Enthalpy cycle (${\sum }_{\mathrm{cycle}}\Delta H=0$)

An enthalpy cycle is a collection of compatible thermodynamic processes that returns the entire material inventory to its initial thermodynamic state. Its signed enthalpy changes sum to zero. This supplies a consistency check and lets a missing change be recovered from the others.

Consider three abstract thermodynamic states $A,B,C$ of the same conserved material inventory at common temperature and pressure. Choose their molar enthalpies, on the same one-mole basis, asThese are constructed values, not material measurements. Endpoint subtraction givesThus $+18-25=-7$: the two-step route and the direct route have the same molar enthalpy change. Closing the route with $C\to A$ gives $+18-25+7=0$. Shifting every $h$ by the same arbitrary reference constant leaves all these differences unchanged.

The short proof is cancellation:Adding any finite number of consecutive endpoint differences works the same way. For a closed enthalpy cycle the final $H_A$ cancels the initial $H_A$. A nonzero experimental residual can therefore expose mismatched states or measurement error; it does not identify which individual input is responsible.