Questions: Carnot Efficiency and Maximum Efficiency Theorem

Question 1

An engineer claims to have built an engine operating between a hot reservoir at 500K and a cold reservoir at 300K with an efficiency of 42%. The Carnot efficiency for these temperatures is 1 − 300/500 = 40%. How should this claim be evaluated?

• A. The engine is impressively efficient and nearly at the theoretical maximum of 40%
• B. The claimed 42% efficiency is physically impossible — it exceeds the Carnot limit and would violate the second law of thermodynamics
• C. The claim is plausible if the engine uses an unusually effective working fluid or clever engineering design
• D. The Carnot limit is a theoretical ideal that applies only to idealized gas cycles, so real engineered systems may exceed it

Answer: (B)

No engine operating between fixed reservoirs can exceed the Carnot efficiency — this is a theorem derived directly from the second law, not an engineering target. The proof is by contradiction: if such an engine existed, you could use it to drive a Carnot refrigerator between the same reservoirs, and the net effect would be a spontaneous heat flow from cold to hot — a second-law violation. The Carnot limit applies regardless of working fluid, design, or engineering ingenuity. A claimed efficiency above the Carnot bound is a claim to violate thermodynamics.

Question 2

A power plant raises its steam temperature from 400K to 800K while keeping its cold reservoir (condenser) fixed at 300K. What happens to the Carnot efficiency?

• A. It doubles from 25% to 50%, because doubling T_H doubles the efficiency
• B. It increases from 25% to 62.5%, but not by a factor of two — the relationship between T_H and efficiency is nonlinear
• C. It stays the same because efficiency depends only on the temperature difference, not the ratio
• D. It doubles only if the cold reservoir temperature also changes proportionally

Answer: (B)

η_Carnot = 1 − T_C/T_H. At 400K: η = 1 − 300/400 = 25%. At 800K: η = 1 − 300/800 = 62.5%. Doubling T_H does not double efficiency because η depends on the ratio T_C/T_H, not on T_H alone. The gain diminishes as T_H grows large: at 400K vs 800K, the ratio T_C/T_H drops from 0.75 to 0.375. The relationship is intrinsically nonlinear in absolute temperature, which is why modest gains become harder to achieve at already-high temperatures.

Question 3

Carnot's theorem states that all reversible engines operating between the same two temperature reservoirs must have the same efficiency, regardless of working fluid or cycle design.

• T. True
• F. False

Answer: True

True. This follows from the same reductio argument used to prove the Carnot limit. If two reversible engines between the same reservoirs had different efficiencies, you could run the less efficient one in reverse (as a refrigerator) driven by the more efficient one, and extract net work while violating the second law. Since both are reversible, either can be run in either direction — so they must have exactly equal efficiency. The Carnot efficiency formula is therefore universal for reversible engines, independent of the specific design.

Question 4

Because fuel cells convert chemical energy directly to electrical energy, they are not bound by the Carnot efficiency limit and can in principle achieve 100% conversion efficiency.

• T. True
• F. False

Answer: True

True. The Carnot limit applies specifically to heat engines — devices that convert thermal energy to work by operating between two temperature reservoirs. Fuel cells are electrochemical devices that do not rely on heat as an intermediate step; they convert Gibbs free energy directly to electrical work. The theoretical maximum efficiency of a fuel cell is determined by the ratio of the Gibbs free energy change to the enthalpy change of the reaction (ΔG/ΔH), which can approach or exceed 100% for some reactions at certain temperatures. This is why fuel cells are not subject to the T_C/T_H thermodynamic bound.

Question 5

Describe the logical structure of the proof of Carnot's theorem: what is assumed, how the machine arrangement exposes the contradiction, and which physical law is violated.

Model answer: Assume, for contradiction, that some engine E has efficiency greater than the Carnot efficiency η_C between reservoirs T_H and T_C. Now run a Carnot engine in reverse as a refrigerator R, driven by the output of E. Calibrate the sizes so that E does just enough work to drive R. Because E is assumed more efficient than the Carnot refrigerator, it extracts less heat from T_H per unit of work output than R pumps back. The net result: heat is transferred spontaneously from the cold reservoir to the hot reservoir with no other effect on the surroundings. This violates the Clausius statement of the second law, which prohibits spontaneous heat flow from cold to hot. The contradiction shows that no engine can exceed Carnot efficiency; an identical argument for two unequal reversible engines shows they must have equal efficiency.

The elegance of the proof is that it requires no detailed knowledge of the engine's internals — only that it converts heat to work between two fixed reservoirs. The second law alone, in its most general statement, determines the maximum efficiency.