Question

Consider two perfectly insulated vessels. Vessel 1 initially contains an ice cube at \(0^{\circ} \mathrm{C}\) and water at \(0^{\circ} \mathrm{C}\). Vessel 2 initially contains an ice cube at \(0^{\circ} \mathrm{C}\) and a saltwater solution at \(0^{\circ} \mathrm{C}\). Consider the process \(\mathrm{H}_{2} \mathrm{O}(s) \rightarrow \mathrm{H}_{2} \mathrm{O}(l).\) a. Determine the sign of \(\Delta S, \Delta S_{\text {sur, }}\) and \(\Delta S_{\text {univ }}\) for the process in vessel 1. b. Determine the sign of \(\Delta S, \Delta S_{\text {sur, }}\) and \(\Delta S_{\text {univ }}\) for the process in vessel 2. (Hint: Think about the effect that a salt has on the freezing point of a solvent.)

Short answer

In conclusion, for both vessel 1 and vessel 2, the signs of ∆S, ∆S_sur, and ∆S_univ are positive, indicating an increase in entropy. However, the ∆S_univ is smaller in vessel 2 compared to vessel 1 due to the presence of the saltwater solution.

Step-by-step solution

Vessel 1: Melting of Ice in Water

In vessel 1, we have water and an ice cube at 0°C. Let's go through the phase change process: - Temperature remains constant, and for ice to melt at 0°C, the surrounding water must absorb the heat of fusion from the ice. - Since melting is a process in which solid changes to liquid, the entropy of the system (∆S) increases, therefore, ∆S > 0. - The surrounding water absorbs the heat released by the ice. Since heat is transferred from the system to the surroundings, the entropy of the surrounding (∆S_sur) also increases, therefore, ∆S_sur > 0. - The increase in entropy for both the system and the surroundings lead to an increase in the total entropy of the universe (∆S_univ), therefore, ∆S_univ > 0.

Vessel 2: Melting of Ice in Saltwater

In vessel 2, we have a saltwater solution and an ice cube at 0°C. Since the freezing point of the saltwater solution is lower than 0°C, the process is slightly different from the one in vessel 1: - The saltwater solution is at 0°C, which is above its freezing point. Therefore, the saltwater provides the heat (absorbs less heat from the ice) needed to melt the ice cube. - Similar to vessel 1, melting of ice involves solid changing to liquid, increasing the entropy of the system (∆S) > 0. - In this case, the surrounding saltwater solution doesn't absorb as much heat as the pure water in vessel 1. The surroundings still have an increased entropy, but to a lesser extent (∆S_sur > 0, but less than in vessel 1). - As both the system and the surroundings have an increase in entropy, the total entropy of the universe (∆S_univ) also increases (∆S_univ > 0). However, since ∆S_sur is smaller compared to vessel 1, ∆S_univ also becomes smaller. In conclusion: - For both vessels, the signs of ∆S, ∆S_sur, and ∆S_univ are positive. - ∆S_univ is smaller in vessel 2 compared to vessel 1 due to the presence of the saltwater solution.

Key concepts

Thermodynamics

Thermodynamics is a field of physics concerned with heat and temperature and their relation to energy and work. It defines macroscopic variables, such as internal energy, entropy, and pressure, that partly describe a body of matter or radiation. It is explained in four fundamental laws, which describe how these quantities behave under various circumstances, and forbid certain phenomena such as perpetual motion.

Within this framework, when discussing phase changes like the melting of ice into water, thermodynamics helps us understand how energy is transferred within the system and surroundings. In the given exercise, the principles of thermodynamics are applied to two different scenarios where ice melts in pure water and saltwater, illustrating how heat transfer affects the melting process.

Heat of Fusion

Heat of fusion is the amount of heat required to convert a solid into a liquid at its melting point without changing its temperature. This is a crucial concept when examining the melting of ice at its freezing point, which is typically 0°C for pure water. For the ice to melt in vessel 1, it must absorb energy in the form of heat from the surrounding water. This energy absorption is crucial for breaking the bonds that hold the water molecules in a solid structure.

The heat of fusion is specific to the substance undergoing the phase change; for water, it is approximately 334 joules per gram. This means every gram of ice requires 334 joules of energy to melt into liquid water at 0°C, without any increase in temperature.

Freezing Point Depression

Freezing point depression is a phenomenon where the freezing point of a liquid is lowered by the addition of a solute, such as salt in water. This colligative property is vital to understand when discussing the melting of ice in a saltwater solution, as seen in vessel 2. The presence of salt means that the freezing point of the solution is below 0°C. Thus, the ice cube will melt even if the solution and the ice cube are both initially at 0°C because the ice is in contact with a liquid that is technically 'supercooled' and not frozen.

This concept has practical applications, such as in deicing roads and producing ice cream. When salt is spread on icy roads, it lowers the freezing point of the ice, causing it to melt at lower temperatures and preventing the formation of dangerous ice patches.

Entropy Change

Entropy is a measure of the disorder or randomness of a system. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. Therefore, during a phase change, such as the melting of ice, where the solid structure breaks down into a more disordered liquid, the entropy of the system increases – \( \Delta S > 0 \).

In the exercise, both vessel 1 and vessel 2 undergo phase changes where the ice melts, resulting in an entropy increase in both systems. However, vessel 2 has a smaller increase in the entropy of the surroundings compared to vessel 1 due to the presence of a saltwater solution, which affects the heat transfer process. A key takeaway is that while the entropy of the system and surroundings will always increase in these melting scenarios, the magnitude of these changes can vary depending on the composition of the surroundings, such as the presence of a solute like salt.