Microstates, Multiplicity and Boltzmann's Entropy · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Counting arrangements: multiplicity and entropy

Science · Thermal & Statistical Physics · ages 20-21
Name ______________________   Date ____________

  1. Which formula links entropy S to multiplicity W?

    • S equals W over k
    • S equals k times W
    • S equals k ln W

  2. What is a microstate?

    • One exact microscopic arrangement
    • The measured pressure
    • The average energy

  3. The second law forbids entropy decrease the way energy conservation forbids free work.

    Circle one:   True   False

  4. Why does entropy use the logarithm of multiplicity?

    • To keep entropy additive when systems combine
    • Because logs look nicer
    • To hide the units

  5. Toss four coins and count heads versus tails. Which outcome owns the most microstates?

    • All heads
    • Two heads and two tails
    • All tails

  6. Could all the air molecules in a room gather in one corner on their own?

    • Forbidden absolutely
    • Happens about every hour
    • Possible but astronomically unlikely

  7. Two independent systems combine and the total multiplicity squares. What happens to the entropy?

    • It doubles
    • It squares
    • It halves

  8. Modern experiments watch just a handful of particles. What do they see?

    • No motion at all
    • Entropy flickering down briefly, as fluctuation theorems quantify
    • The second law failing permanently
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Counting arrangements: multiplicity and entropy W1-mt_55I3VvYMc6-s1

  1. S equals k ln W · Boltzmann's formula is S equals k times the natural log of W.
  2. One exact microscopic arrangement · A microstate pins every molecule to a position with a velocity.
  3. False · Decreases are fantastically improbable, not forbidden.
  4. To keep entropy additive when systems combine · Multiplicities multiply while logs add, so doubled systems double entropy.
  5. Two heads and two tails · The 2-2 split owns six sequences, beating one each for the extremes.
  6. Possible but astronomically unlikely · Nothing bans it, but its pile of microstates is vanishingly tiny.
  7. It doubles · Log of a square is twice the log, so entropy doubles.
  8. Entropy flickering down briefly, as fluctuation theorems quantify · Small counts let entropy dip briefly before the pile reclaims it.
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