IMAT Physics · Heat and Thermodynamics

The Second Law, Entropy and Heat Engines

Heat flows by itself only from hot to cold, no engine turns all its heat into work, and the entropy of an isolated system never decreases.

All Heat and Thermodynamics notes

Why this matters

One past question, from 2011, touches this page: which process produces almost no entropy. The ministry papers have not asked about it yet, but engine efficiency is standard syllabus and a one-line calculation when it appears.

Concept 1 of 2: Heat engines, efficiency and the Carnot limit

A heat engine takes heat from something hot, turns part of it into work, and must dump the rest into something cold. The bigger the temperature gap between the hot and the cold side, the larger the share that can become work. Even a perfect, frictionless engine cannot beat the Carnot limit, and that limit reaches 100% only if the cold side is at absolute zero.

Definition

A heat engine takes heat $Q_h$ from a hot reservoir, does work $W$ , and rejects heat $Q_c$ to a cold reservoir: $W = Q_h - Q_c$ .
Efficiency $\eta = W/Q_h = 1 - Q_c/Q_h$ , always less than 1.
The Carnot efficiency $1 - T_c/T_h$ is the maximum for any engine working between those two temperatures, with $T$ in kelvin.
A refrigerator or heat pump runs an engine backwards: work moves heat from cold to hot.

Efficiency and the Carnot limit

\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h} \qquad \eta_{\max} = 1 - \frac{T_c}{T_h}

$Q_h,\ Q_c$ heat taken from the hot reservoir, heat given to the cold one, in J
$W$ useful work done, in J
$T_h,\ T_c$ reservoir temperatures, in K

Worked example

Each cycle, an engine takes 2000 J from a reservoir at 500 K and gives 1400 J to a reservoir at 300 K. Find its efficiency and compare it with the Carnot limit.

Practice this conceptself-check · 3 quick reps

The Carnot formula needs kelvin

Putting Celsius temperatures into

1 - T_c/T_h

gives a far too large efficiency. Convert both temperatures to kelvin first.

No real engine reaches the Carnot efficiency

The Carnot value is a ceiling for an ideal, reversible engine. Any claim that an engine beats it, or reaches 100% with a cold side above absolute zero, breaks the second law.

Concept 2 of 2: The second law and entropy

Hot coffee cools in a room, but a cool coffee never warms itself by taking heat from the room. Nature runs one way: energy spreads out and order turns into disorder. Entropy measures that spreading. Processes with friction, mixing or heat flowing across a temperature gap create entropy; a perfectly smooth, frictionless motion creates almost none.

Definition

Clausius statement: heat does not flow by itself from a colder body to a hotter one.
Kelvin-Planck statement: no engine working in a cycle can turn all the heat it takes in into work.
Entropy measures disorder, or how spread out energy is. In an isolated system the total entropy never decreases; it rises in every real (irreversible) process.
The entropy of one part can fall, as long as the surroundings gain at least as much.

Process	Entropy change	Why
Ice melting	Increases	An ordered crystal becomes a disordered liquid
Liquid evaporating	Increases	Molecules spread out into a much larger volume as gas
Friction slowing a sliding block	Increases	Ordered motion becomes random thermal motion
Heat flowing from hot to cold	Increases overall	The cold body gains more entropy than the hot body loses
Frictionless swing of a pendulum in a vacuum	Close to zero	No energy is turned into heat, so the motion is reversible

Practice this conceptself-check · 3 quick reps

Entropy can fall locally

The second law is about the total entropy of an isolated system. A living cell, a freezer or a growing crystal can lower its own entropy, as long as it raises the entropy of its surroundings by more.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Heat engines, efficiency and the Carnot limit
Efficiency and the Carnot limit
$\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h} \qquad \eta_{\max} = 1 - \frac{T_c}{T_h}$

Reference tables (1)

The second law and entropy5 rows

Process	Entropy change	Why
Ice melting	Increases	An ordered crystal becomes a disordered liquid
Liquid evaporating	Increases	Molecules spread out into a much larger volume as gas
Friction slowing a sliding block	Increases	Ordered motion becomes random thermal motion
Heat flowing from hot to cold	Increases overall	The cold body gains more entropy than the hot body loses
Frictionless swing of a pendulum in a vacuum	Close to zero	No energy is turned into heat, so the motion is reversible

Watch out for (3)

The Carnot formula needs kelvin
→ Heat engines, efficiency and the Carnot limit
No real engine reaches the Carnot efficiency
→ Heat engines, efficiency and the Carnot limit
Entropy can fall locally
→ The second law and entropy

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