What is free energy?

Free energy is (Delta G = Delta H – T Delta S) — but where does it comes from and why does a negative (Delta G) mean a reaction will happen?

Total entropy

The Second Law of Thermodynamics states that entropy (of everything) must increase for a reaction or process to happen. Since we’re thinking in terms of systems and surroundings, then the entire universe is just the sum of both of these:

[Delta S_{universe}=Delta S_{system:}+Delta S_{surroundings}]

Entropy of systems

The entropy change of a chemical reaction can be calculated from standard molar entropies. These can be looked up in a table, and then the difference between products and reactants found:

[Delta S_{system}=sum nS_{products}-sum nS_{reactants}]

It’s important that if you do this, you need to multiply by the number of moles of reactant involved.

Entropy of surroundings

The definition of entropy for a reversible process is the heat transferred divided by temperature:

[frac{q_{rev}}{T}=Delta S]

And the heat transferred for a reaction is the enthalpy change:

[-frac{Delta H_{system}}{T}=Delta S_{surroundings}]

Note: 1) This is the entropy change of surroundings, and not related to the entropy change of the system and (2) the negative sign, as we’re looking at surroundings, which have absorbed the heat released by a reaction, or provided the heat consumed by it. So the sign of the heat transfer is opposite.

Free energy

To get the overall change of entropy for the whole universe, these need to be added together:

[Delta S_{universe}=Delta S_{system}-frac{Delta H_{system}}{T}]

This rearranges to the equation for Gibb’s free energy:

[Delta G=-TDelta_{universe:}=Delta H_{system}-TDelta_{system}]

This gives us a function for the entropy change of the entire universe (system + surroundings) that depends only on the system and what it does to the surroundings.

Temperature dependence

Looking at the equation for free energy, there’s a clear temperature dependence:

[Delta G=Delta H-TDelta S]

Whether ΔG is positive or negative will depend on the sign (+ or -) of the enthalpy and entropy changes, and temperature: