A 4.504.50-kg block of ice at 0.000.00°C falls into the ocean and melts. The average temperature of the ocean is 3.503.50°C, including all the deep water. By how much does the change of this ice to water at 3.503.50°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?)

Ch 20: The Second Law of Thermodynamics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 20: The Second Law of Thermodynamics

Problem 20

Chapter 20, Problem 20

A $4.50$ -kg block of ice at $0.00$ °C falls into the ocean and melts. The average temperature of the ocean is $3.50$ °C, including all the deep water. By how much does the change of this ice to water at $3.50$ °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?)

Verified step by step guidance

Step 1: Understand the problem. The block of ice melts and warms up to the ocean's temperature. The entropy change involves two processes: (1) the melting of ice at 0°C, and (2) the warming of the resulting water from 0°C to 3.5°C. The ocean's temperature is assumed constant due to its large thermal mass.

Step 2: Calculate the entropy change for the melting of ice. Use the formula ΔS = Q/T, where Q is the heat absorbed during melting and T is the temperature in kelvins. The heat absorbed during melting is Q = mL, where m is the mass of the ice and L is the latent heat of fusion of ice (L = 334,000 J/kg). Convert the temperature to kelvins: T = 273.15 K.

Step 3: Calculate the entropy change for warming the water. Use the formula ΔS = ∫(dQ/T), where dQ = mcΔT. Here, m is the mass of the water, c is the specific heat capacity of water (c = 4,186 J/(kg·K)), and ΔT is the temperature change (3.5°C - 0°C). Integrate over the temperature range from 273.15 K to 276.65 K.

Step 4: Combine the entropy changes. Add the entropy change from the melting process to the entropy change from warming the water to find the total entropy change of the ice-water system.

Step 5: Consider the ocean's entropy change. The heat lost by the ocean is equal to the heat gained by the ice-water system, but the ocean's temperature remains constant. Use ΔS = Q/T for the ocean, where T is the ocean's temperature in kelvins (276.65 K). Compare the entropy changes of the ice-water system and the ocean to determine if the total entropy of the world increases or decreases.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Entropy

Entropy is a measure of the disorder or randomness in a system. In thermodynamics, it quantifies the amount of energy in a physical system that is not available to do work. When ice melts into water, the entropy of the system increases because the molecules in liquid water are more disordered than in solid ice. This concept is crucial for understanding how energy transformations affect the overall disorder of a system.

Phase Change

A phase change refers to the transition of a substance from one state of matter to another, such as from solid to liquid. During this process, energy is absorbed or released, but the temperature of the substance remains constant until the phase change is complete. In this scenario, the melting of ice at 0°C to water at 0°C involves a phase change that requires heat energy, which is essential for calculating the change in entropy.

Heat Transfer

Heat transfer is the process of thermal energy moving from one object or substance to another due to a temperature difference. In this case, the heat from the warmer ocean water is absorbed by the ice, causing it to melt. Understanding heat transfer is vital for analyzing how the melting ice affects the temperature and entropy of the surrounding environment, particularly in relation to the ocean's average temperature.

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Textbook Question

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