The Entropy as a State Function

Entropy is a state function because its value depends only on the initial and final states of a system, not the path taken between them. When an arbitrary reversible process between states A and B is analyzed using the integral of dq/T, this integral yields the same value regardless of whether the process follows path I or path II. This path independence is the defining characteristic of a state function, similar to internal energy.

A reversible cyclic process between two states can be broken into many small Carnot cycles. Each Carnot cycle contains two isothermal processes where heat is exchanged and two adiabatic processes where no heat is exchanged. For each cycle, the summation of dq/T equals zero. When these cycles are combined, the integral of dq/T over any complete path yields the same entropy change, proving path independence.

The integral of dq/T defines the entropy change between two states. During reversible isothermal processes in a Carnot cycle, the ratio of heat exchanged to temperature remains constant. When dq/T terms are summed over a complete cycle, they equal zero. This mathematical relationship establishes that entropy change depends only on the initial and final states, not the specific path or intermediate steps taken.

As a reversible cyclic process is divided into smaller and smaller steps, the summation sign converts into an integral. This mathematical transformation allows for precise calculation of entropy changes over continuous paths. The integral of dq/T over the complete cycle still equals zero, and when the cycle is split into two distinct paths between states A and B, each path integral yields identical entropy changes.

In a Carnot cycle, two of the four processes are reversible and adiabatic, meaning no heat is exchanged during these steps. Since entropy change is defined by the integral of dq/T, adiabatic processes contribute zero to this integral because dq equals zero. This allows the overall entropy change to depend only on the two isothermal processes, simplifying the analysis of path independence.

Reversible processes are essential because they allow the system to be analyzed using well-defined thermodynamic relationships, particularly the Carnot cycle framework. For reversible processes, the integral of dq/T between two states yields a unique value independent of the path taken. Irreversible processes do not satisfy these conditions, so entropy's state function property is rigorously demonstrated only through reversible pathways.

When a reversible process between states A and B follows two different paths, the overall integral of dq/T separates into two parts corresponding to each path. Mathematical simplification shows both path integrals evaluate to the same quantity. This equality demonstrates that entropy change between A and B is identical regardless of which path is taken, establishing entropy as a state function like internal energy.