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Enthalpy cycle 2026-10-18

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, as

h_{A} = 0, h_{B} = 18 kJ mo l^{- 1}, h_{C} = - 7 kJ mo l^{- 1} .

(1)

These are constructed values, not material measurements. Endpoint subtraction gives

Δ h_{A B} Δ h_{BC} Δ h_{A C} = h_{B} - h_{A} = + 18 kJ mo l^{- 1}, = h_{C} - h_{B} = - 25 kJ mo l^{- 1}, = h_{C} - h_{A} = - 7 kJ mo l^{- 1} .

(2)

Thus

+ 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.

Figure 1.
Two routes between the same thermodynamic states give the same enthalpy change
. Original abstract states with molar enthalpies $0$ , $18$ , and $- 7 kJ mo l^{- 1}$ on one common basis. Arrows show signed endpoint differences, not an activation energy curve, a reaction mechanism, or elapsed time. **Details**.

The short proof is cancellation:

(H_{B} - H_{A}) + (H_{C} - H_{B}) = H_{C} - H_{A} .

(3)

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.

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