Carnot Engine and Carnot Theorem - UNSOLVED PRACTICE SET

Q7. State Carnot's theorem in your own words. Why is it considered the most important result in the theory of heat engines?

Q8. Draw a P-V diagram for a Carnot cycle and label the four processes. Indicate the direction of the cycle for a heat engine.

Q9. Explain why a Carnot engine must be reversible. What would happen to its efficiency if it were irreversible?

Q10. A Carnot engine operates between 500 K and 300 K. Calculate its efficiency. If the source temperature is doubled while keeping the sink temperature the same, what happens to the efficiency?

Q11. Why is the Carnot cycle not used in practical engines? Give two reasons.

Q12. Explain the significance of the Carnot engine in establishing an absolute thermodynamic temperature scale.

Q13. Describe the four processes of a Carnot cycle in detail with a neat P-V diagram:

(i) Isothermal expansion

(ii) Adiabatic expansion

(iii) Isothermal compression

(iv) Adiabatic compression

For each process, state what happens to P, V, T, Q, W, and ΔU. Explain why the Carnot cycle is reversible.

Q14. State and prove Carnot's theorem. Show that:

(i) No engine operating between two given temperatures can be more efficient than a reversible engine (Carnot engine) operating between the same temperatures.

(ii) All reversible engines operating between the same two temperatures have the same efficiency, regardless of the working substance.

Use the method of coupling two engines and show that any assumption to the contrary leads to a violation of the Second Law.

Q15. A Carnot engine operates between T₁ and T₂. The same engine is reversed to work as a refrigerator between the same temperatures.

(i) Draw the P-V diagram for the Carnot engine (heat engine mode).

(ii) Draw the P-V diagram for the Carnot refrigerator.

(iii) Show that the efficiency of the engine and the COP of the refrigerator are related by: η = 1/(1 + β) or β = (1 – η)/η

(iv) Verify this relation numerically for T₁ = 400 K and T₂ = 300 K.

Q16. A Carnot engine operates between 800 K and 400 K. The engine absorbs 2000 J of heat from the source per cycle.

(i) Calculate the efficiency of the Carnot engine.

(ii) Calculate the work done per cycle.

(iii) Calculate the heat rejected to the sink per cycle.

(iv) If the engine is reversed to work as a Carnot refrigerator between the same temperatures, calculate its COP.

(v) For the refrigerator, if it extracts 1000 J from the cold reservoir, calculate the work required and heat rejected.

Q17. A proposed engine design claims to operate between 600 K and 300 K with the following specifications:

Heat absorbed per cycle: 1500 J

Heat rejected per cycle: 500 J

The designer claims the engine uses a special "non-ideal" working substance that bypasses Carnot's limitations.

Analyze this claim:

(i) Calculate the claimed efficiency.

(ii) Calculate the Carnot efficiency for these temperatures.

(iii) Is the claimed engine possible? Use Carnot's theorem to justify.

(iv) If the engine were possible, what would be the consequences for the Second Law?

(v) What is the maximum work this engine can actually produce per cycle?

Q18. A Carnot engine uses an ideal monoatomic gas as the working substance. The engine operates between 600 K (source) and 300 K (sink). During the isothermal expansion at 600 K, the gas absorbs 1200 J of heat and the volume doubles.

(i) Calculate the efficiency of the engine.

(ii) Calculate the work done during the isothermal expansion.

(iii) Calculate the heat rejected during the isothermal compression at 300 K.

(iv) Calculate the ratio of volumes after adiabatic expansion to the volume after isothermal expansion (use γ = 5/3).

(v) Calculate the total work done in the complete cycle.