Entropy of a crystal as T→0

Problem

Discuss why the entropy of a perfect crystal tends to zero as $T\to 0$ (third law of thermodynamics).

Solution

As $T\to 0$ thermal agitation vanishes and a perfect crystal settles into its single ground state: there is only one microstate compatible with the macrostate, $\Omega=1$. From Boltzmann’s formula: $S=k_B\ln\Omega=k_B\ln 1=0$ This is the content of the third law: once thermal agitation is zeroed out, there is perfect order and zero “ignorance” about the microscopic state, hence zero entropy. $\ev{S\to 0\ \text{as}\ T\to 0}$

Links

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