Carnot cycles and absolute temperature

A heat engine absorbs magnitude $Q_h>0$ from a hot reservoir and rejects magnitude $Q_c>0$ to a cold reservoir. Reservoirs are ideal bodies whose temperatures remain fixed during the specified exchanges. Over a cycle the working substance returns to its initial state, giving zero net internal-energy change. The net heat into it is $Q_h-Q_c$, so work done on it is the negative of that quantity. Work output is positive when the device operates as an engine.

A Carnot cycle has two reversible isothermal exchanges and two reversible adiabatic connections. During each isotherm the working substance differs only infinitesimally in temperature from its reservoir; the adiabatic stages exchange no heat. Every step must be reversible, including mechanical boundaries and thermal contacts. A cycle drawn with two constant-temperature curves is not automatically Carnot if friction or finite thermal gaps remain. The device must also complete the cycle; a one-time expansion that consumes a changing resource is a different situation.

Suppose an engine exceeded the efficiency of a reversible engine operating between the same two reservoirs. Use its work output to run the reversible device backward, scaling the devices so their work magnitudes cancel. The combined cyclic apparatus would transfer heat from cold to hot without external work or another compensating effect. That violates the Clausius statement of the second law. Reversibility is essential because the benchmark must be able to run backward with exactly reversed exchanges.

Applying the argument in both directions shows that all reversible engines between the same reservoirs have the same efficiency, independent of working substance. Cascading reversible engines through an intermediate reservoir then makes their heat ratios multiply. This multiplicative structure permits a thermodynamic temperature scale with heat ratio equal to the cold-to-hot temperature ratio. For the ordinary positive-temperature reservoirs considered here, use kelvin values; Celsius offsets destroy temperature ratios even when temperature differences are correctly represented.

A practical engine can lose useful work to friction, finite-temperature heat transfer, leakage, or other irreversibilities. Carnot efficiency is an upper bound under the stated reservoir conditions, not the predicted efficiency of any particular apparatus. A real heat source that cools substantially is not one fixed-temperature reservoir; its extraction process must be analyzed with changing temperatures or an appropriate availability calculation. Increasing hot temperature or decreasing cold temperature raises the ideal engine bound, subject to the physical system’s other constraints.

Running a reversible cycle backward produces a refrigerator. It absorbs heat from the cold reservoir, requires positive work on the device, and rejects their sum to the hot reservoir. Its coefficient of performance can exceed one because the numerator includes heat transported rather than only work energy created. This does not violate conservation. For a fixed positive temperature gap the refrigerator COP is finite; as the ideal gap tends to zero the formal COP diverges while the finite-rate engineering problem remains outside the reversible model.