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A Carnot engine has an efficiency of 66% and performs 2.5 × 104 J of work in each cycle. (a) How much heat does the engine extract from its heat source in each cycle? (b) Suppose the engine exhausts heat at room temperature (20.0C). What is the temperature of its heat source?

Short Answer

Expert verified
(a) Heat extracted is approximately 3.79 × 10⁴ J. (b) Temperature of the heat source is about 862.21 K.

Step by step solution

01

Understanding Efficiency

The efficiency of a Carnot engine is given by the formula η=1TcTh, where Tc and Th are the temperatures of the cold and hot reservoirs, respectively. Given the efficiency η=0.66, which means 66%, we'll use this information later to find the temperature of the heat source.
02

Calculating Heat Energy Input

Efficiency η is also defined as the ratio of work done W to the heat absorbed from the heat source Qh. The equation is η=WQh. Given W=2.5×104 J and η=0.66, we solve for Qh using 0.66=2.5×104Qh. Rearranging gives Qh=2.5×1040.663.79×104 J.
03

Understanding Temperatures in Kelvin

Convert the exhaust temperature from Celsius to Kelvin since thermodynamic calculations require the temperature in Kelvin. The conversion is given by T(K)=T(C)+273.15. For a room temperature of 20.0°C, we have Tc=20+273.15=293.15 K.
04

Finding the Temperature of the Heat Source

Using the Carnot efficiency equation η=1TcTh, where η=0.66 and Tc=293.15 K, we can find Th. Solving 0.66=1293.15Th yields 293.15Th=0.34, and rearranging gives Th=293.150.34862.21 K.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Efficiency of Heat Engines
In the world of thermodynamics, the efficiency of a heat engine is a crucial aspect to understand. A heat engine extracts heat from a source, performs work, and exhausts some of the heat to a sink. The efficiency (η) measures how well an engine converts the absorbed heat (Qh) into work (W). It can be calculated using the formula:
η=WQh.
For a Carnot engine, which is a theoretical perfect engine, the efficiency is also given by the expression:
η=1TcTh.
Here, Tc is the temperature of the cold reservoir and Th is the temperature of the hot source. All these temperatures must be in Kelvin.
A Carnot engine represents an ideal because, in reality, 100% efficiency is unattainable due to the second law of thermodynamics. This means that no engine can convert all the heat it absorbs into work without losing some to the surroundings.
Thermodynamic Temperature Conversion
When dealing with thermodynamic processes, it's essential to use the Kelvin scale for temperatures. The Kelvin scale begins at absolute zero, the coldest possible temperature, which helps eliminate negative values that can complicate calculations.
To convert Celsius to Kelvin, simply add 273.15 to the Celsius temperature. For example, converting the room temperature of 20.0°C, we do:
  • T(K)=T(C)+273.15
  • T(K)=20+273.15=293.15 K
In thermodynamic calculations involving engines, using Kelvin ensures consistency in calculating formulas like the Carnot efficiency. This temperature conversion is important because avoiding negative values allows for smoother and more precise calculations when using scientific equations.
Work and Heat Energy Relationship
In the study of heat engines, understanding how work and heat energy are interconnected is indispensable. The energy absorbed as heat from the heat source (Qh) and the energy lost to the cold sink influence the amount of work (W) the engine can perform.
An important relationship here is: W=QhQc, where Qc is the heat expelled to the cold sink. This equation demonstrates that the work done by the engine is the difference between the heat absorbed from the source and the heat lost to the sink.
Moreover, we explored the efficiency calculation η=WQh, concluding that efficiency helps determine how much of the absorbed heat is converted to useful work. By improving efficiency, an engine can perform more work for the same amount of heat input, but natural limitations prevent reaching the theoretical 100% efficiency.

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Most popular questions from this chapter

A gasoline engine has a power output of 180 kW(about 241 hp). Its thermal efficiency is 28.0%. (a) How much heat must be supplied to the engine per second? (b) How much heat is discarded by the engine per second?

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