A Carnot engine is operated between two heat reservoirs at temperatures of 520520 K and 300300 K. If the engine receives 6.456.45 kJ of heat energy from the reservoir at 520520 K in each cycle, how many joules per cycle does it discard to the reservoir at 300300 K?

Ch 20: The Second Law of Thermodynamics

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 20: The Second Law of Thermodynamics

Problem 14a

Chapter 20, Problem 14a

A Carnot engine is operated between two heat reservoirs at temperatures of $520$ K and $300$ K. If the engine receives $6.45$ kJ of heat energy from the reservoir at $520$ K in each cycle, how many joules per cycle does it discard to the reservoir at $300$ K?

Verified step by step guidance

Understand the concept of a Carnot engine: A Carnot engine is an idealized heat engine that operates between two heat reservoirs and is the most efficient possible engine. It follows the Carnot cycle, which consists of two isothermal processes and two adiabatic processes.

Identify the temperatures of the heat reservoirs: The high-temperature reservoir (T1) is at 520 K, and the low-temperature reservoir (T2) is at 300 K.

Use the Carnot efficiency formula to find the efficiency of the engine: The efficiency (η) of a Carnot engine is given by η = 1 - (T2/T1), where T1 and T2 are the temperatures of the hot and cold reservoirs, respectively.

Calculate the work done by the engine per cycle: The work done (W) by the engine is the difference between the heat absorbed from the hot reservoir (Q1) and the heat discarded to the cold reservoir (Q2). Use the formula W = η * Q1, where Q1 is the heat absorbed from the hot reservoir.

Determine the heat discarded to the cold reservoir: Rearrange the formula for work done to find Q2, which is the heat discarded to the cold reservoir. Use Q2 = Q1 - W, where Q1 is the heat absorbed from the hot reservoir and W is the work done by the engine.

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Key Concepts

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

Carnot Engine

A Carnot engine is an idealized heat engine that operates on the reversible Carnot cycle. It is the most efficient engine possible between two heat reservoirs, with its efficiency determined solely by the temperatures of the reservoirs. The Carnot cycle consists of two isothermal processes and two adiabatic processes.

Thermodynamic Efficiency

The efficiency of a heat engine is defined as the ratio of the work output to the heat input. For a Carnot engine, the efficiency is given by 1 - (T_cold/T_hot), where T_cold and T_hot are the absolute temperatures of the cold and hot reservoirs, respectively. This formula highlights that efficiency depends only on the temperatures of the reservoirs.

Heat Transfer in Carnot Cycle

In a Carnot cycle, the heat absorbed from the hot reservoir and the heat discarded to the cold reservoir are related by the efficiency of the engine. The heat discarded (Q_cold) can be calculated using the formula Q_cold = Q_hot * (T_cold/T_hot), where Q_hot is the heat absorbed from the hot reservoir. This relationship is crucial for determining the heat discarded in each cycle.

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