Residual entropy measures disorder at absolute zero, challenging the Third Law's assumption of zero entropy in perfect crystals. This concept arises from systems with multiple ground states, like ice and carbon monoxide, which maintain randomness even at the lowest temperatures.

Calculating residual entropy involves the Boltzmann equation, relating entropy to the number of microstates. Real-world examples include ice and carbon monoxide, which have non-zero residual entropy due to their molecular structures allowing multiple configurations at absolute zero.

Residual Entropy and the Third Law

Residual entropy and Third Law

• Residual entropy measures the disorder or randomness remaining in a system at absolute zero temperature (0 K)
• Third Law of Thermodynamics states entropy of a perfect crystal approaches zero as temperature approaches absolute zero
• Systems with non-zero entropy at absolute zero have residual entropy which is an exception to the Third Law rather than a contradiction
• Third Law assumes a perfect crystal with a unique ground state while systems with residual entropy have multiple degenerate ground states (ice, carbon monoxide)

Degenerate ground states

• Degenerate ground states are multiple distinct configurations of a system with the same lowest energy level
• These states are equally probable and contribute to the system's entropy at absolute zero
• Presence of degenerate ground states leads to non-zero residual entropy as the system can randomly occupy any of these states resulting in disorder even at the lowest possible temperature
• Number of degenerate ground states determines the magnitude of residual entropy with a larger number resulting in higher residual entropy

Calculating residual entropy

• Boltzmann equation relates entropy to the number of microstates in a system: $S = k_B \ln \Omega$
  • $S$ is the entropy
  • $k_B$ is the Boltzmann constant
  • $\Omega$ is the number of microstates
• For a system with degenerate ground states, residual entropy can be calculated by substituting the number of degenerate ground states for $\Omega$
  • If a system has two degenerate ground states, the residual entropy is $S_0 = k_B \ln 2$
• Boltzmann equation provides a simple way to quantify residual entropy based on the number of degenerate ground states

Examples of non-zero residual entropy

• Ice has non-zero residual entropy due to oxygen atoms forming a regular lattice while hydrogen atoms can occupy two different positions relative to each oxygen atom
  • Leads to a large number of degenerate ground states
  • Residual entropy of ice is approximately $3.4 \text{ J/(mol·K)}$
• Carbon monoxide (CO) has residual entropy as molecules can orient themselves in two different ways in solid form leading to degenerate ground states
  • Residual entropy of CO is approximately $4.4 \text{ J/(mol·K)}$
• These examples demonstrate residual entropy is not just a theoretical concept but has real-world manifestations in various materials