Helmholtz Equation: Derivation, Thermodynamics, Applications

Helmholtz Equation is the linear partial differential equation that is named after Hermann von Helmholtz. In this equation, we deal with three functions mainly- Laplacian, Wavenumber, and Amplitude. The Helmholtz equation is the eigenvalue equation that is solved by separating variables only in coordinate systems.

Key Terms- Helmholtz Equation, differential equation, Laplacian, Amplitude, Wave equation, Thermodynamics

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What is Helmholtz Equation

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Helmholtz equation is a Physics and Mathematics equation named after Hermann von Helmholtz.

• It is a partial differential equation with the given mathematical formula. 
• We need to consider three important functions mainly- Laplacian (which can be denoted as ∇²), the Wave Number (which is denoted by k), and the Amplitude (which is denoted as A). 
• This is an Eigenvalue Equation, and can be solved by separating variables in 11 coordinated systems.
• Helmholtz equation is used to solve problems in physics such as seismology, electromagnetic radiation, and acoustics.

∇²A + k²A = 0

Where,

• ∇² = Laplacian
• k = Wave Number
• A = Amplitude

Also Read: Oscillations and Waves Important Notes

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Derivation of Helmholtz Equation

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The equation of wave is given by,

(∇² – 1/c² ∂²/∂x²) u(r, t) = 0 (Wave Equation) (Eq.1)

After separating the variables, we get,

u(r, t) = A(r) T(t) (Eq.2)

Now substituting (2) in (1):

∇² A/A = 1/c² T*d² T/dt²

In this equation, the expression on LHS depends on r and RHS depends on t. But the equations are valid only if both the sides are equal to some constant value.

After solving linear partial differential equations by separating the variables we will get 2 equations – one for A (r) and other for T (t).

∇² A/A = - k² (Eq.3)

And, 1/c² T. d²T/dt² = - k²  s (Eq.4)

After rearranging we get the Helmholtz equation which is:

Where,

-k² = Separation constant

Also Read:

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Derivation of Helmholtz Free Energy Equation

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The Helmholtz function is given by,

F = U - TS

Where,

U = Internal energy

T = Temperature

S = Entropy

Now, F_i = Initial Helmholtz function

And, F_r = Final function

During the isothermal reversible process, when the temperature is constant the work done will be:

W ≤ F_i - F_r

As per the above statement, the Helmholtz function is converted into work. Therefore, this function is also known as free energy in thermodynamics.

Suppose an isolated system acquires “δQ” heat from surroundings, while the temperature is constant.

So,

Entropy gained by the system will be “dS”

& Entropy lost by surroundings will be “δQ/T”

As per the 2^nd law of thermodynamics, net entropy = positive

From Classius inequality:

dS - δQ/T ≥ 0

dS ≥ δQ/T

Now, multiply both the sides by T,

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, we get:

W ≤ T (S_r - S_i) - (U_r - U_i)

W ≤ (U_i - TS_i) - (U_r - TS_r)

Now, if we will see the equation, the term (U_i - TS_i) is the initial Helmholtz function and the term (U_r - TS_r) is the final Helmholtz function.

Therefore, we can say that:

W ≤ F_i - F_r

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Helmholtz Equation Thermodynamics

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Gibbs free energy is a function of temperature & pressure and it is given by,

G = G (T, P)

And, G(T) = H(T) - T S(T)

Where,

H = Enthalpy

S = Entropy

Now, divide LHS and RHS by T:

G(T)/T = H(T)/T - T S(T)/T

Now, doing the partial differentiation on both the sides:

At constant pressure, P

(∂G(T)/T)p = - H(T)/2 + 1/T (∂H(T)P /∂T - (∂S(T)/∂T)P

Since 1/T (∂H(T))p /∂T and (∂S(t)/∂t)P are equal you can cancel them out. After cancellation you will get the equation as:

(∂S/∂T)P = CP (T)/T = 1/T (∂H/∂T)P

CP(T) = (∂S/∂T)P

(∂S/∂T)P = - H/T²

This known as the Gibbs-Helmholtz equation in thermodynamics.

Also Read: Standard Enthalpy of Form

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Applications of Helmholtz Equation

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Helmholtz equation is important for various applications. Some of them are as follows:

1. It is used in seismology which is the scientific study of earthquakes and elastic waves.
2. Exlplaining and analysing natural disaster like Tsunamis.
3. This equation also plays an important role in Medical imaging.
4. Through this equation Volcanic eruptions can be explained and predicted.
5. This equation is also important for the calculation of Electromagnetism.
6. CHELS: CHELS stands for combined Helmholtz equation-least squares. This method is used to recreate acoustic radiation from any arbitrary object.
7. Gibbs-Helmholtz equation: It is used for calculating the change in enthalpy by using the change in the Gibbs energy when the temperature is different at constant pressure.

Also Read:

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Things to Remember

• Helmholtz Equation is named after Hermann von Helmholtz. It is a linear, partial, differential equation. 
• It is used in Physics and Mathematics. Its mathematical formula is : ∇²A + k²A = 0
• Helmholtz Free energy can be defined as the work done, extracted from the system, keeping the temperature and volume constant. 
• Gibbs Helmholtz Equation has applications in the calculation of temperature change effect on the equilibrium constant and the calculation of enthalpy change for reactions when the temperature is not 298K. 

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