The Second Law of Thermodynamics and the Carnot Cycle

Two central theorems connect the Carnot cycle directly back to the Second Law of Thermodynamics, and both can be stated together in a single, classic sentence: 'The Carnot engine is the most efficient heat engine. Also, all Carnot engines operating between the same two temperatures have the same efficiency, irrespective of the nature of the working substance.'

This single statement is actually making two distinct, important claims:

1. The Carnot engine is the most efficient possible heat engine — no heat engine operating between a given pair of temperatures $T_H$ and $T_C$ can EVER exceed the Carnot efficiency, $\eta = 1 - T_C/T_H$ (Eq. 4.29); the Carnot cycle represents the theoretical upper bound that the Second Law itself allows.

2. The efficiency of a Carnot engine depends only on temperature, never on the working substance used. Whether the Carnot cycle uses an ideal monatomic gas, a diatomic gas, or in principle any other working substance at all, its efficiency between two given temperatures $T_H$ and $T_C$ is always exactly $1 - T_C/T_H$ — nothing about the substance's own specific properties (its $\gamma$, its $C_V$, its molecular structure) enters the final efficiency formula at all. …