A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In 55 minutes of operation, the heat rejected by the engine melts 0.04000.0400 kg of ice. During this time, how much work WW is performed by the engine?

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 18

Chapter 20, Problem 18

A Carnot heat engine uses a hot reservoir consisting of a large amount of boiling water and a cold reservoir consisting of a large tub of ice and water. In $5$ minutes of operation, the heat rejected by the engine melts $0.0400$ kg of ice. During this time, how much work $W$ is performed by the engine?

Verified step by step guidance

Understand the concept of a Carnot heat engine: It operates between two thermal reservoirs, absorbing heat from the hot reservoir and rejecting some of it to the cold reservoir, while performing work.

Identify the key information: The heat rejected by the engine melts 0.0400 kg of ice. The latent heat of fusion for ice is approximately 334,000 J/kg.

Calculate the heat rejected to the cold reservoir: Use the formula for the heat required to melt ice, $ Q_c = m \cdot L_f $, where $ m $ is the mass of ice and $ L_f $ is the latent heat of fusion.

Apply the Carnot efficiency formula: $ \eta = 1 - \frac{T_c}{T_h} $, where $ T_c $ and $ T_h $ are the temperatures of the cold and hot reservoirs, respectively. This will help determine the relationship between the heat absorbed $ Q_h $, the heat rejected $ Q_c $, and the work done $ W $.

Use the relationship $ W = Q_h - Q_c $ to find the work done by the engine, knowing $ Q_c $ and using the efficiency to find $ Q_h $.

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Key Concepts

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

Carnot Heat Engine

A Carnot heat engine is an idealized engine that operates between two thermal reservoirs, converting heat into work with maximum efficiency. It follows the Carnot cycle, which consists of two isothermal processes and two adiabatic processes. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy required to change a substance from solid to liquid at its melting point without changing its temperature. For ice, this value is approximately 334,000 J/kg. In the context of the problem, this concept helps determine the heat rejected by the engine, as it is used to melt the ice.

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed. In the context of a heat engine, it implies that the work done by the engine is equal to the heat absorbed from the hot reservoir minus the heat rejected to the cold reservoir. This principle is crucial for calculating the work performed by the engine.

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Related Practice

Textbook Question

A Carnot refrigerator is operated between two heat reservoirs at temperatures of $320$ K and $270$ K. If in each cycle the refrigerator receives $415$ J of heat energy from the reservoir at $270$ K, how many joules of heat energy does it deliver to the reservoir at $320$ K?

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Textbook Question

A sophomore with nothing better to do adds heat to $0.350$ kg of ice at $0.0$ °C until it is all melted. What is the change in entropy of the water?

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Textbook Question

A Carnot refrigerator is operated between two heat reservoirs at temperatures of $320$ K and $270$ K. What is the coefficient of performance of the refrigerator?

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Textbook Question

You decide to take a nice hot bath but discover that your thoughtless roommate has used up most of the hot water. You fill the tub with $195$ kg of $30.0$ °C water and attempt to warm it further by pouring in $5.00$ kg of boiling water from the stove. Is this a reversible or an irreversible process? Use physical reasoning to explain.

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Textbook Question

A Carnot refrigerator is operated between two heat reservoirs at temperatures of $320$ K and $270$ K. If the refrigerator completes $165$ cycles each minute, what power input is required to operate it?

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Textbook Question

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?)

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