Heat Engines & Entropy

Classical Mechanics & Thermodynamics

Course 1 · Chapter 8 · Heat Engines & Entropy

Chapter 7 named Sadi Carnot's 1824 work as the true historical root of the Second Law — discovered a full quarter-century before the First Law that now outranks it in the numbering. This chapter gives Carnot's own real story, and the Second Law itself, the full treatment.

Sadi Carnot: A Real Story of Delayed Recognition

In June 1824, French military engineer Nicolas Léonard Sadi Carnot published, at his own expense, a slim 118-page book titled Reflections on the Motive Power of Fire. It asked a genuinely important practical question: is there a real, theoretical limit to how efficiently a heat engine can convert heat into useful work? The question mattered enormously at the time — contemporary steam engines, central to the Anglo-French industrial competition of the era, typically achieved only 5–7% efficiency, and no one yet knew whether that number reflected poor engineering or a genuine, unavoidable physical limit.

⚠ A Book That Nearly Vanished Carnot's own book attracted almost no attention during his lifetime, and real historical accounts describe it as having "virtually disappeared from booksellers and libraries" within a few years. Carnot himself died young — on 24 August 1832, at just 36 years old, of cholera, while under treatment at a sanatorium in Ivry. Real recognition came only in 1834, two years after his death, when engineer Émile Clapeyron published a detailed commentary reviving Carnot's own ideas — which Lord Kelvin and Rudolf Clausius then built directly on to develop absolute temperature and entropy, and to formally establish the Second Law that now carries the number "two," even though Carnot's own original insight came first of all four laws.

By a genuine, striking coincidence, Carnot's death anniversary — 24 August — falls on the exact same real calendar date this chapter is being written.

The Carnot Cycle: Four Real Stages

Carnot analysed an idealised heat engine operating between a hot reservoir (temperature T_H) and a cold reservoir (temperature T_C), moving a gas through four distinct stages:

1. Isothermal expansion: the gas absorbs heat from the hot reservoir at constant temperature T_H, expanding and doing work on its surroundings.
2. Adiabatic expansion: now thermally isolated (no heat in or out), the gas keeps expanding, cooling from T_H down to T_C as its own internal energy converts directly into further work.
3. Isothermal compression: external work compresses the gas at constant temperature T_C, releasing heat into the cold reservoir.
4. Adiabatic compression: thermally isolated once more, the gas is compressed the rest of the way, its temperature rising back from T_C to T_H — returning it exactly to its starting state, ready to repeat the cycle.

Carnot Efficiency: A Real Theoretical Ceiling

Carnot's own real, central result is that this idealised cycle sets an absolute upper limit on efficiency for any heat engine operating between two given temperatures — not just his own idealised one:

η = 1 − T_C/T_H

Both temperatures must be measured on an absolute (Kelvin) scale. No real engine — regardless of its design, materials, or era — can ever exceed this theoretical maximum operating between the same two temperatures, because a real engine always has some friction, some unwanted heat loss, and can never be perfectly reversible the way Carnot's idealised cycle assumes.

Worked Example: A Power Plant's Theoretical Maximum

A power plant's steam operates between a hot reservoir at 823 K (roughly 550°C) and a cold reservoir (cooling water) at 313 K (roughly 40°C). What is the maximum possible Carnot efficiency?

η = 1 − T_C/T_H
η = 1 − 313/823
η = 1 − 0.380
η = 0.620, or 62.0%

No real power plant operating between these exact two temperatures can exceed 62.0% efficiency, no matter how well-engineered — a genuine physical ceiling, not merely an engineering shortfall.

Entropy: A Name Chosen on Purpose

The quantity underlying the Second Law — entropy — got its real name from Rudolf Clausius in 1865, and he chose the word deliberately. He formed "entropy" from the Greek word for "transformation," specifically to echo the structure of the word "energy," writing that he considered the two quantities "analogous in their physical significance" and wanted names that reflected that kinship.

dS = δQ_rev / T

For a reversible process, the change in entropy (dS) equals the heat transferred (δQ) divided by the absolute temperature (T) at which the transfer happens. The Second Law, in these terms, says the total entropy of an isolated system can only stay the same (for a perfectly reversible process, like Carnot's own idealised cycle) or increase (for any real, irreversible process) — it can never decrease.

💡 A Deeper Explanation, a Few Decades Later In the 1870s, Ludwig Boltzmann gave entropy a genuinely different kind of explanation, grounded in statistics rather than heat flow directly: entropy is proportional to the natural logarithm of the number of possible microscopic arrangements (of individual atoms and molecules) consistent with a system's observable large-scale state. A tidy, ordered arrangement has comparatively few equivalent microscopic configurations; a disordered one has vastly more — which is the real, underlying reason why systems drift toward disorder over time. It isn't that nature actively prefers disorder; there are simply, overwhelmingly, more ways for a system to be disordered than ordered.

The Carnot Cycle, Stage by Stage

Stage	Process	Heat	Temperature
1	Isothermal expansion	Absorbed from hot reservoir	Constant at T_H
2	Adiabatic expansion	None (isolated)	Falls from T_H to T_C
3	Isothermal compression	Released to cold reservoir	Constant at T_C
4	Adiabatic compression	None (isolated)	Rises from T_C to T_H

Hands-On Exercises

Exercise 1

An engine operates between a hot reservoir at 600 K and a cold reservoir at 300 K. Calculate its maximum possible (Carnot) efficiency.

→ Solution

Exercise 2

400 J of heat is transferred to a system at a constant temperature of 320 K, in a reversible process. Using dS = Q/T, calculate the change in entropy.

→ Solution

Exercise 3

Explain, in your own words, why NO real engine - however well-engineered, however advanced its materials - can ever exceed the Carnot efficiency for a given pair of hot and cold reservoir temperatures. What real-world factor does Carnot's own idealised cycle assume away that every real engine actually has?

→ Solution

Quick Reference

Sadi Carnot's 1824 book set the real theoretical foundation for heat engine efficiency, decades before it was widely recognised
Carnot cycle: isothermal expansion → adiabatic expansion → isothermal compression → adiabatic compression
Carnot efficiency: η = 1 − T_C/T_H (absolute/Kelvin temperatures), a real theoretical maximum no engine can exceed
Entropy: named by Clausius in 1865, from Greek for "transformation"; dS = δQ/T for a reversible process
Second Law, restated: total entropy of an isolated system never decreases
Boltzmann's statistical view: entropy reflects the number of microscopic arrangements consistent with a system's observed state

Next chapter: Kinetic Theory of Gases — where the ideal gas law gets its own real, derived statistical-mechanics foundation, connecting individual molecular motion directly to the macroscopic pressure and temperature this chapter has treated as given.