Heat capacity during phase equilibrium: What is the molar heat capacity of water in equilibrium with ice at constant pressure (at the melting point)?

Introduction / Context:
Heat capacity at constant pressure Cp describes how much heat is required to raise the temperature of a mole of substance by 1 degree at constant pressure. At a first-order phase transition (e.g., melting), added heat changes phase at essentially constant temperature, which has implications for Cp behavior.

Given Data / Assumptions:

• System: water in equilibrium with ice at the melting point and 1 atm.
• Quasi-static addition of heat.
• No superheating or supercooling effects.

Concept / Approach:
At phase equilibrium, an infinitesimal addition of heat does not increase temperature; it goes into latent heat of fusion. In the mathematical definition Cp = (dQ/dT)_p per mole, as dT approaches zero while dQ remains finite (latent heat), Cp tends to infinity.

Step-by-Step Solution:
At the melting point, temperature remains constant during phase change.A finite dQ = L_f is absorbed with dT ≈ 0.Therefore Cp = dQ/dT → ∞ at the transition.Select '∞' to represent divergence.

Verification / Alternative check:
Thermodynamic treatments of first-order transitions predict discontinuities or divergences in response functions like Cp at coexistence lines, consistent with calorimetric observations.

Why Other Options Are Wrong:
0 or 1 do not reflect latent-heat absorption at constant temperature; 'None of these' is unnecessary given a correct option is available.

Common Pitfalls:

• Confusing Cp at the phase boundary with Cp of a single phase just off the boundary.
• Ignoring that 'infinite' here means unbounded in the limit, not a numerically measurable infinity.

Final Answer:
∞