Second Law of Thermodynamics

Equivalent forms

\oint \frac{\delta Q}{T} \leq 0

dS \geq \frac{\delta Q}{T}

\eta = 1 - \frac{T_c}{T_h} \leq \eta_{\text{Carnot}}

A single inequality that explains why heat flows downhill, why engines can't be perfect, and why time has an arrow.

Symbol	Name	SI unit	Dimension	Range
$ΔS$	Entropy change of universeoutput Total entropy produced; the arrow of time	J/K	M L² T⁻² Θ⁻¹	0 – 1
$Q$	Heat transferred Heat passing between reservoirs	J	M L² T⁻²	10 – 200
$T_h$	Hot temperature Temperature of hot reservoir	K	Θ	300 – 900
$T_c$	Cold temperature Temperature of cold reservoir	K	Θ	100 – 500

Unit systems

SI:: S in J/K
natural:: S dimensionless in units of k_B
CGS:: S in erg/K

Where it holds

Holds for macroscopic systems; statistically, entropy can fluctuate down for tiny systems over short times (fluctuation theorems quantify the exponentially small probability).

Discovery

Rudolf Clausius / William Thomson (Lord Kelvin) · 1865

Clausius coined 'entropy' (Greek for transformation) in 1865, summarizing the universe in two laws: 'The energy of the universe is constant; the entropy of the universe tends to a maximum.'

Try this

Why can't you un-stir milk from your coffee?

Energy is conserved either way, so the First Law allows un-mixing. The Second Law is what forbids it: the entropy of an isolated system never decreases, giving time its direction.

Research status: stable

Common misconceptions

Entropy is not 'disorder' in a literal tidiness sense, and the Second Law does NOT forbid local order (life, crystals) — those are paid for by larger entropy increases elsewhere.

Derivation

For any cyclic process Clausius' theorem gives

\oint \delta Q/T \le 0

, with equality for reversible cycles.

Defining

dS = \delta Q_rev/T

makes S a state function; for an isolated system

\delta Q = 0

so

dS \ge 0

.

For heat Q crossing from T_h to T_c the surroundings change

by -Q/T_h + Q/T_c

, giving

\Delta S = Q(1/T_c - 1/T_h) \ge 0

.

Limiting cases

T_h = T_c

⟶

\Delta S = 0

: reversible, no net entropy produced (equilibrium)

Reversible process⟶ Equality

dS = \delta Q/T

holds; the bound is saturated

Free expansion / mixing⟶

\Delta S

> 0 with no heat or work: pure irreversibility

Entropy Change→Carnot Efficiency→Boltzmann Entropy→First Law of Thermodynamics→