Entropy and disorder

Problem

Explain why the entropy of a messy room is NOT greater than that of a tidy room in the thermodynamic sense, despite what common sense suggests.

Solution

Thermodynamic entropy counts molecular microstates, i.e. the possible configurations of positions and velocities of the roughly $10^{25}$ molecules present, not the macroscopic arrangement of objects. Moving books and clothes around changes only very few macroscopic “states” and shifts a negligible amount of molecules: the contribution to $S=k_B\ln\Omega$ is trivial compared with the enormous number of molecular microstates, which remains practically unchanged. The “disorder” of a room is therefore a popular metaphor, not Boltzmann entropy: the two concepts do not coincide. $\ev{\text{macroscopic disorder}\neq S_\text{thermodynamic}}$

Links

Topics: Entropy and the second law Concepts: Entropy Skills: Micro-macro interpretation