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1.2 Reversible Carnot cycle:
The reversible Carnot cycle is an idealized model for processe in heat
engines
Consider a cyclic process which is represented
by the closed path P → Q → P.
The mechanical work done in the cycle is
In the P-V diagram, this work is pesented by the area within the closed path.
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The Carnot cycle consists of 4 steps:
1) A → B: isothermal expansion to
volume V
B
at temperature T
2
2) B → C: adiabatic expansion to volume
V
C
, tempeature drops to T
1
< T
2
3) C → D: isothermal compression to
volume V
D
at constant temperature T
1
4) D → A: adiabatic compression to the
original state of the system
Exchanges of heat and work in each step: P-V diagram of
a Carnot cycle
• During s-1: the system absorbs heat Q
2
(Q
2
> 0)
and does work W
AB
(W
AB
> 0)
A
B
AB
V
V
nRTWQ ln
22
An ideal gas is used for
working substance
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P-V diagram of
a Carnot cycle
• During s-3: the system rejects heat Q
1
,
Q
1
< 0 and the work is done on
the system, W
CD
< 0:
D
C
CD
V
V
nRTWQ ln
11
• For two adiabatic steps s-2 & s-4:
1
1
1
2
CB
VTVT
1
1
1
2
DA
VTVT
&
1
1
1
1
D
C
A
B
V
V
V
V
D
C
A
B
V
V
V
V
2
1
2
1
2
1
)/ln(
)/ln(
T
T
VV
VV
T
T
Q
Q
AB
DC
1
1
2
2
||||
T
Q
T
Q
The ratio of the heat rejected and the heat absorbed is just equal to
the ratio of the temperatures.
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What is the results of a cycle ABCDA ?
- The system absorbs heat Q
2
from the
surroundings at temperature T
2
(hot reservoir) , deposits heat Q
1
in the
surroundings at temperature T
1
(cold
reservoir).
- The internal energy of the system
remains unchanged.
- The net mechanical work done by the
system in one cycle is represented by the
area enclosed and is positive. From the 1
st
law:
P-V diagram of
a Carnot cycle
W
net
= |Q2| - |Q1| > 0 (since T
2
> T
1
)
By continously repeating the cycle it is possible to do mechanical work
by continously taking heat energy from a hot reservoir and depositing
a smaller amount of heat energy in a cold reservoir.
Such a device is called a heat engine.
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Consider the cycle in the reverse direction A→D→C→B→A:
The result is as follows:
Mechanical work is put into the system in order to take heat energy from
the cold reservoir and deposit heat in a higher-temperature reservoir.
In this case the device is a refrigerator.
We see that heat energy can’t flow from the cold reservoir to the hot
reservoir without action of an external mechanical work.
1.3 Efficiency of a Carnot engine:
Efficiency =
Mechanical work done by engine
Heat energy supplied from the hot reservoir
For a heat engine the efficiency represents the fraction of heat
that is converted to work
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From the general definition we can calculate the efficiency of a Carnot
engine:
2
1
2
1
2
12
2
1
||
||
1
||
||||
T
T
Q
Q
Q
QQ
Q
W
e
Carnot
Remark:
The efficiency of Carnot engine depends only on the temperatures
of two heat reservoirs
The efficiency is large when the temperature difference is large, and
the efficiency is very small when the temperatures are nearly equal
The efficiency can never be exactly unity unless T
1
= 0. This means
heat can never be converted completely into mechanical work.
Although the Carnot engine is an idealized model, but the conclusions
derived from it give an good orientation for investigation of other heat
engines that we usually encounter.
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§2. The second law of thermodynamics:
Experiment evidence suggests that it is impossible to build a heat engine
that converts heat completely to work, that is, an engine with 100% thermal
efficiency. This impossibility is the basis of the following statement of
The second law of thermodynamics.
2.1 “Engine” statement of the second law:
“It is impossible for any system to undergo a process in which it absorbs
heat from a reservoir at a single temperature and converts the heat
completely into mechanical work, with the system ending in the same
state in which it began”.
This statement is also called the Kelvin-Planck statement of the second law.
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Some interpretation:
The difference between mechnical energy and heat energy is that they
relate to two different kinds of motion: organized and random motions,
respectively. One can convert the organized motions of molecules to
random motions, but it is difficult to control the random motions of
molecules, thus one can’t convert completely random motions to
organized motions, one can convert a part of it.
Can we build an automobile or power plan by cooling the surrounding
air? It is impossible in practice.
The reason is as follows: the engine receives heat from one body (hot
reservoir), but at the same time needs another body (cold reservoir)
in which it deposits the remaining heat (heat can’t be converted 100%).
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2.2 “Refrigerator” statement of the second law:
The second law has also an alternative statement:
“It is impossible for any process to have as its sole result the transfer of
heat from a cooler to a hotter body”.
This is called the “refrigerator” statement, or also the Clausius statement
of the second law of thermodynamics.
This statement means that in order to transfer heat from a cooler to a
hotter body it ís absolutely neccesary to spend a work which does on the
system.
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2.3 The equivalency of two statements of the second law:
Two statements my seem to have no relation each to other. But, in fact,
they are completely equivalent.
Suppose that we have a refrigerator which can transfer heat from a cooler
to a hotter body without any external work (the “refrigerator” statement
is violated).
Then we can use this refrigerator in conjuntion with a heat engine,
pumping the heat rejected by the engine back to the hot reservoir
→ violate the “engine” statement.
Suppose that we have a heat engine which can run by receive heat from
the hot reservoir (the “engine” statement is violated).
Then we can use this heat engine in conjunction with a refrigerator. The
output work from the engine can be used for driving the refrigerator that
pumps heat from the cool reservoir and deliver to the hot reservoir without
requiring any input of work → violate the “refrigerator” statement.
Violation of
the “refrigerator” statement
Violation of
the “engine” statement
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