# Helmholtz Free Energy

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.