The Third Law and Absolute Entropies

The statement

The entropy of a perfect crystalline substance approaches zero as the temperature approaches absolute zero.

The statistical justification is direct: entropy is proportional to the logarithm of the number of accessible microstates. A perfect crystal at absolute zero has exactly one arrangement, and the logarithm of one is zero.

Consequences

• Absolute entropies can be tabulated, since integrating heat capacity from absolute zero gives a real value rather than a difference.
• Entropy changes of reaction can be computed directly from those absolute values, without needing formation entropies.
• Heat capacity must approach zero as temperature approaches zero, or the entropy integral would diverge.
• Absolute zero is unattainable in a finite number of steps, which is an alternative statement of the law.

The heat capacity consequence is worth noting because it is experimentally confirmed and was not obvious beforehand — classical theory predicts a constant heat capacity at low temperature, which would make the integral diverge.

Calculating an absolute entropy

Integrate the heat capacity divided by temperature from absolute zero to the temperature of interest, adding the entropy of each phase change encountered:

Each phase change contributes a discontinuous jump, since the transition occurs at constant temperature. Fusion and vaporisation both contribute, and vaporisation contributes far more because the entropy change is much larger.

Residual entropy

Some substances retain measurable entropy at very low temperature. This happens when the crystal has more than one arrangement of essentially equal energy, so it is frozen into a disordered state as it cools rather than settling into one arrangement.

Molecules whose two orientations in the lattice differ very little in energy are the standard case: as the crystal cools, molecules become locked in whichever orientation they happened to have, leaving a distribution rather than a single arrangement.

The measured residual entropy matches the value calculated from the number of possible arrangements, which is a striking confirmation of the statistical interpretation of entropy. Explaining residual entropy is therefore a good question, because it tests whether entropy is understood as a count of arrangements rather than as a vague measure of disorder.

Standard entropy trends

All four trends reduce to the same principle: more accessible energy levels means more microstates means higher entropy. Being able to state that single reason rather than four separate ones is what a good answer does.

Frequently asked questions

Why can entropy have an absolute value when enthalpy cannot?

Because the third law provides a genuine zero point — a perfect crystal at absolute zero has exactly one microstate. There is no comparable natural zero for enthalpy.

What causes residual entropy?

Multiple arrangements of nearly equal energy that become frozen in as the crystal cools, so it does not reach a single ordered state.

Why must heat capacity go to zero at absolute zero?

Because otherwise the integral of C_p/T would diverge and the entropy would be infinite rather than zero, contradicting the third law.

Why is absolute zero unattainable?

Because each cooling step removes a decreasing fraction of the remaining entropy, so an infinite number of steps would be required to reach zero exactly.