Question

Molar heat capacity of water in equilibrium with ice at constant pressure is: (a) Zero (b) Infinity (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 is infinity due to the phase change at constant temperature.

Step-by-step solution

Defining the System

Identify the system we are working with, which in this case is water in equilibrium with ice. When water and ice are in equilibrium, it is at their melting point.

Understanding Molar Heat Capacity

Molar heat capacity refers to the amount of heat needed to increase the temperature of one mole of a substance by 1 Kelvin at constant pressure.

Analyzing the Equilibrium Condition

Since water and ice are in equilibrium at the melting point, any heat added to the system goes into changing the phase of the substance (from ice to water) rather than raising the temperature.

Realizing the Implication of Phase Change

During the phase change, the temperature remains constant even though heat is being added to the system. Therefore, the molar heat capacity is theoretically infinite because the temperature does not change.

Conclusion

In systems where phase change occurs without a temperature change, the molar heat capacity is considered to be infinite as heat added does not raise the temperature. In this case, the correct answer is (b) Infinity.

Key concepts

Phase Change

In thermodynamics, a phase change is a process where a substance transitions between different states of matter, such as from solid to liquid, liquid to gas, and vice versa. A classic example of phase change is the melting of ice into water. During this process, the temperature of the system remains constant even though heat is consistently added. This is because the added energy is utilized for breaking the intermolecular bonds, which changes the state of the matter. Phase change requires specific heat, known as latent heat, different from the heat needed to change the temperature of the substance. Common types of phase changes include:

• Melting: Transition from solid to liquid.
• Freezing: Transition from liquid to solid.
• Evaporation: Transition from liquid to gas.
• Condensation: Transition from gas to liquid.
• Sublimation: Transition from solid to gas without passing through an intermediary liquid state.

Understanding phase change is crucial for analyzing systems like water in equilibrium with ice at their melting point.

Equilibrium at Melting Point

Equilibrium at the melting point refers to a balanced state where the rate of melting equals the rate of freezing. For water and ice, this occurs at 0 degrees Celsius under standard atmospheric pressure. At this point, both the solid and liquid phases coexist in equilibrium, meaning no net change in the amount of ice or water is observed until external conditions shift. In such equilibrium:

• The system's temperature remains constant because it is at the exact temperature where the solid and liquid phases can coexist stably.
• The heat absorbed or released is used for altering the phase rather than changing the temperature.

This concept is vital in understanding why the temperature remains unchanged during phase transitions, contrary to situations without phase changes where temperature directly correlates with heat addition.

Constant Pressure Heat Transfer

When heat is transferred at constant pressure, the process alters the system's internal energy and can cause the system to do work on its surroundings or vice versa. However, when water is in equilibrium with ice specifically at its melting point, the heat added does not lead to an increase in temperature due to the phase change. Key points about heat transfer at constant pressure include:

• Molar heat capacity plays a critical role in understanding how much heat causes a temperature change under constant pressure.
• During a phase change like melting, the entire energy input goes into the phase transition, maintaining temperature uniformity.

In this unique scenario, where temperature remains constant despite heat addition, the molar heat capacity is effectively infinite because theoretically, it would require an infinite amount of heat to alter the temperature during the phase transition.