The transition for which the first derivative of the chemical potential with respect to temperature is continuous but its second derivative of the chemical potential with respect to temperature is discontinuous is classified as:  

CONCEPT:

Phase Transitions and Classification

• A phase transition occurs when a substance changes from one phase to another (e.g., solid to liquid, liquid to gas) and is characterized by changes in thermodynamic quantities.
• The order of a phase transition is determined by the behavior of the derivatives of the thermodynamic potential (such as chemical potential, Gibbs free energy, etc.) with respect to temperature or pressure:
  • First-order phase transitions: The first derivative of the thermodynamic potential is discontinuous (e.g., latent heat).
  • Second-order phase transitions: The first derivative is continuous, but the second derivative is discontinuous.
  • Zero-order phase transitions: Rare cases where the thermodynamic potential itself is discontinuous.

EXPLANATION:

• In the given question, the transition is described as having the first derivative of the chemical potential with respect to temperature being continuous, but the second derivative is discontinuous.
• This behavior aligns with the definition of a second-order phase transition.
• In second-order phase transitions:
  • There is no latent heat involved (no discontinuity in entropy or enthalpy).
  • Examples include the transition in superconductors and ferromagnetic materials near critical points.

Therefore, the correct answer is Option 2: Second-order phase transition.