Residual Entropy and the Third Law of Thermodynamics

Residual Entropy

The Third Law of Thermodynamics states that the entropy of a pure, perfectly ordered crystalline substance approaches zero as the temperature approaches absolute zero (0 K).

However, some crystalline substances can retain a degree of configurational or orientational disorder even at very low temperatures. Consequently, their entropy may approach a finite value rather than zero. This finite entropy is called residual entropy.

Important: Residual entropy does not mean that the Third Law fails. The Third Law refers to a perfectly ordered crystal in its unique ground state. Residual entropy occurs when more than one microscopic arrangement remains accessible in the actual solid.

Boltzmann Equation for Residual Entropy

Residual entropy can be related to the number of accessible microscopic configurations by the Boltzmann equation:

S = k ln W

where k is the Boltzmann constant and W represents the number of accessible microscopic arrangements or configurations of the system.

Example: Residual Entropy of Carbon Monoxide

Carbon monoxide (CO) provides a classic example of residual entropy. In an idealized CO crystal, each CO molecule can adopt two orientations, CO or OC, with approximately equal probability.

For a crystal containing N CO molecules, the number of possible arrangements is:

W = 2^N

Therefore, according to the Boltzmann equation:

S_res = k ln W

S_res = k ln(2^N)

S_res = Nk ln 2

For one mole of CO, Nk = R. Hence:

S_res = R ln 2

Using R = 8.314 J K⁻¹ mol⁻¹:

S_res = (8.314)(ln 2)

S_res ≈ 5.76 J K⁻¹ mol⁻¹

Origin of Residual Entropy

The magnitude of residual entropy depends on the number of accessible microscopic arrangements that remain possible in the solid. If molecules, atoms, or groups can occupy more than one orientation or configuration, the crystal can retain configurational disorder at very low temperature.

Thus, residual entropy is associated primarily with orientational or configurational disorder that remains frozen into the structure of the solid.

Residual Entropy of Ice

Ice provides another important example. In ordinary ice, the hydrogen atoms are subject to structural constraints, but more than one arrangement of hydrogen atoms can satisfy the hydrogen-bonding requirements of the ice structure. This proton disorder gives rise to a residual entropy.

Why Does Residual Entropy Not Violate the Third Law?

The Third Law concerns a perfect crystal with a unique, perfectly ordered ground state. Such a crystal has zero entropy at absolute zero.

A real crystal exhibiting residual entropy has multiple accessible microscopic arrangements or a disordered ground-state configuration. Therefore, its finite residual entropy reflects the nature of its ground-state disorder rather than a failure of the Third Law.

Limitations and Practical Considerations

The Third Law provides the basis for assigning absolute entropy values, but real substances may contain defects, impurities, orientational disorder, or other forms of structural disorder. Consequently, the experimentally observed entropy at very low temperature may include a residual contribution.

• Perfect crystals are ideal thermodynamic reference states.
• Real crystals may contain defects or configurational disorder.
• Some solids retain orientational disorder at very low temperatures.
• Such disorder can produce a finite residual entropy.
• Residual entropy does not contradict the Third Law when the crystal is not a unique perfectly ordered ground state.