Path function

A path function is a quantity associated with a thermodynamic process rather than with a thermodynamic state . Its value for a change of state depends on the particular path taken through state space, so it cannot be expressed as the difference of a single-valued function of state.

Path functions are commonly written with an inexact differential symbol $\delta$ to emphasize that there is no underlying state function whose differential it is. Canonical examples are heat $\delta Q$ and work $\delta W$.

Physical interpretation. Path functions quantify transfer across the boundary during a process—energy carried as heat, mechanical work, electrical work, or energy advected with matter in an open system . Because transfers depend on how the process is executed (with friction, with gradients, rapidly vs slowly, etc.), they are not fixed by endpoint states alone.

Key relations

• First law bookkeeping: the first law relates path-dependent transfers to the change in internal energy (a state function ) via $dU = \delta Q - \delta W$, with signs set by the work sign convention (and, for $P\,dV$ work, the pressure–volume work convention ).
• Cycles need not vanish: for a cycle one has $\Delta U=0$, but generally $\oint \delta Q \neq 0$ and $\oint \delta W \neq 0$ (indeed, they are equal in magnitude with consistent signs).
• Reversible limit: along a reversible process , heat transfer is tied to entropy by $\delta Q_{\mathrm{rev}} = T\,dS$, linking the path function $\delta Q$ to the state function entropy and the state variable temperature .
• Irreversibility constraint: in irreversible cycles the Clausius inequality implies $\oint \delta Q/T \le 0$, expressing that the same endpoints can be connected by paths with different heat/work transfers.