# 4.2 Heat engines  (Page 2/6)

 Page 2 / 6

Heat engines operate by carrying a working substance through a cycle. In a steam power plant, the working substance is water, which starts as a liquid, becomes vaporized, is then used to drive a turbine, and is finally condensed back into the liquid state. As is the case for all working substances in cyclic processes, once the water returns to its initial state, it repeats the same sequence.

For now, we assume that the cycles of heat engines are reversible, so there is no energy loss to friction or other irreversible effects. Suppose that the engine of [link] goes through one complete cycle and that ${Q}_{\text{h}},{Q}_{\text{c}},$ and W represent the heats exchanged and the work done for that cycle. Since the initial and final states of the system are the same, $\text{Δ}{E}_{\text{int}}=0$ for the cycle. We therefore have from the first law of thermodynamics,

$W=Q-\text{Δ}{E}_{\text{int}}=\left({Q}_{\text{h}}-{Q}_{\text{c}}\right)-0,$

so that

$W={Q}_{\text{h}}-{Q}_{\text{c}}.$

The most important measure of a heat engine is its efficiency ( e )    , which is simply “what we get out” divided by “what we put in” during each cycle, as defined by $e={W}_{\text{out}}\text{/}{Q}_{\text{in}}.$

With a heat engine working between two heat reservoirs, we get out W and put in ${Q}_{\text{h}},$ so the efficiency of the engine is

$e=\frac{W}{{Q}_{\text{h}}}=1-\frac{{Q}_{\text{c}}}{{Q}_{\text{h}}}.$

Here, we used [link] , $W={Q}_{\text{h}}-{Q}_{\text{c}},$ in the final step of this expression for the efficiency.

## A lawn mower

A lawn mower is rated to have an efficiency of $25.0\text{%}$ and an average power of 3.00 kW. What are (a) the average work and (b) the minimum heat discharge into the air by the lawn mower in one minute of use?

## Strategy

From the average power—that is, the rate of work production—we can figure out the work done in the given elapsed time. Then, from the efficiency given, we can figure out the minimum heat discharge ${Q}_{\text{c}}={Q}_{\text{h}}\left(1-e\right)$ with ${Q}_{\text{h}}={Q}_{\text{c}}+W.$

## Solution

1. The average work delivered by the lawn mower is
$W=P\text{Δ}t=3.00\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}60\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}1.00\phantom{\rule{0.2em}{0ex}}\text{J}=180\phantom{\rule{0.2em}{0ex}}\text{kJ}.$
2. The minimum heat discharged into the air is given by
${Q}_{\text{c}}={Q}_{\text{h}}\left(1-e\right)=\left({Q}_{\text{c}}+W\right)\left(1-e\right),$

${Q}_{\text{c}}=W\left(1\text{/}e-1\right)=180\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(1\text{/}0.25-1\right)\phantom{\rule{0.2em}{0ex}}\text{kJ}=540\phantom{\rule{0.2em}{0ex}}\text{kJ}.$

## Significance

As the efficiency rises, the minimum heat discharged falls. This helps our environment and atmosphere by not having as much waste heat expelled.

## Summary

• The work done by a heat engine is the difference between the heat absorbed from the hot reservoir and the heat discharged to the cold reservoir, that is, $W={Q}_{\text{h}}-{Q}_{\text{c}}.$
• The ratio of the work done by the engine and the heat absorbed from the hot reservoir provides the efficiency of the engine, that is, $e=W\text{/}{Q}_{\text{h}}=1-{Q}_{\text{c}}\text{/}{Q}_{\text{h}}.$

## Conceptual questions

Explain in practical terms why efficiency is defined as $W\text{/}{Q}_{\text{h}}.$

## Problems

An engine is found to have an efficiency of 0.40. If it does 200 J of work per cycle, what are the corresponding quantities of heat absorbed and rejected?

In performing 100.0 J of work, an engine rejects 50.0 J of heat. What is the efficiency of the engine?

0.667

An engine with an efficiency of 0.30 absorbs 500 J of heat per cycle. (a) How much work does it perform per cycle? (b) How much heat does it reject per cycle?

It is found that an engine rejects 100.0 J while absorbing 125.0 J each cycle of operation. (a) What is the efficiency of the engine? (b) How much work does it perform per cycle?

a. 0.556; b. 125.0 J

The temperature of the cold reservoir of the engine is 300 K. It has an efficiency of 0.30 and absorbs 500 J of heat per cycle. (a) How much work does it perform per cycle? (b) How much heat does it reject per cycle?

The Kelvin temperature of the hot reservoir of an engine is twice that of the cold reservoir, and work done by the engine per cycle is 50 J. Calculate (a) the efficiency of the engine, (b) the heat absorbed per cycle, and (c) the heat rejected per cycle.

a. 0.50; b. 100 J; c. 50 J

A coal power plant consumes 100,000 kg of coal per hour and produces 500 MW of power. If the heat of combustion of coal is 30 MJ/kg, what is the efficiency of the power plant?

in the 2nd example, for chapter 8.2 on page 3/3, I don't understand where the value 48uC comes from, I just couldn't get that value in my calculator.
are you talking about the capacitance combination problem
sam
please write the problem or send a snap of th page....I don't have the book in my vicinity.
SABYASACHI
what is electromagnetic force. do electric and magnetic force happen differently
yes
yes
Pranay
why
Godson
how?
Godson
derived the electric potential due to disk of charge
how can we derived potential electric due to the disk
aron
how can you derived electric potential of a disk
aron
how can you derived electric potential due to disk
aron
where is response?
aron
what is difference between heat and temperature?
temperature is the measure of degree of hotness or coldness. on the other hand, heat is the form of energy, which causes temperature. So we can safely say, heat is the reason and temperature is its consequence.
SABYASACHI
how many liquid metals do we have
do we have gasses as metals
Jeffery
who knows should please tell us
yes...gallium & cesium
Idris
Hg is liquid. No metal gasses at standard temp and pressure
Shane
I don't ever understand any of this formulae
which formula
How to determine a temperature scale
what is the formula for absolute error
Nyro
define kelvin planck statement
using p-v diagram, explain what takes place during each four processes of a carnot cycle
Masia
using p-v diagram, explain what takes place during each four processes of a carnot cycle
Masia
state zeroths law of thermodynamics
The zeroth law of thermodynamics states that if twothermodynamic systems are each in thermal equilibrium with a third one, then they are in thermal equilibrium with each other
Seza
zeroth law of thermodynamic state that when a body A and B are in thermal equilibrium to body C ,a state is attained when body A,B and C are in thermal equilibrium.
akanbi
sorry guys I got no any idea on the law can someone help me please?
SNAHR
hi
Muhsin
The zeroth law of thermodynamics states that if two thermodynamic systems are each in thermal equilibrium with a third one, then they are in thermal equilibrium with each other. Accordingly, thermal equilibrium between systems is a transitive relation. Two systems are said to be in the relation of
Muhsin
I can help u Snahr as possible
Vinayaka
it simply states that, if two bodies are separately in thermal equilibrium with a third body, then those two bodies will also be in thermal equilibrium with each other
Vinayaka
What mass of steam of 100 degree celcius must be mixed with 150g of ice at 0 degree celcius, in a thermally insulated container, to produce liquid water at 50 degree celcius
sorry I dont know
Bamidele
thank you
Emmanuel
What is the pressure?
SHREESH
To convert 0°C ice to 0°c water. Q=M*s=150g*334J/g=50100 J.......... Now 0° water to 50° water... Q=M*s*dt=150g*4.186J/g*50= 31395 J....... Which adds upto 81495 J..... This is amount of heat the steam has to carry. 81495= M *s=M*2230J/g..therefore.....M=36.54g of steam
SHREESH
This is at 1 atm
SHREESH
If there is change in pressure u can refer to the steam table ....
SHREESH
instrument for measuring highest temperature of a body is?
Thermometer
Umar
how does beryllium decay occur
Photon?
Umar
state the first law of thermodynamics
Its state that "energy can neither be created nor destroyed but can be transformed from one form to another. "
Ayodamola
what about the other laws can anyone here help with it please
Sandy
The second law of thermodynamics states that the entropy of any isolated system always increases. The third law of thermodynamics states that the entropy of a system approaches a constant value as the temperature approaches absolute zero.
sahil
The first law is very simple to understand by its equation. The law states that "total energy in thermodynamic sytem is always constant" i.e d¶=du+dw where d¶=total heat du=internal energy dw=workdone... PLEASE REFER TO THE BOOKS FOR MORE UNDERSTANDING OF THE CONCEPT.
Elia
what is distance.?
what is physics?
Ali
Physics is a scientific phenomenon that deals with matter and its properties
Ayodamola
physics is the study of nature and science
John
Chater1to7
min
Physics is branch of science which deals with the study of matters in relation with energy.
Elia
What is differential form of Gauss's law?