Derive ΔG = ΔH − TΔS. Why useful work of system is called free energy?

Derivation of ΔG = ΔH − TΔS

The equation ΔG = ΔH − TΔS is the Gibbs free energy equation. It is derived from the first and second laws of thermodynamics.

First Law of Thermodynamics: The change in internal energy (ΔU) of a system is equal to the heat (q) added to the system minus the work (w) done by the system.
ΔU = q − w

Second Law of Thermodynamics: For a reversible process, the change in entropy (ΔS) is the heat absorbed (q_rev) divided by the absolute temperature (T).
ΔS = q_rev/T
q_rev = TΔS

We can divide the total work (w) into two types: expansion work (PΔV) and non-expansion work (w_non−exp). So we can write,
w = PΔV + w_non−exp

Substitute the value of w into the first law equation, we get-
ΔU = q − (PΔV + w_non−exp)
or, q = ΔU + PΔV + w_non−exp

At constant pressure and for a reversible process, the change in enthalpy (ΔH) is defined as the heat absorbed (q_p), which is equal to the change in internal energy plus the expansion work.
ΔH = q_p = ΔU + PΔV

Since, the heat absorbed in a reversible process is
q_p = TΔS

We can rewrite the equation for q at constant pressure and for a reversible process:
q_rev = TΔS = ΔU + PΔV + w_non−exp

Since, ΔH = ΔU + PΔV
We can substitute ΔH into the equation:
TΔS = ΔH + w_non−exp

Rearranging this to solve for the maximum non-expansion work, which is the useful work:
−w_non−exp = ΔH − TΔS

The Gibbs free energy change (ΔG) is defined as the negative of the maximum non-expansion work that can be done by a system at constant temperature and pressure.

ΔG = −w_non−exp

Substituting the value of −w_non−exp with ΔG into the above equation, we get the final form:
ΔG = ΔH − TΔS

This equation is also known as the Gibbs-Helmholtz equation.

Why Useful Work of a System Is Called Free Energy ?

The term free energy is used because it represents the portion of a system's total energy that is free or available to do useful, non-expansion work. This useful work can be electrical, mechanical, or chemical, but it does not include the work done by the system to expand against the surrounding pressure.

Think like this to understand free energy: Your total salary (enthalpy) is the energy you receive. However, some of that money is taken out for taxes and other mandatory deductions (entropy, or energy lost as heat). The money left in your bank account is the free energy—the amount you have available to spend on things you want or need.

The total energy of a system is given by its enthalpy (ΔH). However, not all of this energy can be converted into useful work. Some of it is inevitably lost as heat, which is related to the system's entropy (TΔS).

Therefore, the Gibbs free energy (ΔG) represents the difference between the total energy (ΔH) and the energy that is unavailable to do work (TΔS). It's the maximum amount of free or usable energy that a system can provide at a constant temperature and pressure.