Carnot Cycle Explained
The Four Steps of the Carnot Cycle
The Carnot cycle begins with isothermal expansion. The working fluid (typically an ideal gas) is placed in thermal contact with a hot reservoir at temperature T{sub}H{/sub}. The gas expands slowly, doing work on a piston while absorbing heat Q{sub}H{/sub} from the reservoir. Because the temperature remains constant, the internal energy of an ideal gas does not change, and all the absorbed heat is converted to work during this step.
Next comes adiabatic expansion. The gas is isolated from all heat reservoirs and continues to expand. With no heat entering or leaving, the gas cools as it does work at the expense of its internal energy. The temperature drops from T{sub}H{/sub} to T{sub}C{/sub} (the temperature of the cold reservoir). The relationship between temperature and volume during this step is governed by TVgamma-1 = constant, where gamma is the heat capacity ratio.
The third step is isothermal compression. The gas is placed in contact with the cold reservoir at T{sub}C{/sub} and compressed slowly. Heat Q{sub}C{/sub} is rejected to the cold reservoir while the gas temperature remains constant. Finally, adiabatic compression returns the gas to its original state. Isolated from heat reservoirs, the gas is compressed, its temperature rising from T{sub}C{/sub} back to T{sub}H{/sub}. The cycle is now complete and can be repeated indefinitely.
Carnot Efficiency
The efficiency of the Carnot cycle is eta = 1 - T{sub}C{/sub}/T{sub}H{/sub}, where both temperatures are in Kelvin. This formula has several remarkable features. It depends only on the temperatures of the hot and cold reservoirs, not on the working fluid, the size of the engine, or any other details. It sets an absolute upper bound that no heat engine can exceed. And it shows that perfect efficiency (eta = 1) requires either an infinitely hot source or a cold reservoir at absolute zero, neither of which is achievable.
For typical operating conditions, Carnot efficiency provides a sobering reality check. A power plant using steam at 600 K and cooling water at 300 K has a Carnot efficiency of 50 percent. A car engine operating between combustion gases at 2000 K and exhaust at 800 K has a Carnot efficiency of 60 percent. Real efficiencies are always lower due to friction, heat losses, finite-rate processes, and other irreversibilities.
The Carnot theorem, proved rigorously from the second law, states two things. First, no engine operating between two reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. Second, all reversible engines operating between the same two reservoirs have the same efficiency. These results establish the Carnot efficiency as a universal limit, independent of engineering details.
The Carnot cycle proves that thermodynamics imposes absolute efficiency limits on heat engines, determined solely by the temperatures of the hot and cold reservoirs.
The Carnot Cycle and Entropy
The Carnot cycle played a central role in the development of the concept of entropy. During isothermal expansion, the engine absorbs heat Q{sub}H{/sub} at temperature T{sub}H{/sub}, and the entropy of the hot reservoir decreases by Q{sub}H{/sub}/T{sub}H{/sub}. During isothermal compression, the engine rejects heat Q{sub}C{/sub} at temperature T{sub}C{/sub}, and the entropy of the cold reservoir increases by Q{sub}C{/sub}/T{sub}C{/sub}. For a reversible Carnot cycle, these entropy changes exactly cancel: Q{sub}H{/sub}/T{sub}H{/sub} = Q{sub}C{/sub}/T{sub}C{/sub}.
This equality of entropy changes is the defining feature of reversibility. The total entropy of the universe remains unchanged after one complete Carnot cycle. During the adiabatic steps, no heat is exchanged, so no entropy is transferred. The entropy of the working fluid changes during each step but returns to its original value after the complete cycle, consistent with entropy being a state function.
For any real (irreversible) engine, Q{sub}C{/sub}/T{sub}C{/sub} > Q{sub}H{/sub}/T{sub}H{/sub}, meaning more entropy is generated than in the reversible case. The excess entropy production represents lost work potential, the difference between the work a Carnot engine would produce and the work the real engine actually produces. Minimizing entropy production is therefore equivalent to maximizing efficiency.
Practical Significance and Real Engines
No real engine operates on the Carnot cycle because its reversible processes would require infinitely slow operation, producing zero power output. Real engines use cycles like the Otto cycle (gasoline engines), Diesel cycle (diesel engines), Rankine cycle (steam power plants), and Brayton cycle (gas turbines). Each of these cycles has its own theoretical efficiency, always less than or equal to the Carnot efficiency for the same temperature range.
The Carnot cycle remains valuable as a benchmark. When engineers report that a power plant achieves 45 percent of Carnot efficiency, this immediately communicates how much room remains for improvement. The comparison also reveals whether the dominant losses come from operating at temperatures below the theoretical maximum (reducing the Carnot limit itself) or from irreversibilities within the cycle (reducing the fraction of the Carnot limit achieved).
Carnot analysis also applies in reverse to refrigerators and heat pumps. A Carnot refrigerator has the maximum possible coefficient of performance COP = T{sub}C{/sub}/(T{sub}H{/sub} - T{sub}C{/sub}), and a Carnot heat pump has COP = T{sub}H{/sub}/(T{sub}H{/sub} - T{sub}C{/sub}). These limits guide the design of cooling and heating systems, showing that efficiency improves as the temperature difference between hot and cold reservoirs decreases.