Carnot Cycle
The Carnot Cycle is an idealized, reversible thermodynamic cycle consisting of two isothermal and two adiabatic processes, first described by Sadi Carnot in 1824 as the most efficient possible heat engine operating between two fixed temperatures. No real engine can exceed the efficiency of a Carnot engine operating between the same hot and cold reservoirs, making it the theoretical upper bound for heat engine performance. The Carnot efficiency depends only on the absolute temperatures of the reservoirs and sets the fundamental limit imposed by the Second Law of Thermodynamics.
Key Formula
η_Carnot = 1 − (T_C / T_H)
LaTeX: \eta_{Carnot} = 1 - \frac{T_C}{T_H}
| Symbol | Meaning | Unit |
|---|---|---|
| η_Carnot | Maximum (Carnot) thermal efficiency | dimensionless |
| T_C | Absolute temperature of cold reservoir | K |
| T_H | Absolute temperature of hot reservoir | K |
Worked Example
Problem
A Carnot engine operates between a hot reservoir at 500°C and a cold reservoir at 25°C. Calculate: (a) the Carnot efficiency, and (b) the heat rejected to the cold reservoir if 1000 J is absorbed from the hot reservoir.
Solution
Step 1: Convert temperatures to Kelvin. T_H = 500 + 273.15 = 773.15 K. T_C = 25 + 273.15 = 298.15 K. Step 2: Carnot efficiency: η = 1 − (T_C / T_H) = 1 − (298.15 / 773.15) = 1 − 0.3857 = 0.6143 ≈ 61.4%. Step 3: Work output: W = η × Q_H = 0.6143 × 1000 J = 614.3 J. Step 4: Heat rejected: Q_C = Q_H − W = 1000 − 614.3 = 385.7 J.
Answer
(a) η_Carnot ≈ 61.4%; (b) Q_C ≈ 385.7 J rejected to cold reservoir
The Four Processes of the Carnot Cycle
| Process | Type | Description | ΔU | Q | W |
|---|---|---|---|---|---|
| A → B | Isothermal expansion | Gas expands at T_H, absorbs Q_H | 0 | +Q_H | +W₁ |
| B → C | Adiabatic expansion | Gas expands, cools from T_H to T_C | −ΔU | 0 | +W₂ |
| C → D | Isothermal compression | Gas compressed at T_C, rejects Q_C | 0 | −Q_C | −W₃ |
| D → A | Adiabatic compression | Gas compressed, warms from T_C to T_H | +ΔU | 0 | −W₄ |
| Net cycle | Complete cycle | Returns to initial state | 0 | Q_H − Q_C | W_net |
Interactive Tools
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Related Terms
Heat Engine
A heat engine is a device that converts thermal energy into mechanical work by exploiting the temperature difference between a high-temperature heat source (hot reservoir) and a low-temperature heat sink (cold reservoir). The engine absorbs heat Q_H from the hot reservoir, converts part of it to useful work W, and rejects the remainder Q_C to the cold reservoir, operating in a cyclic process. The thermal efficiency of a heat engine is always less than 100% due to the Second Law of Thermodynamics, and the maximum theoretical efficiency is set by the Carnot efficiency.
Second Law of Thermodynamics
The Second Law of Thermodynamics states that in any spontaneous process, the total entropy of an isolated system can only increase or remain constant, never decrease. This gives thermodynamics a preferred direction of time, explaining why heat flows from hot to cold, why mechanical energy converts irreversibly to heat, and why perpetual motion machines of the second kind are impossible. It is the thermodynamic basis for the arrow of time and sets fundamental efficiency limits on all heat engines.
Entropy
Entropy is a thermodynamic state function that quantifies the degree of disorder, randomness, or the number of microstates available to a system at a given macrostate. Macroscopically, it is defined via the Clausius inequality as the ratio of reversible heat exchange to absolute temperature; microscopically, Boltzmann's formula connects it to the number of microscopic configurations. Entropy always increases in irreversible processes in isolated systems, driving systems toward equilibrium and explaining the thermodynamic arrow of time.
Named after Nicolas Léonard Sadi Carnot (1796–1832), a French military engineer who published "Réflexions sur la puissance motrice du feu" in 1824. His name Sadi was inspired by the Persian poet Sa'di of Shiraz. The cycle concept was later mathematically formalized by Rudolf Clausius and Lord Kelvin.