Problem 23 - University Physics with Modern Physics

Question

A 15.0-kg block of ice at 0.0\(^\circ\)C melts to liquid water at 0.0\(^\circ\)C inside a large room at 20.0\(^\circ\)C. Treat the ice and the room as an isolated system, and assume that the room is large enough for its temperature change to be ignored. (a) Is the melting of the ice reversible or irreversible? Explain, using simple physical reasoning without resorting to any equations. (b) Calculate the net entropy change of the system during this process. Explain whether or not this result is consistent with your answer to part (a).

Step-by-Step Solution

Answer

(a) The process is irreversible. (b) The net entropy change is 1,250 J/K, consistent with irreversibility.

Step 1: Understanding the Scenario

We are given a 15.0-kg block of ice at 0°C melting to liquid water at the same temperature inside a room at 20°C. The system is considered isolated, with the room's temperature change being negligible.

Step 2: Conceptual Analysis for Reversibility (Part a)

Melting of ice at 0°C in a room at 20°C is irreversible. In a reversible process, the system and surroundings can be returned to their original states without any net change. This melting involves heat absorption from a warmer environment, which cannot be reversed without external work being performed. Hence, heat flows spontaneously in one direction, indicating irreversibility.

Step 3: Apply the Entropy Formula (Part b)

Entropy change in the system can be calculated using the formula for entropy change during a phase change: \[\Delta S = \frac{Q}{T}\]where \(Q\) is the heat absorbed and \(T\) is the temperature in Kelvins. Here, \(Q = mL_f\), with \(m = 15.0 \text{ kg}\) and \(L_f = 334,000 \text{ J/kg}\) as the latent heat of fusion for ice.

Step 4: Calculate Heat Absorbed

The heat absorbed during melting is calculated as:\[Q = mL_f = 15.0 \times 334,000 = 5,010,000 \text{ J}\]

Step 5: Calculate Entropy Change for Ice

The entropy change for the ice as it melts is:\[\Delta S_{\text{ice}} = \frac{Q}{T} = \frac{5,010,000}{273.15} = 18,340 \text{ J/K}\]

Step 6: Consider Entropy Change of the Room

The room loses the same amount of heat \(Q\) but at a different temperature. Therefore, its entropy change is:\[\Delta S_{\text{room}} = -\frac{5,010,000}{293.15} = -17,090 \text{ J/K}\]

Step 7: Compute Net Entropy Change of the System

The net entropy change \(\Delta S_{\text{net}}\) of the system is:\[\Delta S_{\text{net}} = \Delta S_{\text{ice}} + \Delta S_{\text{room}} = 18,340 - 17,090 = 1,250 \text{ J/K}\] This positive value indicates an increase in entropy.

Step 8: Consistency with Irreversibility

A positive net entropy change confirms the process is irreversible. According to the second law of thermodynamics, entropy of an isolated system never decreases, reflected here by the net increase in entropy.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, energy, and work. It is fundamental to understanding how energy moves and transforms in physical systems. This field is governed by four primary laws that describe how energy behaves within these systems.

An important concept in thermodynamics is the idea of an isolated system. In such a system, no energy or matter is exchanged with the surroundings. This makes it useful for analysis as external variables remain constant. In our problem, the melting of ice happens in an isolated setting, simplifying calculations.

The second law of thermodynamics plays a major role in understanding energy distribution. It states that in any natural process, the total entropy of an isolated system can never decrease over time. This law helps explain why certain processes are irreversible and why heat flows spontaneously from hot to cold regions.

Irreversible Process

An irreversible process is one where the system and its environment cannot be returned to their original states without a net change in the universe. These processes are marked by spontaneous progression and often involve dissipative effects, such as friction or resistance to changes in energy states.

In the scenario of melting ice in a warm room, the process is clearly irreversible. Heat naturally flows from the warmer room to the colder ice. This heat transfer increases the disorder, or entropy, of the system. Once the ice has melted without external intervention, it cannot revert to its initial solid state spontaneously.

Understanding irreversible processes is crucial, as many real-world phenomena, like the aforementioned, do not spontaneously reverse. By recognizing the direction of entropy change, we can identify irreversible processes and predict how systems evolve over time.

Phase Transition

A phase transition is the transformation of a substance from one state of matter to another, such as solid to liquid or liquid to gas. During these transitions, the substance absorbs or releases heat but remains at a constant temperature. This absorbed or released energy is called latent heat.

In the example of ice melting, the phase transition occurs from solid to liquid while the temperature remains at 0°C. The energy absorbed from the surrounding room increases the internal energy of the ice, allowing it to change state without altering its temperature.

Phase transitions are significant because they involve energy changes without temperature changes. Understanding these dynamics is crucial in thermodynamics, as it helps explain energy distribution during changes of state, as seen in our exercise.

Latent Heat

Latent heat is the amount of heat needed to change the phase of a substance without changing its temperature. It is a key factor in phase transitions, such as the melting of ice or boiling of water.

For the melting ice, the latent heat of fusion is the energy required to change the ice at 0°C to liquid water at the same temperature. In our problem, the latent heat of fusion for ice is given as 334,000 J/kg. Thus, a significant amount of heat from the room, calculated as 5,010,000 J, is absorbed by the ice to initiate the phase transition.

Understanding latent heat is essential in analyzing phase transitions, as it quantifies the energy involved without affecting the substance's temperature. Recognizing this helps evaluate the energy needs and adjustments in systems undergoing phase changes.