Entropy

Entropy is a thermodynamic state function, denoted S and measured in joules per kelvin, defined through its differential dS = dQ_rev/T: the reversible heat added to a system divided by the absolute temperature at which it is added. Because the Clausius inequality makes ∮ dQ_rev/T vanish around any reversible cycle, S depends only on the state of the system, not on the path taken to reach it.

The entropy difference between two states is found by integrating dQ_rev/T along any reversible path connecting them — even an imaginary one. This makes entropy computable for irreversible processes too: invent a reversible route between the same endpoints and integrate along it. For an ideal gas, isothermal expansion gives ΔS = nR ln(V₂/V₁); reversible adiabatic processes are isentropic, ΔS = 0.

The second law, in its most general form, states that the entropy of an isolated system never decreases. Microscopically — as Boltzmann showed — entropy counts the number of microscopic arrangements consistent with a system's macroscopic state, so its increase is the overwhelming statistical tendency of systems to move toward more probable, more disordered configurations. This is the deep reason behind the arrow of time.

Clausius coined the word entropy in 1865 from the Greek for 'transformation', deliberately shaping it to rhyme with energy. Boltzmann (1877) and Gibbs later gave it a statistical interpretation, and the formula S = k log W is carved on Boltzmann's tombstone in Vienna.