Free Energies and Legendre Transforms
Free energies are thermodynamic potentials obtained by Legendre-transforming the internal energy, each minimized at equilibrium under the constraints natural to it.
Definition
A free energy is a thermodynamic potential formed by Legendre-transforming the internal energy with respect to one or more extensive variables, yielding a function of the conjugate intensive variables that is minimized at equilibrium under the corresponding constraints.
Scope
This topic covers the construction of the enthalpy, Helmholtz free energy, Gibbs free energy, and grand potential as Legendre transforms of the internal energy, their natural variables, the differential forms that define their conjugate pairs, and the extremum principles that select equilibrium states. The interpretation of free energy as available work is included.
Core questions
- How does a Legendre transform exchange an extensive variable for its conjugate intensive variable?
- Why is each free energy minimized at equilibrium for its own natural variables?
- How does free energy quantify the maximum work extractable under given constraints?
- What are the natural variables of the enthalpy, Helmholtz, Gibbs, and grand potentials?
Key concepts
- Legendre transform and conjugate variables
- Enthalpy and natural variables
- Helmholtz free energy at fixed temperature and volume
- Gibbs free energy at fixed temperature and pressure
- Grand potential and the chemical potential
Clinical relevance
Free energies determine the spontaneity and equilibrium of chemical reactions and phase changes, set the maximum useful work obtainable from a process, and provide the bridge to statistical mechanics through the partition function.
History
The free-energy concept arose with Helmholtz's 1882 work on the energy available for work and Gibbs's earlier formulation of the function now bearing his name, establishing the potentials minimized under common experimental constraints.
Key figures
- J. Willard Gibbs
- Hermann von Helmholtz
Related topics
Seminal works
- callen1985
Frequently asked questions
- Why is it called 'free' energy?
- It measures the portion of a system's energy that is free to do useful work under the given constraints, as opposed to the part bound up in the entropy term that cannot be extracted as work at finite temperature.