the third law of thermodynamics
/ Nernst: nairnst /
The first two laws leave the zero of entropy undetermined — they only ever talk about entropy differences. The third law nails it down at the coldest possible temperature. As a system is cooled toward absolute zero, its entropy stops depending on anything and settles onto a single fixed value; for a perfect crystal that value is zero. A striking corollary follows: you can never actually reach absolute zero in a finite number of steps.
Nernst's form states that the entropy change of any reversible isothermal process approaches zero as T -> 0. Planck's stronger form fixes the constant: the entropy of a perfect, defect-free crystalline substance tends to zero as T -> 0, so S(T=0) = 0. Immediate consequences are that the heat capacities C_p and C_v and the thermal expansion coefficient all vanish as T -> 0. The statistical basis is the Boltzmann relation S = k_B ln W: a system with a unique, nondegenerate quantum ground state has W = 1, hence S = 0 at T = 0.
The law governs all of low-temperature physics and explains why cooling gets harder and harder near absolute zero (unattainability). Honest caveat: the entropy tends to zero only if the ground state is unique. Systems frozen into a degenerate or disordered ground state keep a residual entropy — ice, for example, retains about R ln(3/2) per mole from the many allowed proton arrangements, and glasses freeze in disorder.
Cooling a copper block, its heat capacity falls off as C ~ T at the lowest temperatures (the electronic part) plus a T^3 phonon part, both vanishing as T -> 0 exactly as the third law demands. Because each stage of cooling removes less and less entropy, an infinite number of steps would be needed to reach 0 K — absolute zero is approached but never attained.
Heat capacities vanishing at T -> 0 and the unattainability of absolute zero are both third-law consequences.
S -> 0 as T -> 0 requires a nondegenerate ground state; substances with ground-state degeneracy (e.g. water ice, or a glass frozen out of equilibrium) retain a nonzero residual entropy, so the third law's zero is an idealization for a perfect crystal.