Helmholtz Equation

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

Here, 

U = Internal energy

T = Temperature

S = Entropy

F_i is the initial helmholtz function and F_r being the final function.

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

W   ≤    F_i - F_r

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

Derivation:

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 2^nd 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 (1^st law of thermodynamics)                 

TdS ≥ (dU + δW)

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

Integrating both the sides:  

w S_r U_r \[\int\] δW ≤ T\[\int\]dS - \[\int\] dU 0 S_i U_i W ≤ T (S_r - S_i) - (U_r - U_i) W ≤ (U_i - TS_i) - (U_r - TS_r)

Now, if we observe the equation.  The terms  (U_i - TS_i) and (U_r - TS_r) are the initial and the final Helmholtz functions.Therefore, we can say that: W  ≤    F_i - F_r

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

Applications:

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.