Irreversibility and spontaneity
The first law establishes that energy is conserved — but it does not explain why some processes are spontaneous in one direction and never the other. An ink drop dilutes in water; it never spontaneously reconcentrates. Warm air in a room never spontaneously piles up in a corner.
The second law answers this question through the concept of entropy S.
Thermodynamic definition — Clausius
For a reversible (quasi-static) transformation, Clausius (1865) defines the entropy change:
dS = δQ_{rev} / T
where δQ_{rev} is the infinitesimal heat exchanged reversibly at temperature T.
For any transformation, the Clausius inequality states:
dS ≥ δQ / T
The equality holds for reversible processes; the strict inequality for irreversible ones. In an isolated system (δQ = 0): dS ≥ 0.
"The entropy of the universe tends towards a maximum." — Rudolph Clausius
Statistical definition — Boltzmann
Boltzmann (1872) connects entropy to thermodynamic probability Ω:
S = k_B ln Ω
where k_B = 1.381 × 10⁻²³ J·K⁻¹ is the Boltzmann constant and Ω is the number of microstates compatible with the system's macrostate.
Interpretation: S measures disorder or the dispersal of energy at the molecular scale. An expanded gas has more microstates than a compressed one → higher entropy.
For N indistinguishable particles distributed over Ω microstates, use Stirling's approximation: ln N! ≈ N ln N − N.
Third law — absolute zero
The third law (Nernst, 1906) states:
For a perfect crystal at T = 0 K, S = 0.
At T = 0 K there is only one possible microstate (Ω = 1) → S = k_B ln 1 = 0.
This allows absolute standard molar entropies S°(298 K) to be defined for all substances, tabulated like formation enthalpies.
ΔS°_{reaction} = Σ S°(products) − Σ S°(reactants)
Entropy and Gibbs energy
Only the total entropy of the universe (system + surroundings) determines spontaneity. For a system at constant T and P, combining the first and second laws gives the Gibbs function:
G = H − TS → ΔG = ΔH − TΔS
| ΔH | ΔS | Reaction spontaneity |
|---|---|---|
| − | + | Spontaneous at all T |
| + | − | Non-spontaneous at all T |
| − | − | Spontaneous if T < ΔH/ΔS |
| + | + | Spontaneous if T > ΔH/ΔS |
ΔG < 0: spontaneous; ΔG = 0: equilibrium; ΔG > 0: non-spontaneous.
Entropy is central to modern thermodynamics, with applications ranging from physical chemistry to information theory (Shannon, 1948: H = −Σ p_i log p_i) and cosmology.