Entropy: The Ledger and the Count

Two faces of the same quantity. Above, heat moves between two reservoirs and the entropy ledger is kept in J/K — one side loses, the other gains, and only the total decides whether the process is allowed. Below, the same idea counted out: every way N particles can sit in a two-sided box, with the fattest macrostate marked.

Total entropy change
+0.800 J/K
Allowed — the total rises, so the second law permits this.
Hot side — entropy change
-1.20 J/K
ΔShot = -Q / Thot
Cold side — entropy change
+2.00 J/K
ΔScold = +Q / Tcold
Heat transferred Q600 J
Hot reservoir Thot500 K
Cold reservoir Tcold300 K
Total microstates  W = 2N
1,024
Wpeak = 252 arrangements
Speak = k ln W = 7.638e-23 J/K
Particle count N10
Boltzmann constant k = 1.380649 × 10-23 J/K · reservoirs are large enough that their temperatures do not change
Tip: slide the cold reservoir up towards the hot one. The gain and the loss close on each other and the total shrinks to nothing — that is the reversible limit.