Question

Entropy of a perfect crystal at 0 K is

• A. Negative
• B. Infinity
• C. Constant
• D. Zero

Hint:

The Third Law of Thermodynamics directly links absolute zero temperature with absolute molecular order. Flawless structure + zero thermal motion = only 1 microstate. Since \(\ln(1) = 0\), entropy must equal zero.

Answer 1

The Correct Option is D

Concept: The Third Law of Thermodynamics establishes an absolute baseline value for entropy. It states that the entropy of a pure, perfectly crystalline substance approaches exactly zero as its absolute thermodynamic temperature drops to absolute zero (\(0\text{ K}\)). Statistical Thermodynamics Formulation:
Entropy (\(S\)) can be understood at a microscopic level using the Boltzmann entropy formula: \[ S = k_B \ln \Omega \] where:
• \(k_B\) is the Boltzmann constant.
• \(\Omega\) represents the number of distinct microscopic configurations (microstates) that correspond to the macrostate of the system. In a perfect crystal, every single atom is arranged in a flawless, repetitive geometric spatial lattice. When this flawless structure is brought down to absolute zero (\(0\text{ K}\)), all internal thermal motion, vibrations, and structural translations stop completely. Because there are no thermal dislocations or alternative configurations available, the system is locked into a single microstate: \[ \Omega = 1 \] Substituting this value into Boltzmann's relation: \[ S = k_B \ln(1) = k_B \times 0 = 0 \] This absolute zero entropy baseline allows for the calculation of absolute third-law entropies for chemical substances at higher temperatures. Reviewing the other options:
• Option (1) is incorrect: According to statistical definitions, entropy cannot drop below zero for a pure substance since \(\Omega \ge 1\), making \(\ln\Omega \ge 0\).
• Option (2) is incorrect: High values approaching infinity occur at extremely high temperatures or under unconstrained volumetric expansion, not at absolute zero.
• Option (3) is incomplete: While zero is technically a constant value, stating "zero" is the precise absolute value required by the Third Law of Thermodynamics. Hence, option (4) is correct.