Helmholtz free energy

Helmholtz free energy is the thermodynamic potential F = U − TS, where U is internal energy, T is absolute temperature, and S is entropy. At fixed temperature and volume, its decrease characterizes spontaneous processes and its minimum identifies equilibrium.

Thermodynamic potential: A state function that combines energy and other state variables to characterize equilibrium or transformations under specified constraints. Helmholtz free energy is one potential tailored to fixed temperature and volume.
Internal energy: The total microscopic kinetic and potential energy contained within a system. It is the energy term from which the entropy-weighted contribution is subtracted.
Canonical ensemble: A statistical-mechanical ensemble describing systems in thermal contact with a reservoir at fixed temperature and fixed particle number. Its partition function determines the Helmholtz free energy directly.
Gibbs free energy: The thermodynamic potential G = H − TS, commonly used for systems at fixed temperature and pressure. Its natural constraints are fixed pressure and temperature, unlike Helmholtz free energy’s fixed volume and temperature.
Fundamental thermodynamic relation: A differential relation connecting a system’s energy changes to entropy, volume, particle number, and their conjugate variables. It yields dF = −S dT − P dV for a simple closed system.
Entropy: A state function measuring the dispersal of energy among microscopic configurations. Its product with temperature accounts for energy unavailable for useful work under fixed-temperature conditions.
Partition function (statistical mechanics): A weighted sum over a system’s microscopic states that encodes its equilibrium statistical properties. For the canonical ensemble, F = −kBT ln Z.
Enthalpy: A thermodynamic state function equal to internal energy plus pressure times volume. Enthalpy is useful at fixed pressure but does not include the explicit entropy subtraction of Helmholtz free energy.
Thermodynamic equilibrium: A state in which macroscopic properties remain stable and no spontaneous change occurs under the system’s constraints. At fixed temperature and volume, equilibrium minimizes Helmholtz free energy.
Temperature: A physical quantity that sets the direction of thermal energy transfer and characterizes thermal equilibrium. It scales the entropic term in Helmholtz free energy.

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