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The second law

Heat and matter · level 8, lesson 2 of 6

A question to start

The first law would allow a cold cup of tea to grow hotter by taking heat from the cooler room around it: energy would still be conserved. Yet that never happens. What rule forbids it?

The idea

The second law of thermodynamics says that heat flows by itself only from hotter to colder. To move heat the other way, as a fridge or heat pump does, you must put in work.

It can be put in numbers with entropy, \(S\). When heat \(Q\) flows into something at temperature \(T\) (in kelvin), its entropy changes by

\[\Delta S = \frac{Q}{T}.\]

The second law says that in any real process, the total entropy never decreases. Heat flowing from hot to cold raises the total; the reverse would lower it, so it does not happen by itself.

300 K200 K1,000 J ΔS = −3.33 J/KΔS = +5.00 J/Ktotal: +1.67 J/K

Worked examples

1. Heat leaking from a habitat. 1,000 J flows from rooms at 300 K into the Mars ground at 200 K: \(\Delta S = -1000/300 + 1000/200 = -3.33 + 5.00 = +1.67\) J/K. The total rises, so this happens by itself.

2. The reverse. 1,000 J flowing from 200 K to 300 K would give \(-5.00 + 3.33 = -1.67\) J/K. Forbidden without work.

3. Melting ice. 334,000 J melts 1 kg of ice at 273 K: its entropy rises by \(334{,}000/273 = 1{,}220\) J/K. The particles go from neat rows to a jumble.

Try it

  1. State the second law in words about heat.
  2. 600 J flows from a 400 K block to a 300 K block. Find the total change in entropy.
  3. Why can a fridge make its inside colder than the kitchen without breaking the second law?
  4. Find the entropy change when 1 kg of water boils at 373 K (\(L_v\) = 2,260 kJ/kg).
Answers
  1. Heat flows by itself only from hotter to colder.
  2. \(-600/400 + 600/300 = -1.5 + 2.0 = +0.5\) J/K.
  3. It uses work (electricity) to pump the heat, and dumps more heat into the kitchen than it takes from the inside, so the total entropy still rises.
  4. \(2{,}260{,}000/373 = 6{,}060\) J/K.

Common mistake

"Entropy can never go down anywhere." It can fall in one place, as in a fridge's inside, as long as it rises by at least as much somewhere else. It is the total that never falls.

Where it leads