Thermodynamic Potentials and Relations
Thermodynamic potentials repackage the laws of thermodynamics into energy-like functions whose natural variables and derivatives yield the equations of state and the Maxwell relations.
Definition
Thermodynamic potentials are state functions, each natural in a particular set of independent variables, whose minimization characterizes equilibrium and whose partial derivatives generate the system's equations of state and material response functions.
Scope
This area covers the thermodynamic potentials -- internal energy, enthalpy, Helmholtz and Gibbs free energies, and the grand potential -- obtained from one another by Legendre transforms, together with the Maxwell relations that follow from their second derivatives. Response functions such as heat capacities and compressibilities, thermodynamic stability and convexity conditions, the chemical potential, and the conditions for phase equilibrium are included. The microscopic computation of these potentials from partition functions belongs to the statistical-ensembles area.
Sub-topics
Core questions
- How does the Legendre transform generate the family of thermodynamic potentials?
- Why is each potential minimized at equilibrium for its natural set of held variables?
- How do the Maxwell relations connect seemingly unrelated derivatives of state variables?
- What stability and convexity conditions must a thermodynamic potential satisfy?
Key concepts
- Internal energy, enthalpy, Helmholtz and Gibbs free energies
- Legendre transforms and natural variables
- Maxwell relations
- Response functions: heat capacity, compressibility, expansivity
- Chemical potential and equilibrium conditions
Key theories
- Legendre transform structure of potentials
- Each thermodynamic potential is a Legendre transform of the internal energy that exchanges an extensive variable for its conjugate intensive variable, giving a function naturally minimized under the corresponding constraints.
- Maxwell relations
- Because mixed second partial derivatives of a potential are equal, derivatives of conjugate thermodynamic variables satisfy equalities such as that between the temperature dependence of volume and the pressure dependence of entropy.
Clinical relevance
Thermodynamic potentials are the working tools of physical chemistry, materials science, and chemical engineering, predicting reaction spontaneity through the Gibbs free energy, phase diagrams through the chemical potential, and material responses through heat capacities and compressibilities.
History
Gibbs's monumental 1870s treatise on heterogeneous equilibrium introduced the free-energy functions and the chemical potential, building on Helmholtz's free energy and Maxwell's relations to give thermodynamics its modern analytical structure.
Key figures
- J. Willard Gibbs
- Hermann von Helmholtz
- James Clerk Maxwell
Related topics
Seminal works
- gibbs1876
- callen1985
Frequently asked questions
- Why are there several different free energies?
- Each is suited to a different set of controlled variables: the Helmholtz free energy is minimized at fixed temperature and volume, the Gibbs free energy at fixed temperature and pressure, so the right potential to minimize depends on which quantities the experiment holds constant.
- What makes the Maxwell relations useful?
- They let hard-to-measure quantities, such as how entropy changes with pressure, be replaced by easily measured ones, such as how volume changes with temperature, turning experimentally inaccessible derivatives into accessible ones.