Problem 20 - University Physics with Modern Physics

Question

A 4.50-kg block of ice at 0.00\(^\circ\)C falls into the ocean and melts. The average temperature of the ocean is 3.50\(^\circ\)C, including all the deep water. By how much does the change of this ice to water at 3.50\(^\circ\)C alter the entropy of the world? Does the entropy increase or decrease? (\(Hint\): Do you think that the ocean temperature will change appreciably as the ice melts?)

Step-by-Step Solution

Answer

The entropy of the world increases by 5,508.82 J/K.

Step 1: Identify the Process

The process involves melting ice and then warming the melted water from 0.00\(^\circ\)C to 3.50\(^\circ\)C. There are two phases: ice melting and temperature change of resulting water.

Step 2: Calculate Entropy Change for Ice Melting

Melting of ice involves a phase change where the temperature remains constant at 0.00\(^\circ\)C. The formula for entropy change during phase change is \( \Delta S = \frac{Q}{T} \), where \( Q \) is the heat absorbed. The heat required to melt ice is calculated as \( Q = m \cdot L_f \), where \( m = 4.50 \text{ kg}\) is the mass of the ice and \( L_f = 334,000 \text{ J/kg} \) is the latent heat of fusion of ice. Hence, \( Q = 4.50 \cdot 334,000 = 1,503,000 \text{ J} \). The entropy change for melting is \( \Delta S_{melt} = \frac{1,503,000}{273.15} = 5,503.85 \text{ J/K} \).

Step 3: Calculate Entropy Change for Heating Water

The water warms from 0.00\(^\circ\)C to 3.50\(^\circ\)C. This is a sensible heat process. The formula for entropy change in this case is: \( \Delta S = m \cdot c_w \cdot \ln\left(\frac{T_f}{T_i}\right) \), where \( c_w = 4,186 \text{ J/kg}^\circ \text{C} \) is the specific heat capacity of water. Substituting the values: \( \Delta S_{heat} = 4.50 \cdot 4,186 \cdot \ln\left(\frac{276.65}{273.15}\right) \approx 4.97 \text{ J/K} \).

Step 4: Calculate Total Entropy Change

The total entropy change for the world is the sum of the ice melting and water heating: \( \Delta S_{total} = \Delta S_{melt} + \Delta S_{heat} = 5,503.85 + 4.97 = 5,508.82 \text{ J/K} \). Entropy change in the ocean is negligible due to its vast size.

Step 5: Analyze Entropy Change

Since the total entropy change \( \Delta S_{total} \approx 5,508.82 \text{ J/K} \) is positive, the entropy of the world increases.

Key Concepts

Entropy Change

Entropy is a measure of disorder in a system. When ice melts and subsequently warms up, it undergoes a change in entropy. This is due to the exchange of heat energy between the ice and its surroundings.

For a phase change or temperature change, the change in entropy, denoted as \( \Delta S \), can be calculated using the formula \( \Delta S = \frac{Q}{T} \) for phase changes, where \( Q \) is the heat absorbed, and \( T \) is the absolute temperature in Kelvin. For temperature changes that involve specific heat, we use \( \Delta S = m \cdot c_w \cdot \ln\left(\frac{T_f}{T_i}\right) \).

In the case of the ice melting into water and warming, both changes contribute to an increase in entropy. This aligns with the second law of thermodynamics, which states that the total entropy of an isolated system can never decrease over time.

Phase Transition

A phase transition refers to the transformation of a substance from one state of matter to another, such as from solid ice to liquid water. During a phase transition, the physical structure of the material changes while its temperature remains constant.

In this exercise, the ice undergoes a phase transition from solid to liquid at 0.00\(^\circ\)C. Despite absorbing heat, the temperature of the ice remains unchanged during this process until it is completely melted. This absorbed heat is used to break the bonds holding the water molecules in a solid structure, rather than increasing kinetic energy, which raises temperature.

• The phase transition requires significant energy input, known as latent heat, which we'll explore further.
• The entropy change during this process signifies an increase in disorder, as the structured, solid ice becomes unstructured, flowing liquid water.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is an intrinsic property of materials and is represented by the symbol \( c \).

For water, which has a high specific heat capacity of approximately 4,186 J/kg\(^\circ\)C, this means a substantial amount of heat is needed to increase its temperature. This property of water plays a crucial role in environmental and climate regulation, as it can absorb or release heat with minimal temperature change.

• In the given problem, once the ice has melted into water, it absorbs heat from the surroundings to warm up.
• This heat absorption changes the temperature from 0.00\(^\circ\)C to 3.50\(^\circ\)C and results in a further incremental increase in entropy.

Latent Heat of Fusion

The latent heat of fusion is the amount of heat needed to convert a substance from a solid to a liquid state without changing its temperature. For ice, the latent heat of fusion is a substantial 334,000 J/kg.

This means that to melt ice, a significant amount of energy must be provided just to break the bonds between molecules in the solid state. However, this occurs without any temperature change until the transition is complete.

• The concept of latent heat is crucial during the melting process of ice, as it determines the energy requirements for phase transition.
• In the exercise, a large chunk of heat energy is used purely for the phase change from solid to liquid, which is reflected in the entropy change calculation.

In practice, the heat for melting is absorbed from the surrounding environment, causing an increase in entropy because the structured crystalline form of ice becomes the more disordered form of liquid water.