Question

Molar heat capacity of water in equilibrium with ice at constant pressure is (a) zero (b) infinity \((\infty)\) (c) \(40.45 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) (d) \(75.48 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\)

Short answer

The molar heat capacity of water in equilibrium with ice is infinity, option (b).

Step-by-step solution

Understand Molar Heat Capacity

Molar heat capacity is the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin at constant pressure. It depends on the physical state of the substance.

State of Water in Equilibrium with Ice

When water is in equilibrium with ice, it is at the melting/freezing point, and this equilibrium means no temperature change occurs when small amounts of heat are added or removed.

Heat Capacity at Phase Equilibrium

Since adding or removing heat does not change the temperature of a system in equilibrium between two phases (like ice and water), the molar heat capacity for water in equilibrium with ice is theoretically infinite. This is because you would need an infinite amount of heat change to make any temperature change, which isn't possible.

Select the Correct Option

Given that the molar heat capacity at equilibrium is indeed infinite, the correct answer is option (b), infinity \(\infty\).

Key concepts

Phase Equilibrium

When a substance exists in two phases simultaneously, such as water and ice coexisting at 0°C, it is said to be in phase equilibrium. For phase equilibrium to occur, conditions must be just right, allowing for both phases to be stable over time without transferring energy in a way that causes further phase change. In the case of ice and water at 0°C (or 32°F), the energy added to or removed from the system is used to convert one phase to the other without changing the temperature. This conversion process is essential to understanding why molar heat capacity at this point is infinite. Under phase equilibrium, any heat added goes into transforming ice to liquid water or vice versa, with no impact on the system's overall temperature. This phenomenon demonstrates that during phase changes, like melting or freezing, temperature remains constant until the entire phase transition is complete. Thus, even if you add a small or large amount of heat, the temperature holds steady until the ice fully melts or water completely freezes.

Temperature Change

Temperature change is a concept closely related to the application of heat. In most situations, when heat is added to a substance, the temperature increases. Conversely, removing heat causes the temperature to decrease. However, during phase equilibrium, as seen with ice and water, heat added or removed does not change the temperature. This is because the energy is used for the phase transition rather than for altering the temperature. Consider an example where you add heat to an ice-water mixture at 0°C. Instead of seeing a temperature increase, the energy facilitates the melting of ice, maintaining the temperature constant throughout the process. This constant temperature fact notably makes phase equilibrium scenarios intriguing cases in thermodynamics, where usual rules about heat causing temperature change seem to bend.

Constant Pressure Heat Capacity

Knowing about constant pressure heat capacity ( C_p ) is crucial because it determines how much heat a substance needs to change temperature when pressure remains unchanged. This concept helps us quantify how heat flows in or out of a substance without altering its physical state. For substances not undergoing phase change, C_p gives a clear indication of the heat requirement for temperature changes. For instance, heating water from 20°C to 30°C involves using the specific molar heat capacity to calculate the heat needed for that temperature rise. During phase changes like melting or boiling, C_p tends to diverge from regular expectations. In the case of water in equilibrium with ice, because no temperature change happens despite heat exchange, C_p theoretically reaches infinity. This means, in theory, you'd need an infinite amount of heat to effectuate a temperature change while remaining in phase equilibrium, highlighting the unique nature at constant pressure during phase transitions.