Free Energy Field Note
What Is Free Energy?
Free energy is the portion of a system's internal energy that can be converted into useful work at constant temperature and volume. It is the energy that is "free" — not tied up in the disordered thermal motion of the system.
$$F = U - TS$$
Where:
- F is the Helmholtz free energy
- U is the internal energy
- T is the temperature
- S is the entropy
Why "Free"?
The term "free" means available. The energy U is the total energy, but some of it is locked away in entropy — in the number of ways the system can rearrange itself. The product TS represents this locked energy. What remains, U - TS, is the portion you can actually extract as work.
At T = 0, F = U. Everything is available. At high T, the entropy term dominates. Most energy is unavailable.
The Minimum Principle
At constant temperature and volume, a system in contact with a heat bath spontaneously evolves to minimize its free energy. This is the analog of "energy is minimized" at zero temperature. Free energy replaces internal energy as the quantity that nature minimizes when heat can flow.
This principle explains everything from chemical reactions to phase transitions. A reaction proceeds if and only if the products have lower free energy than the reactants.
The Connection to the Partition Function
$$F = -k_B T \ln Z$$
This single equation connects the microscopic (Z, the sum over states) to the macroscopic (F, a measurable thermodynamic quantity). From F, you can derive every other thermodynamic observable by differentiation.
Gibbs Free Energy
For processes at constant temperature and pressure (the more common laboratory condition):
$$G = H - TS$$
Where H = U + PV is the enthalpy. G determines chemical reaction spontaneity at constant P and T. ΔG < 0 means the reaction proceeds forward.
Key Equations
- F = U - TS
- F = -k_B T ln Z
- dF = -S dT - P dV (fundamental relation)
- G = H - TS (constant P and T)
- ΔG = ΔH - TΔS (reaction spontaneity)
Practical Significance
Free energy is the currency of chemistry. Enzymes lower the activation free energy of reactions. Batteries store free energy in chemical bonds. Proteins fold to minimize their free energy. Life itself is a free-energy machine — it captures gradients of free energy and uses them to do work.
When ΔG = 0, the system is in equilibrium. Nothing net happens. This is the resting state of chemistry.