Statistical Entropy and the Third Law

Statistical mechanics gives entropy a molecular meaning as a measure of the number of accessible microstates, which in turn explains why the entropy of a perfect crystal approaches zero at absolute zero.

Definition

Statistical entropy is the molecular measure of entropy as proportional to the logarithm of the number of microstates consistent with a system's macrostate, and the third law follows from the uniqueness of the ground state of a perfect crystal at absolute zero.

Scope

This topic covers the statistical definition of entropy and its link to the third law: the Boltzmann relation between entropy and the logarithm of the number of microstates, the Gibbs entropy expression, and the calculation of entropy from the partition function. It develops the third law as the statement that a perfect crystal has a single ground microstate and hence zero entropy at absolute zero, the concept of residual entropy arising from frozen-in disorder, and the consequent calculation of absolute entropies. The general Boltzmann distribution and partition function are treated in sibling topics.

Core questions

• How does the Boltzmann relation connect entropy to the number of microstates?
• How is entropy calculated from the partition function?
• Why does the entropy of a perfect crystal approach zero at absolute zero?
• What is residual entropy, and why does it arise in some substances?

Key concepts

• Boltzmann entropy and microstates
• Gibbs entropy expression
• Entropy from the partition function
• Third law and the perfect crystal
• Residual entropy

Key theories

Boltzmann's entropy relation

Entropy is proportional to the logarithm of the number of microstates compatible with the macroscopic state, giving a molecular foundation for the second law and explaining the spontaneous tendency toward states of higher multiplicity.

Statistical basis of the third law

At absolute zero a perfect crystal occupies a single non-degenerate ground microstate, so its statistical entropy is zero; deviations such as residual entropy reveal disorder frozen in before the system could reach this unique state.

Clinical relevance

The statistical interpretation of entropy provides absolute entropies for thermochemical calculations, explains residual entropy in substances such as carbon monoxide and ice, and gives the molecular basis for understanding spontaneity, mixing, and the limits of cooling toward absolute zero.

History

Boltzmann's relation between entropy and microstates, engraved on his tombstone, dates from the 1870s; Nernst's heat theorem of 1906 became the third law, and Pauling's 1935 explanation of the residual entropy of ice confirmed the statistical picture by linking it to frozen-in proton disorder.

Key figures

• Ludwig Boltzmann
• Walther Nernst
• Linus Pauling

Seminal works

• mcquarrie1997
• atkins2018

Frequently asked questions

What does it mean physically for entropy to count microstates?

A macrostate that can be realized in many microscopic arrangements has high entropy; entropy therefore measures how many indistinguishable molecular configurations correspond to the same observable state, which is why spreading energy and matter out increases it.

Why do some substances have nonzero entropy even at absolute zero?

If a substance freezes into more than one nearly equivalent arrangement before reaching its true ground state, that disorder becomes locked in; the leftover residual entropy reflects the number of frozen-in configurations, as in carbon monoxide and ice.