State Function

A state function is a property of a thermodynamic system determined by its current state rather than by its history. In equilibrium thermodynamics, its change between two specified states depends only on those endpoints, regardless of the process connecting them. Internal energy, enthalpy, and entropy are important examples. By contrast, heat transferred and work performed describe processes, not properties possessed by a system at a particular state. This distinction makes it possible to calculate property changes using convenient hypothetical processes rather than reconstructing an actual process in detail. (ocw.mit.edu)

States, variables, and properties

A thermodynamic state is specified by a sufficient set of macroscopic properties. For a homogeneous, single-phase, simple compressible substance of fixed composition and amount, two independent intensive properties commonly determine the equilibrium state. Temperature and pressure, for example, can serve as independent coordinates in regions where they uniquely identify the state. Other properties can then be expressed as functions of those coordinates. The familiar relation between pressure, volume, and temperature is an equation of state. (ocw.mit.edu)

The terms state variable, state property, and state function overlap in usage. “Variable” often emphasizes a coordinate used to describe the state, while “function” emphasizes a property calculated from selected coordinates. Being a state function does not mean being constant: a property can change substantially as the state changes. Nor does path independence imply that the chosen variables are independent of one another; an equation of state constrains their permissible combinations. (ocw.mit.edu)

Mathematical characterization

For a state function $X$, the change from state $A$ to state $B$ is

Its infinitesimal change is an exact differential, denoted $dX$. Consequently, its line integral is independent of the path between fixed endpoints:

The second equation states that a property returns to its original value after a complete cycle. It does not require the heat or work exchanged during that cycle to vanish separately. (ocw.mit.edu)

If $X$ is a differentiable function of independent coordinates $x$ and $y$, then

For sufficiently smooth functions, equality of mixed partial derivatives gives a consistency condition on these coefficients. This mathematical structure underlies the Maxwell relations, which connect measurable properties through derivatives of thermodynamic potentials. (live.ocw.mit.edu)

Heat, work, and the first law

Heat and work are path-dependent process quantities. Heat is energy transferred because of a temperature difference; work is energy transferred through other mechanisms, such as expansion against an external pressure. Neither represents an inventory of “heat” or “work” stored inside a system. (ocw.mit.edu)

For a closed system with negligible changes in bulk kinetic and potential energy, the first law of thermodynamics can be written

where $Q$ is heat entering the system and $W$ is work done by it. The symbol $\delta$ distinguishes process increments from exact differentials. Some disciplines instead take work done on the system as positive and write $dU=\delta Q+\delta W_{\mathrm{on}}$. Although heat and work individually depend on the process, their signed combination gives the same internal-energy change for identical endpoints. (ocw.mit.edu)

Consider an ideal gas expanding between equilibrium states with the same temperature. Its internal energy is unchanged. A reversible isothermal expansion produces work and requires compensating heat input, whereas an insulated free expansion into a vacuum transfers neither heat nor work. These different exchanges are compatible with the same value of $\Delta U$. (ocw.mit.edu)

Entropy and reversible reference paths

The second law of thermodynamics establishes entropy as a state property. For a reversible process,

Here $T$ is absolute temperature. The reversible path is a computational reference: the actual process between the endpoints need not be reversible. Dividing reversible heat transfer by temperature produces an exact differential even though heat itself is path dependent. (ocw.mit.edu)

An irreversible process can generate entropy without transferring heat. Therefore, an adiabatic process is not necessarily isentropic. For an isolated system, entropy cannot decrease; for a reversible adiabatic process it remains constant. These statements concern entropy change and production, not the existence of a stored heat quantity. (ocw.mit.edu)

Thermodynamic potentials

Several central state functions are combined into thermodynamic potentials:

Here $F$ denotes Helmholtz free energy and $G$ denotes Gibbs free energy. Because these expressions combine state properties, they are also state functions. For a simple compressible system of fixed composition,

These relations connect neighboring equilibrium states; they are not formulas equating actual irreversible heat transfer with $T\,dS$. (live.ocw.mit.edu)

Such potentials organize equilibrium calculations under different constraints. Gibbs free energy is particularly useful at fixed temperature and pressure, including descriptions of chemical equilibrium. Their derivatives provide properties such as entropy and volume, while second derivatives yield response functions. (openstax.org)

Chemical calculations and reference conventions

In chemistry, enthalpy’s path independence gives Hess’s law: if a chemical reaction is expressed as a sum of steps, its enthalpy change equals the sum of the stepwise changes. Intermediate substances cancel, provided the endpoint amounts, phases, temperatures, and pressures match. Under constant pressure with only pressure–volume work, the heat transferred equals the enthalpy change. (openstax.org)

Tabulated quantities commonly use a standard state, a reference convention rather than a universal temperature. Changing an additive reference constant changes numerical property values but not differences calculated consistently. Thus, being a state function does not require an experimentally accessible absolute zero of that property. (old.goldbook.iupac.org)