This is the final exam for Entropy & the Second Law. It pulls the whole course
together: why disorder wins as a matter of counting; microstates, macrostates and the
multiplicity W; Boltzmann’s S = k ln W; the arrow of time and why it emerges from
reversible laws; the difference between energy’s quantity and its quality; the Carnot
limit and why perpetual motion is impossible; how local order is always paid for by
global disorder; entropy as missing information; and the model’s honest limits. Several
questions look easy until you spot the trap — that a freezing pond breaks the second
law, or that life violates it. Reason each one through.
How this exam works
Read carefully — this exam is final. Each question appears one at a time. Once you submit an answer it is locked for good: there’s no going back, no retry, and no restart. Your score is hidden until the end, where you’ll see a pass/fail verdict. The pass mark is 70%. A few questions ask you to select all correct answers.
In the statistical picture, what exactly is entropy a measure of?
Select an answer to continue.
Course Recap
Big picture
Entropy & the second law, in one picture
- Entropy & the Second Law
- Counting the ways
- A macrostate's multiplicity W = the number of microstates that realise it. S = k ln W. Disorder wins not by force but because spread-out macrostates have astronomically more microstates — so a random system lands there. The log makes independent entropies add.
- The arrow of time
- Micro-laws are reversible; the arrow is statistical. Forward = increasing entropy. Backward isn't forbidden, just ~2^(−N) improbable, so large systems are one-way. Rests on a low-entropy early universe. Small systems fluctuate visibly.
- Energy quality, not quantity
- First law conserves quantity ("can't win"); second degrades quality ("can't break even"). Heat engines must dump waste heat. Carnot ceiling η = 1 − Tc/Th, set by temperatures alone. 100% efficiency and perpetual motion are impossible.
- Local order, global cost
- A fridge, a cell, a company build local order only by exporting more disorder to the surroundings — check the TOTAL ledger. Life "feeds on free energy". Order decays on its own, so maintenance is forever: rust, bit rot, technical debt, forgetting.
- Information & limits
- Entropy = missing information (Boltzmann → Shannon); erasing a bit costs kT ln 2 (Landauer). Limits: it's not eyeball-disorder (freezing pond); it's for ISOLATED systems (Earth is open, life breaks nothing); it's statistical not absolute; social "decay" is metaphor, not physics.
- Counting the ways
Key takeaways — the whole course
The second law of thermodynamics says that, left alone, an isolated system
drifts toward its most probable state — and “spread-out and disordered” is
overwhelmingly more probable than “concentrated and ordered.” That’s entropy, and
Boltzmann pinned it down: S = k ln W, the log of the number of microstates. From
that one counting fact everything follows. It gives time its arrow — forward is the
way entropy rises, and the reverse isn’t forbidden, just astronomically improbable for
big systems. It splits energy’s conserved quantity (first law) from its degrading
quality (second law), which is why heat engines waste heat, why the Carnot limit
η = 1 − Tc/Th is a wall, and why perpetual motion is a scam. It means local order
always costs global disorder: a fridge, a cell, or a company builds order in one place
only by dumping more disorder elsewhere, and because order decays on its own,
maintenance is forever. And it’s really about missing information (Boltzmann →
Shannon → Landauer). But hold its edges: entropy is not eyeball-disorder, it governs
isolated systems so life and evolution break nothing, it’s statistical not
absolute, and “society is doomed” is metaphor, not physics. Whenever you see order
appear, ask the one question this course was built around: what’s paying for it, and
where is the disorder going?