Questions: Heat Engine Efficiency and Carnot's Theorem

Question 1

A heat engine absorbs heat from a reservoir at 800 K and exhausts to a cold reservoir at 200 K. What is the maximum possible efficiency?

• A. 25%
• B. 50%
• C. 60%
• D. 75%

Answer: (D)

η_Carnot = 1 − T_C/T_H = 1 − 200/800 = 1 − 0.25 = 0.75 = 75%. Temperatures must be in Kelvin (absolute). The 25% option is T_C/T_H itself — confusing the heat rejected fraction with the efficiency. No engine operating between these reservoirs can exceed 75%, regardless of its design, working fluid, or engineering quality.

Question 2

An engineering team claims their frictionless, perfectly insulated engine operating between 600 K and 300 K achieves 60% efficiency. What can you conclude?

• A. This is plausible — friction and insulation losses are the main practical barriers to high efficiency
• B. This violates the second law of thermodynamics and is physically impossible
• C. This is achievable with advanced materials that reduce irreversibilities
• D. The claim is valid for certain working fluids but not others

Answer: (B)

η_Carnot for this engine = 1 − 300/600 = 50%. A claimed 60% exceeds the Carnot limit. Carnot's theorem states that no engine operating between two fixed reservoirs can exceed the efficiency of a reversible engine between those same reservoirs. This is not an engineering limitation — it is a consequence of the second law. No materials, working fluids, or engineering refinements can overcome it.

Question 3

An engine operating between reservoirs at 1000 K and 500 K can theoretically achieve 80% efficiency with sufficiently advanced engineering.

• T. True
• F. False

Answer: False

η_Carnot = 1 − 500/1000 = 50%. The 50% limit is set by the second law of thermodynamics, not by engineering imperfection. Even a perfectly reversible engine — the Carnot engine — achieves exactly 50% between these reservoirs. Claiming 80% would require violating the entropy non-decrease principle. Advanced engineering can close the gap between real and Carnot efficiency, but it cannot raise the ceiling.

Question 4

To maximize the efficiency of a heat engine, an engineer should both increase the hot reservoir temperature and decrease the cold reservoir temperature.

• T. True
• F. False

Answer: True

η_Carnot = 1 − T_C/T_H. Increasing T_H makes the fraction T_C/T_H smaller (better). Decreasing T_C also makes T_C/T_H smaller (better). Both changes independently improve efficiency, and together they compound. This is why industrial power plants burn fuel as hot as materials allow and exhaust heat through cooling towers to get T_C as close to ambient as possible.

Question 5

Why can a heat engine never achieve 100% efficiency, and why does the Carnot formula use absolute temperatures in Kelvin rather than Celsius or Fahrenheit?

Model answer: Efficiency reaches 100% only when T_C/T_H = 0, which requires either T_C = 0 K (absolute zero, unachievable by the third law) or T_H = ∞ (impossible). The Carnot formula derives from the entropy balance: ΔS_universe = Q_C/T_C − Q_H/T_H ≥ 0, where T must be absolute (Kelvin) because entropy change Q/T is only meaningful on the absolute scale. Celsius and Fahrenheit have arbitrary zeros and would give physically nonsensical results.

The entropy argument is the deep reason for both constraints. The second law requires Q_C/T_C ≥ Q_H/T_H, bounding Q_C from below. Efficiency η = 1 − Q_C/Q_H is therefore bounded above by 1 − T_C/T_H. Both T_C > 0 K (third law) and T_H < ∞ (practical) prevent the bound from reaching 1.