## Free Energy in Thermodynamics

**Free energy**, in thermodynamics, is an energy term of the system and a state function in a thermodynamic equilibrium reaction. Entropy formula measure the unavailable energy for doing useful work but free energy is the calculation of internal energy or enthalpy change that available for doing useful work. The property free energy measures in two forms in thermodynamics reaction, first are Helmholtz free energy or work function (A), and the second is Gibbs free energy (G).

### Definition of Gibbs Free Energy

This is a thermodynamics property defined as,

G = H – TS

Since H and TS are energy terms, G is also energy terms, further heat or enthalpy of reaction, temperature, and entropy are all state functions and perfectly differential quantities. Therefore, G also a state function and perfectly differential quantity.

### Gibbs Free Energy and Work Done

Let us consider a reversible isothermal and isobaric change of the system, thus the change of G

ΔG = ΔH – TΔS

But, ΔH = ΔU + PdV, for isobaric process.

∴ ΔG = ΔU + PdV – TΔS

Again, TΔS = q (heat) = ΔU + w, and the work may be partially mechanical and partially non-mechanical or fully mechanical or fully non-mechanical.

∴ ΔG = ΔU + PΔV – ΔU – w

or, – ΔG = w – PΔV = w_{non-mechanical}

This formula signifies that decreases of G equal to the non-mechanical work done by the system in the reversible isothermal isobaric process.

### Calculation of Gibbs Free Energy Change

We have by the definition of Gibbs free energy,

G = H – TS = U + PV – TS

For a small change of the system,

dG = dU + PdV + VdP – TdS – SdT

But TdS = q = dU + PdV, when the work is mechanical work only.

Then, dG = VdP – sdT

This is another basic thermodynamic equation.

Two cases of this equation may be considered,

- When the process is reversible isothermal, dT = 0 and dG = VdP.
- When the process is reversible isobaric, dP = 0, and dG = – SdT.

### Definition of Helmholtz Free Energy

This is a thermodynamic property defined as,

A = U – TS

Since U and TS are energy terms, A is also energy terms. Further internal energy, temperature, and entropy are all state functions and perfectly differential quantities. Therefore, A also a state function and perfectly differential quantity.

### How to Measure of Work Function?

Let us consider an isothermal reversible change of the system, thus the change of A

ΔA = ΔU – TΔS

But, TΔS = q = ΔU + w_{max}

Because of reversible isothermal process yields maximum work. Therefore,

ΔA = ΔU – (ΔU + w_{max})

or, – ΔA = w_{max}

This formula signifies that decreases of work function equal to the maximum work done by the system. Therefore, work function measures of workability of a system. When the system works, work function decreases,

- If the system isothermal reversible, its value will be equated with w
_{max}. - If the same change is not isothermal and reversible, then A will decreases to the same extent as the system works but – ΔA ≠ w
_{max}. - When the process is irreversible, there will be some drop of A but it will exceed the output work.

### Formula for Change in Helmholtz free energy

We have from the definition of Helmholtz free energy formula,

A = U – TS

For a small change of the system,

dA = dU – TdS – SdT

But TdS = q (heat) = dU + PdV, when the work is mechanical work only.

Then, dA = – PdV – SdT

This is the basic thermodynamic equation.

Two cases of this equation may be considered,

- When the process is reversible isothermal, dT = 0 and dA = – PdV.
- When the process is reversible isochoric, dV = 0, and dA = – SdT.