

Significance of Free Energy
Many real-life based open systems in thermodynamics exist as the process of exchanging heat and work with respect to the environment compared to closed systems. One of the most common examples of that would be of the living systems that attains an inherent reduction in their entropy as they grow mature. The Helmholtz function is the second thermodynamic potential used to measure the work done in a closed system at a constant temperature and volume. The other three thermodynamic potentials are internal energy, enthalpy, and Gibbs free energy.
Helmholtz Free Energy Equation
The equation:
A = U - TS .. eqⁿ1.
U is the internal energy in Joules, T is the absolute temperature in Kelvin, and S is the final entropy in Joules per Kelvin (JK⁻¹).
A stands for 'Arbeit' meaning work and is minimized to the equilibrium. Hermann von Helmholtz formulated it.
Upon differentiating the Helmholtz function, one obtains:
dA = dU - TdS - SdT
As the Thermodynamics’ second law states,
dA = TdS - pdV - SdT
Therefore, dA = -pdV - SdT
For A (T, V) the total differential equation can be:
dA = (\[\frac{∂A}{∂T}\])\[_{v}\] dT + (\[\frac{∂A}{∂V}\])\[_{T}\] dV
To keep the integrity of the thermodynamics' laws in open systems, parameters like dimensions of energy, known as thermodynamic potentials, are described in the system. These formulas take references from Helmholtz and Gibbs Free energy to explain the entire work done in the system. On separating a system from its heat reservoir, it is no longer a part of the system, and therefore the energy change (ΔU) can be represented as:
ΔU = Q - W ....eqⁿ2.
Q is the heat absorbed, and W is the work done, that is generally not state functions, and are path-dependent. But for a reversible isothermal process, the heat would be directly proportional to maximum work,
-W\[_{max}\] = ΔU - TΔS (where ΔS stands for the entropy change in the system held at a constant temperature like a battery) ...eqⁿ3.
In this system, the entropy change of the surroundings is not considered, thus making the U, T, and S as state functions.
Relationship Between Helmholtz Free Energy & Gibbs Free Energy
The above equation's overall work can be considered lesser than that of the ideal maximum that implies with W ≤ −ΔF; the whole process can be a reversible one, thus limiting the case. As the Helmholtz free energy acquires its minimum value, the system becomes an equilibrium, from which no more work can be deduced. Therefore, this equilibrium condition having maximum entropy is the prerequisite for the least Helmholtz free energy at a constant temperature. The Helmholtz free energy concept takes place inside a container and rigid walls. The overall process takes place under continuous pressure, instead of constant temperature. A vegetable growing in a garden takes place at normal atmospheric pressure, and the temperature keeps changing.
Taking from ...eqⁿ3,
-W\[_{max}\] = ΔU - TΔS
If the system is a natural process that involves huge volume change (for steaming of water), the work done by the atmospheric water vapour to push through the surrounding pressure is described as PΔV that can be combined with the total work done:
W\[_{max}\] = W’\[_{max}\] + PΔV (atmospheric work)
Here the W’\[_{max}\] refers to the maximum work done in a system of constant T and P, and PΔV represents the atmospheric work.
Upon comparing with eqⁿ1, and substituting the PΔV in the above equation gives,
-W\[_{max}\] = ΔU + PΔV - TΔS
Which leads to the foundation of the Gibbs Energy,
G = U + PV - TS
That can also be defined as the following equation for isothermal systems and at constant pressure,
ΔG = ΔU + PΔV - TΔS
Here, the negative value represents the reversible maximum work that can be extracted from the system, except for the atmospheric work done. Similar to the Helmholtz Free Energy case, when the Gibbs energy reaches its minimum state, the system moves into an equilibrium state, from where no further work can be extracted.
Here the entropy can further decrease, as long as it is counteracted by the combined value of ΔU + PΔV.
Therefore, for the Gibbs free energy for open systems that are at constant temperature and pressure, the direction for spontaneous change is in the state of lower free energy, often represented by a ball rolling downhill. A real-life example can be of a vehicle battery that, in its charged state, has the maximum Gibbs Energy and, in its discharged state, has a minimum Gibbs Energy. The total work between the two states can be defined as the charge, which can be obtained from the battery at constant temperature and pressure.
The Gibbs Free Energy is represented in an equation as: (taking from eqⁿ1)
G = U - TS + PV
Where TS is the absolute temperature and Final Entropy
PV represents absolute pressure and Final Volume
In the process of Glucose oxidation, the Gibbs free energy change can be calculated as 2870 kJ, which is crucial for energy synthesis in all living cells.
FAQs on Helmholtz Free Energy
1. What is meant by Helmholtz free energy in Physics?
Helmholtz free energy, denoted by the symbol A, is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal) and constant volume (isochoric). The change in Helmholtz energy during a process represents the maximum amount of work that the system can perform.
2. How do you calculate Helmholtz free energy?
The Helmholtz free energy (A) of a system is calculated using the formula:
A = U - TS
Where:
- U is the internal energy of the system.
- T is the absolute temperature in Kelvin.
- S is the entropy of the system.
3. What is the physical significance of Helmholtz free energy?
The primary physical significance of Helmholtz free energy lies in its connection to spontaneity and work. For a process occurring at constant temperature and volume, a decrease in Helmholtz free energy (ΔA < 0) indicates that the process is spontaneous. The magnitude of this decrease, -ΔA, is equal to the maximum possible work that can be extracted from the system during that process. For this reason, it is often referred to as the 'work function'.
4. How does Helmholtz free energy differ from Gibbs free energy?
While both are measures of 'useful' energy, they apply under different conditions:
- Conditions: Helmholtz free energy (A) is used for processes at constant temperature and volume. Gibbs free energy (G) is used for processes at constant temperature and pressure, which is more common in chemistry labs.
- Type of Work: The change in Helmholtz energy (ΔA) represents the maximum total work (including any pressure-volume work). The change in Gibbs energy (ΔG) represents the maximum non-expansion work (e.g., electrical work in a battery).
5. Under what specific conditions is the concept of Helmholtz free energy most useful?
The concept of Helmholtz free energy is most useful in situations where reactions or processes occur in a closed, rigid container, ensuring the volume remains constant. Examples include:
- Analysing processes inside a bomb calorimeter.
- Deriving fundamental equations of state for fluids and solids in statistical mechanics.
- Theoretical modelling of systems where volume, not pressure, is the natural independent variable.
6. Is Helmholtz free energy a state function? Explain why.
Yes, Helmholtz free energy is a state function. This is because its value is determined by other state functions: internal energy (U), temperature (T), and entropy (S). A state function's value depends only on the current equilibrium state of the system, not on the path taken to reach that state. Therefore, the change in Helmholtz energy (ΔA) between two states is always the same, regardless of the process connecting them.
7. What are some real-world applications of Helmholtz free energy?
While Gibbs energy is more common in applied chemistry, Helmholtz energy is fundamental in several areas:
- Fluid Dynamics: It is used to develop highly accurate equations of state for industrial fluids, predicting their behaviour under different conditions.
- Materials Science: It helps in understanding the elasticity and deformation of solids.
- Explosives Analysis: The energy released during a detonation in a confined space is best analysed using Helmholtz free energy since the process occurs at a nearly constant volume.
8. Can the Helmholtz free energy of a system be negative?
Yes, the absolute value of Helmholtz free energy (A) can be negative. Its value is relative and depends on the zero points defined for internal energy (U) and entropy (S). The more physically meaningful quantity is the change in Helmholtz free energy (ΔA). A negative ΔA signifies that a process is spontaneous and can proceed without external energy input, which is a key principle in thermodynamics.

















