtz equation. That’s why it is also called an eigenvalue equation.

Here, we have three functions namely:

  • Laplacian denoted by a symbol ⛛2

  • The wavenumber symbolized as k

  • Amplitude as A.

The relation between these functions is given by:

Here, in the case of usual waves, k corresponds to the eigenvalue and A to the eigenfunction which simply represents the amplitude.

Helmholtz’s free energy is used to calculate the work function of a closed thermodynamic system at constant temperature and constant volume. It is mostly denoted by (f). 

The formula for Helmohtlz free energy can be written as :

                       F = U - TS

  • Where F = the helmholtz free energy. It is sometimes denoted as A.

  • U = internal energy of the system

  • T= The absolute temperature of the surrounding area.

  • S= Entropy of the given system.

In contrast to this particular free energy, there is another free energy which is known as Gibbs free energy.

Gibbs free energy can be defined as a thermodynamic potential that is used under constant pressure conditions. 

The equation of the Gibbs free energy is described as 

                      ∆G= ∆H - T∆S

  • ∆G = change in Gibbs free energy in a system

  • T = the absolute temperature of the temperature.

  • ∆S = change in entropy of a system.

  • ∆H = change in the enthalpy of a system.

Helmholtz Equation Derivation

The wave equation is given by,

Separating the variables, we get, u(r , t) = A(r) T(t)...(2)

Now substituting (2) in (1):  

Here, the expression on LHs depends on r.  While the expression on RHS depends on t.These two equations are valid only if both sides are equal to some constant value. On solving linear partial differential equations by separation of variables. We obtained two equations i.e., one for A (r)  and the other for T(t).

Hence, we have obtained the Helmholtz equation where - is a separation constant.      

Helmholtz Free Energy Equation Derivation

Helmholtz function is given by,

 F = U - TS


U = Internal energy

T = Temperature

S = Entropy

Fi is the initial helmholtz function and Fr being the final function.

During the isothermal (constant temperature) reversible process,  work done will be:                    

W   ≤    Fi - Fr

This statement says that the helmholtz function gets converted to the work. That’s why this function is also called free energy in thermodynamics.


Let’s say an isolated system acquires a δQ heat from surroundings, while the temperature remains constant. So, Entropy gained by the system = dS

Entropy lost by surroundings = δQ/T

Acc to 2nd law of thermodynamics, net entropy =  positive

From Classius inequality:                                

dS - δQ/T ≥ 0                               

dS  ≥ δQ/T

Multiplying by T both the sides, we get                      

 TdS  ≥  δQ

Now putting  

δQ = dU + δW (1st law of thermodynamics)                 

TdS ≥ (dU + δW)

Now,    TdS ≥  dU + δW       Or,     δW   ≤ TdS - dU           

Integrating both the sides:  


w Sr Ur \[\int\] δW ≤ T\[\int\]dS - \[\int\] dU 0 Si Ui W ≤ T (Sr - Si) - (Ur - Ui) W ≤ (Ui - TSi) - (Ur - TSr)

Now, if we observe the equation.  The terms  (Ui - TSi) and (Ur - TSr) are the initial and the final Helmholtz functions.Therefore, we can say that: W  ≤    Fi - Fr

By whatever magnitude the Helmholtz function is reduced, gets converted to work.


The application of Helmholtz's equation is researching explosives. It is very well known that explosive reactions take place due to their ability to induce pressure. Helmholtz’s free energy helps to predict the fundamental equation of the state of pure substances. This is the main application of Helmholtz’s free energy.

Apart from the described application above, there are some other applications also with Helmholtz energy shares. This can be listed as written below:

  • In the Equation of State: 

Helmholtz’s free energy equation is highly used in refrigerators as it is able to predict pure substances. So these are highly used for industrial applications.

  • Ine Auto-Encoder:

Helmholtz’s free energy is also very helpful to encode data. Due to its ability to analyze so precisely, it acts as a wonderful autoencoder in artificial neural networks. It proves helpful in the calculation of total code codes and reconstructed codes.

  • Due to its high precision, it is an excellent analyzer of pure substances.

Points to Remember about Helmohtlz Free Energy:

  • Internal energy, enthalpy, Gibbs free energy, and Helmholtz’s free energy are thermodynamically potential.

  • No more work can be done once Helmholtz’s free energy reaches its lowest point.

Helmholtz Equation Thermodynamics

The Gibbs-Helmholtz equation is a thermodynamic equation. This equation was named after Josiah Willard Gibbs and Hermann von Helmholtz. This equation is used for calculating the changes in Gibbs energy of a system as a function of temperature. Gibbs free energy is a function of temperature and pressure given by,

Applications of Helmholtz Equation

There are various applications where the helmholtz equation is found to be important. They are hereunder:

  • Seismology:  For the scientific study of earthquakes and its propagating elastic waves.

  • Tsunamis

  • Volcanic eruptions

  • Medical imaging

  • Electromagnetism: In the science of optics, the Gibbs-Helmholtz equation: Is used in the calculation of change in enthalpy using change in Gibbs energy when the temperature is varied at constant pressure.

  • CHELS: A combined Helmholtz equation-least squares abbreviated as CHELS. 

This method is used for reconstructing acoustic radiation from an arbitrary object.

FAQs on Helmholtz Equation

1. What is Helmholtz's function in thermodynamics?

In a closed thermodynamics system, a potential, the Helmohtlz energy works as a thermodynamic potential to obtain work. During a reversible isothermal process, the decrease in function is equal to the maximum amount of work done. This function is termed Helmholtz free energy.

2. Is Helmholtz free energy 0 at equilibrium?

Most of the processes in the universe are irreversible. So, Helmholtz’s free energy is also irreversible. As we know that it acts in constant temperature and volume, the value of this thermodynamic factor is going to be less than 0. So, at equilibrium, Helmholtz’s free energy will be less than 0 and not exact 0.

3. What is the major difference shared between the Gibbs free energy and Helmholtz free energy?

The maximum work done reversibly in a system is known as Gibbs free energy. The major difference which the Gibbs free energy and Helmholtz free energy share is If the maximum work done in a system is under constant pressure, then it is known as Gibbs free energy. On the other hand, if the maximum work done in a system is under constant volume and temperature then it is known as Helmholtz free energy. 

4. Why is it called Helmohtlz free energy?

The name is suggested after the names of the two scientists who found this energy. The scientists were Josiah Williard Gibbs and Hermann Von Helmholtz. 

5.  What did Helmholtz discover?

A German physician and physicist named Helmholtz had interests in the physiology of senses. For which he revolutionized the field of ophthalmology with the invention of the ophthalmoscope. An ophthalmoscope is an instrument that is used to examine the inside of a human eye.

6. How is Helmholtz's free energy calculated?

We know that U is the internal energy of a system.PV = pressure-volume product. TS = The temperature-entropy product. Where T is the temperature above absolute zero. Then by Helmholtz free energy equation:       

F = U − TS, and G = H- TS. 

Where  H = U + PV. So we get that: G = U + PV - TS

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This is how we can calculate the Helmholtz free energy.

7. Can Helmholtz free energy be negative?

Since we know that work is done, W = Fi - Fr. The final Helmholtz function is always lesser than the initial one. Therefore, the ΔF difference between  Fr and  Fi is negative.

8. What is the difference between Helmholtz's free energy and Gibbs's free energy?

In a closed thermodynamic system at constant temperature and pressure, Gibbs free energy is available to do non-PV work while Helmholtz free energy is the maximum useful non-PV work that can be extracted from a thermodynamically closed system at constant temperature and volume.

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