Molar heat capacity of water in equilibrium with ice at constant pressure is (a) zero (b) infinity (c) \(40.45 \mathrm{~kJ} / \mathrm{K}\) -mol (d) \(75.48 \mathrm{~J} / \mathrm{K}-\mathrm{mol}\)

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In other words, it studies how energy transitions occur and how energy affects matter, especially in terms of heat exchange. A key concept in thermodynamics is the conservation of energy, which means that energy can neither be created nor destroyed, only transformed from one form to another.

Under this framework, we can understand the molar heat capacity as a way to measure how much energy in the form of heat is required to raise the temperature of a mole of a substance by one Kelvin. It's a thermodynamic property that gives us vital clues about how a substance behaves when it interacts with heat. For substances in phase equilibrium, such as water and ice, thermodynamics tells us that when they are at their melting point, heat does not increase temperature but instead changes the phase of the substance.

Phase equilibrium occurs when a substance is at a state where multiple phases (such as solid, liquid, and gas) exist together without a net change over time. At equilibrium, the rates of transition between phases are equal and opposite, meaning as much ice melts into water as water freezes into ice. This concept is crucial when dealing with systems at the border of phase changes, such as the co-existence of ice and water at 0°C.

Understanding phase equilibrium helps us grasp why the molar heat capacity at this point appears to be undefined or infinite. It's because any heat added to the system induces a phase change rather than a temperature change. The system absorbs or releases heat as it maintains equilibrium, which is why, during this state, traditional definitions of heat capacity break down.

The heat capacity of water is remarkably high compared to many other substances. This physical property means that water can absorb or release a large amount of heat with little change in temperature. Specifically, the molar heat capacity is a detailed measure reflecting this characteristic for a mole of water.

However, at the phase transition point where ice and water are in equilibrium, the concept of molar heat capacity changes significantly. Because the added heat is utilized in altering the phase rather than raising the temperature, we can't quantify the molar heat capacity in usual terms. In traditional contexts, adding heat to water increases its temperature until it reaches the boiling point. But at 0°C, the addition of heat results in melting, and since temperature does not increase, the heat capacity, if defined by temperature change, is effectively infinite.