Second Law of Thermodynamics and Entropy

Break an egg, stir cream into coffee, let perfume fill a room — you never see the reverse, though no law of mechanics forbids it. The second law is a law of probability: disordered arrangements so overwhelmingly outnumber ordered ones that, with $10^{23}$ players, "unlikely" becomes "never". Entropy is the arrow on time’s compass.

The second law determines the direction of natural processes.

Statements

• Clausius: Heat cannot spontaneously flow from cold to hot.
• Kelvin-Planck: No engine can convert heat entirely into work.

Entropy measures disorder:

$$\Delta S = \int \frac{dQ_{\text{rev}}}{T} \geq 0 \quad\text{(for an isolated system)}$$

Carnot engine — The most efficient heat engine operating between $T_H$ and $T_C$:

$$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H}$$

No real engine can exceed Carnot efficiency.

Entropy changes

Statistical interpretation — Boltzmann:

$$S = k_B\ln\Omega$$

where $\Omega$ is the number of microstates. Systems evolve toward the macrostate with the largest $\Omega$.

Key insight: Entropy always increases in isolated systems. This defines the arrow of time — the past had lower entropy than the future.

Common pitfall: Entropy can decrease locally — your freezer does it nightly — as long as more entropy is exported elsewhere (the warm air behind it). The law binds only the total: system plus surroundings, never a subsystem alone.