How Does the Helmholtz Equation Apply in Waves and Optics?
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
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.
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 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.
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.
FAQs on Helmholtz Equation Explained: Meaning, Formula & Solutions
1. What is the Helmholtz equation and what does it represent?
The Helmholtz equation is a linear partial differential equation that describes the spatial behaviour of time-independent waves. It is written as ∇²A + k²A = 0, where ∇² is the Laplacian operator, A is the amplitude (eigenfunction), and k is the wavenumber (related to the eigenvalue). It is derived from the general wave equation and is used to analyse phenomena where the time variation can be separated, such as in acoustics, seismology, and electromagnetism.
2. What is Helmholtz free energy in thermodynamics?
Helmholtz free energy (F) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (T) and volume (V). It is defined by the formula F = U - TS, where U is the internal energy of the system and S is its entropy. The decrease in Helmholtz free energy during a process represents the maximum amount of work that can be extracted from the system.
3. What is the main difference between Helmholtz free energy and Gibbs free energy?
The primary difference lies in the conditions under which they measure available work. Helmholtz free energy (F) applies to systems at constant temperature and volume, representing the total work available. In contrast, Gibbs free energy (G) applies to systems at constant temperature and pressure, representing the maximum non-PV (pressure-volume) work available. This makes Gibbs energy more suitable for chemical reactions in open beakers, while Helmholtz energy is ideal for processes in rigid, sealed containers.
4. What are some important applications of the Helmholtz equation and Helmholtz free energy?
These concepts have wide-ranging applications in different fields of physics and engineering:
- Helmholtz Equation: Used in modelling wave phenomena like acoustic radiation (e.g., speaker design), seismology (studying earthquake waves), medical imaging (e.g., ultrasound), and analysing electromagnetic waves in waveguides.
- Helmholtz Free Energy: Essential for predicting equations of state for pure substances, which is critical in industrial applications like refrigeration. It is also used in computational chemistry and in machine learning as a basis for certain types of autoencoders.
5. How is the Helmholtz equation derived from the more general wave equation?
The Helmholtz equation is derived using a technique called separation of variables. Starting with the standard wave equation, ∇²u = (1/c²) ∂²u/∂t², we assume the solution u(r, t) can be separated into a space-dependent part A(r) and a time-dependent part T(t). By substituting u(r,t) = A(r)T(t) into the wave equation and rearranging, the equation separates into two independent ordinary differential equations. The spatial part results in the Helmholtz equation, ∇²A + k²A = 0, effectively removing time from the wave analysis.
6. Why is the change in Helmholtz free energy (ΔF) often negative for spontaneous processes?
For a spontaneous process occurring at constant temperature and volume, the system moves towards a state of thermodynamic equilibrium. According to the second law of thermodynamics, this means the system seeks to minimise its available energy for doing work. The Helmholtz free energy (F) quantifies this available energy. Therefore, a spontaneous change leads to a decrease in F, making the change, ΔF = F_final - F_initial, a negative value. When ΔF reaches its minimum (and is no longer decreasing), the system is at equilibrium.
7. How does the Helmholtz equation relate to quantum mechanics?
The Helmholtz equation is mathematically identical to the time-independent Schrödinger equation, which is a cornerstone of quantum mechanics. The Schrödinger equation is written as (-ħ²/2m)∇²ψ + Vψ = Eψ. If the potential V is constant (or zero), this equation takes the exact form of the Helmholtz equation. In this context, the eigenfunction 'A' corresponds to the wavefunction 'ψ', and the eigenvalue 'k²' corresponds to the energy 'E' of the particle. This mathematical link highlights the wave-like nature of particles described in quantum theory.
8. What is the distinction between the Helmholtz equation and the Gibbs-Helmholtz equation?
These are two different, though related, equations in thermodynamics. The Helmholtz equation (∇²A + k²A = 0) is a wave equation in physics. The Gibbs-Helmholtz equation, on the other hand, is a thermodynamic relationship used to calculate how the Gibbs free energy (G) of a system changes with temperature (T) at constant pressure. It connects the change in the G/T ratio to the system's enthalpy (H), and it does not involve wave mechanics or spatial derivatives.
