QCC Notes
CLASS 11 · PHYSICSJEE MAIN × NEETहिंदी
§ 11.5NCERT Class 11 · Physics · Chapter 11

Carnot Engine, Carnot Cycle & Carnot's Theorem

QCC Notes · Class 11 Physics NCERT-aligned Updated 26 Sep 2026 5 min read

In 1824, French engineer Nicolas Léonard Sadi Carnot proposed an idealized, theoretical heat engine operating on a completely reversible cycle. The Carnot Engine establishes the theoretical upper limit of efficiency that any heat engine can ever achieve operating between two given temperatures.

1. Reversible vs. Irreversible Processes

Reversible Process

A process that can be reversed such that both the system and the surroundings return precisely to their initial states without leaving any change in the universe. Necessary conditions:

  • Must be quasi-static (infinitely slow).
  • Completely free from dissipative forces like friction, viscosity, and electrical resistance.
  • Thermal exchange occurs across infinitesimal temperature differences.

Irreversible Process

A process that cannot be reversed along the same path. Natural processes in nature are all irreversible due to:

  • Presence of friction, viscosity, and inelasticity.
  • Spontaneous, finite temperature differences.
  • Unrestrained expansion (free expansion into vacuum).
  • Spontaneous chemical reactions.

2. The Four Stages of the Carnot Cycle

The Carnot cycle operates with an ideal gas enclosed in a cylinder with perfectly insulating walls and a perfectly conducting base, fitted with a frictionless, insulating piston. The cycle comprises four successive reversible processes:

Step 1 → Reversible Isothermal Expansion (State A to State B at T1)

The cylinder base is placed in thermal contact with the source at temperature T1. The gas expands slowly from (P1, V1, T1) to (P2, V2, T1). Heat Q1 is absorbed reversibly from the source:

W1 = Q1 = n R T1 ln(V2 / V1)    (ΔU1 = 0)

Step 2 → Reversible Adiabatic Expansion (State B to State C from T1 to T2)

The cylinder is placed onto a perfectly insulating stand. The gas continues to expand adiabatically from (P2, V2, T1) to (P3, V3, T2) until temperature drops to sink temperature T2:

W2 = n R (T1 − T2) / (γ − 1)    (Q = 0)

Step 3 → Reversible Isothermal Compression (State C to State D at T2)

The cylinder is placed on the sink at temperature T2. The gas is compressed slowly from (P3, V3, T2) to (P4, V4, T2). Heat Q2 is rejected to the sink:

W3 = − Q2 = n R T2 ln(V4 / V3) = − n R T2 ln(V3 / V4)

Step 4 → Reversible Adiabatic Compression (State D to State A from T2 to T1)

The cylinder is placed back onto the insulating stand. The gas is compressed adiabatically from (P4, V4, T2) back to initial state (P1, V1, T1), raising temperature from T2 to T1:

W4 = − n R (T1 − T2) / (γ − 1)    (Q = 0)

3. Net Work Done & Efficiency of Carnot Engine

Notice that the adiabatic work terms cancel each other: W2 + W4 = 0. Hence, the net work done in one complete cycle is:

Wnet = W1 + W3 = n R T1 ln(V2 / V1) − n R T2 ln(V3 / V4)

From the adiabatic relationships for stages B→C and D→A:

T1 V2γ−1 = T2 V3γ−1    and    T1 V1γ−1 = T2 V4γ−1

Dividing the two equations yields: (V2 / V1)γ−1 = (V3 / V4)γ−1 ⇒ V2 / V1 = V3 / V4.

Therefore, the ratio of heat exchanged is simply:

Q2 / Q1 = T2 / T1

Substituting into the thermal efficiency formula η = 1 − Q2 / Q1 gives the famous Carnot Efficiency Formula:

ηcarnot = 1 − (T2 / T1) = (T1 − T2) / T1

4. Crucial Properties of Carnot Efficiency

  • Depends Only on Temperatures: ηcarnot depends solely on the absolute temperatures of the source (T1) and sink (T2). It is completely independent of the nature of the working substance.
  • Condition for η = 1 (100%): Possible only if T2 = 0 K (absolute zero) or T1 = ∞. Since absolute zero is unattainable according to the Third Law of Thermodynamics, 100% efficiency is impossible.
  • Methods to Increase Efficiency:
    • Increasing source temperature T1.
    • Decreasing sink temperature T2.
    • Decreasing T2 by ΔT is always more effective in boosting efficiency than increasing T1 by the same ΔT.

5. Carnot Refrigerator COP

When the Carnot cycle is operated in reverse, it functions as a Carnot refrigerator:

βcarnot = T2 / (T1 − T2)

As the cold space temperature T2 approaches absolute zero, β → 0, meaning an infinite amount of work is required to extract even an infinitesimal amount of heat.

6. Carnot's Theorem

Statement of Carnot's Theorem:
  1. No heat engine operating between two given constant temperatures T1 and T2 can have a greater efficiency than a reversible Carnot engine operating between the same two temperatures.
  2. All reversible heat engines operating between the same two temperatures have the same efficiency, regardless of the working substance used.
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