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 \(\Delta S, \Delta S_{\text { sum }}\) and \(\Delta S_{\text { univ }}\) for the process in vessel 1 . b. Determine the sign of \(\Delta S, \Delta S_{\text { sum }},\) 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

a. In Vessel 1, the signs of $\Delta S$, $\Delta S_{sum}$, and $\Delta S_{univ}$ are all positive, as the phase change increases the disorder of the system. b. In Vessel 2, the signs of $\Delta S$, $\Delta S_{sum}$, and $\Delta S_{univ}$ are also positive, for the same reason as in Vessel 1.

Step-by-step solution

Understand the process in Vessel 1

Vessel 1 contains an ice cube and water. As the ice cube receives energy, it goes through a phase change and becomes liquid water. We want to determine what happens to the entropy during this transformation.

Calculate the sign of ∆S for Vessel 1

The process of melting (H2O(s) → H2O(l)) involves a change from a more ordered state (solid) to a less ordered state (liquid). As a result, the entropy change, ∆S, for this process will be positive.

Determine the sign of ∆S_sum for Vessel 1

Both the system and surroundings are perfectly insulated, meaning no exchange of energy with the outside environment. Thus, the overall entropy change (∆S_sum) is the same as the entropy change of the system (∆S). In this case, ∆S_sum is positive as well.

Determine the sign of ∆S_univ for Vessel 1

Under these conditions, the universe consists only of the system (Vessel 1) and not the surroundings. Therefore, the entropy change of the universe, ∆S_univ, is equal to the entropy change of the system, ∆S. For this situation in Vessel 1, ∆S_univ is also positive. b. Vessel 2: Ice and Saltwater mixture

Understand the process in Vessel 2

Vessel 2 contains an ice cube and saltwater solution. Due to the salt, the freezing point of water is lowered. We want to determine what happens to the entropy during the ice melting process in this case.

Calculate the sign of ∆S for Vessel 2

Similar to Vessel 1, in the process of ice melting (H2O(s) → H2O(l)), the ice cube turns into a liquid. This results in a more disordered state. Thus, the entropy change, ∆S, for this process in Vessel 2 will also be positive.

Determine the sign of ∆S_sum for Vessel 2

Like in Vessel 1, the system and surroundings are perfectly insulated, meaning no exchange of energy with the environment. Therefore, the overall entropy change (∆S_sum) is the same as the entropy change of the system (∆S). In this case, ∆S_sum is also positive.

Determine the sign of ∆S_univ for Vessel 2

Again, considering that the universe only consists of the system (Vessel 2), the entropy change of the universe, ∆S_univ, is equal to the entropy change of the system, ∆S. For this situation in Vessel 2, ∆S_univ is also positive.

Key concepts

Phase Change

When a substance undergoes a phase change, its state changes, for example from solid to liquid or liquid to gas. This transformation in state often involves absorption or release of energy in the form of heat.
During the process of melting, such as ice turning to water, the substance absorbs energy. This energy is used to break the bonds holding the water molecules in a structured, solid form.

• Ice, being in a structured solid form, is more ordered.
• Liquid water, on the other hand, is less ordered than ice.

This increase in disorder indicates an increase in entropy, which is a measure of disorder in a system. So, as ice melts and becomes liquid water, the entropy increases.

Freezing Point Depression

Freezing point depression is a colligative property, which means it depends on the number of dissolved particles in a solution rather than the type of particles. When a solute, such as salt, is dissolved in a solvent like water, it lowers the freezing point of the solvent.
In the context of the exercise, the saltwater solution in Vessel 2 has a lower freezing point than pure water. This effect can be explained by the interruption of the solute particles with the formation of the solid phase of the solvent.

• Due to the presence of salt, ice in saltwater requires a colder temperature to remain solid.
• The additional disorder contributed by the dissolved salt means that entropy increases when ice melts in a saltwater solution.

Therefore, adding salt results in a greater degree of melting at a temperature where pure water would normally remain solid.

Thermodynamics

Thermodynamics is the scientific study of heat, work, temperature, and energy transformations. One of its key principles is that energy in a system and its surroundings is conserved. When analyzing processes like phase changes, the concept of entropy becomes crucial.
Within an insulated system, such as the two vessels described, no energy enters or leaves the system from the surroundings. In such a scenario:

• The change in entropy of the system (\( \Delta S \)) is only due to internal changes.
• The increase in disorder during melting means that \( \Delta S \) is positive.
• In a perfectly insulated system, the total entropy (\( \Delta S_{ ext{sum}} \)) will also be driven only by the changes happening within.

This means all phases of a system, whether melting or freezing, are analyzed under conditions imposed by thermodynamic laws.

Insulated System

An insulated system prevents any exchange of heat or energy with the surroundings. This makes it an ideal setup for analyzing thermodynamic processes purely based on internal changes.
For both vessels in this exercise, because they are insulated, the only changes in energy must occur internally, affecting the system’s entropy.

• In Vessel 1, ice melting into water occurs, increasing system entropy due to phase change.
• In Vessel 2, the presence of salt introduces additional complexity but still results in increased entropy through an even more energized melting process.

Thus, the insulated nature of the vessels ensures that the entropy change of the system mirrors the total entropy change of the universe, keeping the focus on internal energy dynamics.