Class 12 > Unit # 17:Second Law of Thermodynamics > Kelvin's Statement and Working of Heat Engines

Class 12 Physics • Thermodynamics

Kelvin's Statement & Efficiency of a Heat Engine

Detailed explanation of Kelvin's statement of the Second Law of Thermodynamics, essentials and working principle of heat engines, and step-by-step derivation of thermal efficiency (η).

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Q. State Kelvin’s Statement of the 2nd law of thermodynamics and describe the working principle of a heat engine. Also, derive an expression for the efficiency of a heat engine.

1. Kelvin's Statement of 2nd Law of Thermodynamics

Statement: According to Lord Kelvin:

"It is impossible to construct an engine, operating continuously in a cycle, that can take heat from a source and convert it completely into work."

Explanation: When converting heat into mechanical work, a cyclic engine cannot convert 100% of the input heat energy into work. A fraction of the absorbed heat must always be rejected to a lower-temperature reservoir (the exhaust/sink). Thus, even though the First Law of Thermodynamics is fully satisfied, it is impossible to build an engine with an efficiency of 100%.

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2. Heat Engines & Their Working Principle

Definition: Any device that transforms thermal energy (heat) into mechanical energy (work) is called a heat engine.

Essentials of a Heat Engine: Every heat engine consists of three essential components:

• Hot Reservoir (Source): Maintains a high temperature (T₁) and supplies heat energy (Q₁).
• Working Substance: The material (such as gas or steam) inside the engine that absorbs heat and performs mechanical work.
• Cold Reservoir (Sink): Maintains a lower temperature (T₂) and absorbs waste heat (Q₂).

Working Mechanism

All cyclic heat engines function through a repeating thermodynamic process:

1. The working substance absorbs heat Q₁ from the high-temperature source (T₁).
2. It expands and performs net mechanical work W on the surroundings.
3. It discards the remaining waste heat Q₂ to the low-temperature sink (T₂) through exhaust or cooling systems to return to its initial state.

From energy conservation, the net mechanical work done W in one complete cycle is:

Q_net = Q₁ − Q₂   &implies;   W = Q₁ − Q₂

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3. Thermal Efficiency of a Heat Engine

The effectiveness of a heat engine is measured by its **thermal efficiency** (η).

Definition: Thermal efficiency (η) of a heat engine is defined as the ratio of the net work done (W) per cycle to the total heat absorbed (Q₁) from the source in that cycle.

Mathematically:

η = Net work done (W)Heat absorbed (Q₁) = WQ₁

Derivation of Formula

Substituting W = Q₁ − Q₂ into the efficiency expression:

η = Q₁ − Q₂Q₁  &implies;  η = 1 − Q₂Q₁

In percentage form:

% η = (1 − Q₂Q₁) × 100%

In terms of absolute temperatures of the source (T₁) and sink (T₂):

η = 1 − T₂T₁

Key Point for Board Exams: For 100% efficiency (η = 1), Q₂ must equal 0 (no heat rejected to the sink). However, experimental facts and Kelvin's statement prove that Q₂ > 0 always. Therefore, the efficiency of a real cyclic heat engine is always strictly less than 100% (η < 1).