Fundamental thermodynamic relation

The differential relation dU = T dS − P dV + μ dN expresses changes in a simple system’s internal energy through entropy, volume, particle number, and their conjugate variables.

Internal energy: The total microscopic kinetic and potential energy of a system, excluding its bulk motion and external potential energy. The relation expresses how this energy changes when the system’s state changes.
Thermodynamic state function: A property whose value depends only on a system’s state, not on the process used to reach it. Internal energy and entropy are state functions, so their differentials describe state changes.
Equation of state: A relation among thermodynamic state variables that describes a material’s equilibrium states. Derivatives of the fundamental relation can supply pressure and temperature equations of state.
Enthalpy: A thermodynamic potential defined as internal energy plus pressure times volume. Its natural variables replace volume with pressure, making it useful at constant pressure.
Entropy: A thermodynamic state function measuring energy dispersal and the number of microscopic configurations compatible with a state. Its differential is weighted by temperature in the energy relation.
Exact differential: A differential of a state function whose integral depends only on its endpoints. The energy differential is exact, unlike heat and work along a process.
Maxwell relations: Identities linking thermodynamic response derivatives, derived from equality of mixed partial derivatives of potentials. Its exact differentials generate identities between measurable derivatives.
Helmholtz free energy: A thermodynamic potential defined as internal energy minus temperature times entropy. It replaces entropy with temperature as a natural variable for fixed-temperature systems.
Temperature: A thermodynamic state variable that determines the direction of heat transfer and is conjugate to entropy. It is the coefficient multiplying entropy change.
Thermodynamic potential: A state function, such as internal energy or enthalpy, used to characterize equilibrium under specified constraints. The relation identifies internal energy’s natural variables and conjugate coefficients.

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