Physics 262/266
George Mason University
Prof. Paul So
Chapter 20: The 2nd Law of
Thermodynamics






Preferential Direction in
Thermodynamic Processes
Heat Engine and Efficiency
The 2nd Law of
Thermodynamics
The Carnot Cycle (the most
efficient heat engine)
Entropy
Entropy and Disorder
Preferred Direction of Natural
Processes

Processes not observed in nature:
Example 1:
Ball absorbing heat energy
from surrounding
Then, converts it into mechanical
energy and starts to bounce and roll
Note: energy is conserved (1st Law is NOT violated): heat  mechanical eng.
BUT, we don’t observe this process in nature while the reverse occurs!
Preferred Direction of Natural
Processes
Example 2:
cold
Q
hot
Two objects are in thermal contact and heat
flows from the cold object to the hot object.
Again, energy is conserved (1st Law is NOT violated):
Qhot (absorbed )  Qcold (release)  0
BUT, we don’t observe this process in nature while the reverse occurs!
Disorder and Thermodynamic
Processes

The 2nd Law of Thermodynamics is the physical
principle which will delineate the preferred direction
of natural processes.

We will also see that
The preferred
direction of
natural processes
The degree of
randomness (disorder) of
the resulting state
All natural processes in isolation will tend toward a
state with a larger degree of disorder !
The 2nd Law of Thermodynamics


Historically, there are more than one but equivalent
way to state the 2nd Law:
To address the condition in example #2, here is the
Clausius Statement on the 2nd Law:
cold
Q
hot
“It is impossible for any process to have as its sole
result the transfer of heat from a cooler to a hotter
body.”
The 2nd Law of Thermodynamics

There is also the Kelvin-Planck’s Statement:
“It is impossible for any system to undergo a cyclic
process in which it absorbs heat from a reservoir at a
given temperature and converts the heat completely
into mechanical work.”

This implies that all heat engines have limited efficiency !
(efficiency of real mechanical engines ~ 15 to 40%)
To understand this form of the 2nd Law, we need to look at a toy model:
heat engine
Heat Engines


Definition: A device that converts a
given amount of heat into
mechanical energy.
All heat engines carry some working
substance thru a cyclic process:
Engine absorbs heat from hot
reservoir at TH
Mechanical work is done by engine
Engine releases residual heat to cold
reservoir at TC
Dstering
Work Done by an Heat Engine
The engine works in a cyclic process,
U  0
1st Law gives,
U  Qnet  W  0
Qnet  W
where,
Qnet  QH  QC  QH  QC
explicit signs for heats
Efficiency for a Heat Engine

Thermal Efficiency e is defined as the ratio of the
mechanical energy output to the heat energy input,
W what you get out

e
QH what you put in
Substituting W  QH  QC , we have
e 
using explicit signs here
Q
Q
W QH  QC

 1 C  1 C
QH
QH
QH
QH
A “perfect” (100% efficient) heat engine
|QH|
A “perfect” heat engine means 100%
efficiency (e=1). This means that
QC
e  1
 1 means QC  0
QH
All heat absorbed from reservoir TH is
converted into mechanical work W.
No residual heat is released back.
e perfect  1
The Kelvin-Planck’s statement of the 2nd Law does not allow this !
erealistic  1
Ddrinking bird
2nd Law, Disorder, & Available Energy
Two Forms of Energy in any Thermal Process:
Internal Energy
Macroscopic Mechanical Energy
In the Kinetic-Molecular Model,
this consists of the KE and PE
associated with all the randomly
moving microscopic molecules.
The piston’s motion in an
automobile engine results from the
coordinated macroscopic motion
of the molecules.
(One typically cannot control the
individual random motions of all
these molecules.)
(Energy associated with this
coordinated [ordered] motion can
be used for useful work.)
2nd Law, Disorder, & Available Energy
In a natural process (a block sliding to a stop),
stopped
e.g.
v
f
slightly warmer
due to friction
 The coordinated motion of the block is converted into the KE & PE of the
slightly more agitated random motions of the molecules in the block/table.
 Macroscopic Mechanical Energy (KE of the block) is converted into
Internal Energy through heat as a result of friction.
2nd Law, Disorder, & Available Energy
 Now consider the reverse situation… it is unlikely that one can coordinate
ALL the randomly moving molecules in a concerted fashion.
 In other words, one typically cannot convert the internal energy of a system
completely back into macroscopic mechanical energy.
 However, this does not mean that internal energy is not accessible. An Heat
Engine is exactly the machine that can perform this conversion but only
partially.
The 2nd Law of Thermodynamics is basically
a statement limiting the availability of internal
energy for useful mechanical work.
The Carnot Cycle (The Most Efficient Heat
Engine)



A reversible cycle described by Sadi Carnot in
1824.
The Carnot Theorem gives the theoretical limit
to the thermal efficiency of any heat engine.
The Carnot cycle consists of:




A cycle operating between two temperatures: TH and TC
2 reversible isothermal processes in which Q 
2 reversible adiabatic processes in which TH  TC
An Ideal Gas as its working substance
Steps of the Carnot Cycle
animation
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/carnot.htm
Steps of the Carnot Cycle
animation
http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/carnot.htm
Details on the Carnot Cycle
The isothermal expansion (ab) and compression (cd):
U isothermal  0
(T is constant and U(T) is a function
of T only for an Ideal Gas.)
V 
QH  Wab  nRTH ln  b 
 Va 
(ab : isothermal expansion)
QC  Wcd
 Vd 
 Vc 
 nRTC ln    nRTC ln   (Vc  Vd )
 Vc 
 Vd 
(cd : isothermal compression)
Details on the Carnot Cycle
The adiabatic expansion (bc) and compression (da):
Qadiabatic  0
(by definition)
From Section 19.8, we learned that TV  1  const
for adiabatic processes.
TH Vb 1  TCVc 1
(bc: adiabatic expansion)
TH Va 1  TCVd 1
(da: adiabatic compression)
Dividing these two equations gives,
Vb Vc

Va Vd
Efficiency of the Carnot Cycle
From definition, we have e  1 
QC
QH
Using our results for QC and QH from the isothermal processes,
nRTC ln Vc Vd 
TC ln Vc Vd 
e  1
 1
nRTH ln Vb Va 
TH ln Vb Va 
From the adiabatic processes, we have Vb Va  Vc Vd
ln Vc Vd 
1
ln Vb Va 
e  1
QC
QH
 1
TC
(T must be in K)
TH
(Carnot Cycle only)
Efficiency of the Carnot Cycle
ecarnot
TC
 1
TH
(Carnot Cycle)
General Comments:
 Higher efficiency if either TC is lower and/or TH is higher.
 For any realistic thermal process, the cold reservoir is far
above absolute zero and TC > 0.
 Thus, a realistic e is strictly less than 1! (No 100%
efficient heat engine)
 Realistic heat engines must take in energy from the high T
reservoir for the work that it produces AND some heat
energy must be released back to the lower T reservoir.
(Kelvin-Planck’s Statement)
skip
Internal Combustion Engine
The Otto Cycle
A fuel vapor can be compressed, then detonated to rebound
the cylinder, doing useful work.
animation
http://auto.howstuffworks.com/engine1.htm
The Otto Cycle
intake
intake
exhaust
ab
b, c
cd
For the two constant V processes: 2 and 4,
we can calculate,
QH  U 2  nCV (Tc  Tb )  0
QC  U 4  nCV (Ta  Td )  0
r is the compression ratio (8 to 13)
exhaust
The Otto Cycle
Applying the definition of efficiency,
nCV Ta  Td 
QC
Ta  Td
e  1
 1
 1
QH
nCV Tc  Tb 
Tc  Tb
Now, we can utilize the two adiabatic processes: 1 and 3,
TaVa 1  TbVb 1
Ta  rV 
 1
 TbV
Ta r  1  Tb
 1
and
TcVc 1  TdVd 1
TcV
 1
 Td  rV 
Tc  Td r  1
 1
The Otto Cycle
Substituting Tb and Tc into the efficiency equation, we have,
e  1
Ta  Td
Ta  Td
Ta  Td
1
 1


Tc  Tb
Td r  1  Ta r  1
Ta  Td  r  1
e  1
1
r  1
Using a typical value for the compression ratio r = 8 and 
= 1.40 gives,
e  0.56 (or 56%)
Note: This is a theoretical value.
Realistic gasoline engine typically
has e ~ 35%.
The Diesel Cycle
Key difference:

No fuel in cylinder at the beginning of the
compression stroke (process 1)

Fuel is injected only moments before
ignition in the power stroke

No fuel until the end of the adiabatic
compression can avoid pre-ignition

Compression ratio r value can be higher (15
to 20)

Higher temperature can be reached during
the adiabatic compression

Higher e and no need for spark plugs
Dsteam engine
Refrigerators
Refrigerators are basically heat engine running in reverse.
 Heat from inside the refrigerator (cold T reservoir) is absorbed and released
into the room (high T reservoir) with the input of mechanical work.
Refrigerators
From 1st Law,
U cycle  0   QC  QH     W
QH  QC  W

explicit signs
(Note: we have put in the explicit signs according to our sign convention.)
 |QC| (absorbed)  positive
 |QH| (released)  negative
 |W| (work is done on working substance by motor)  negative
Coefficient of Performance for a Refrigerator
QC
QC
what you get
K


what you put in W
QH  QC
A “perfect” Refrigerator
K
QC
W

QC
QH  QC
A “perfect” refrigerator means ( K   ).
This means that
QH  QC
or
W 0
No mechanical work W is needed to
transfer heat from the cold reservoir
to the hot reservoir.
The Clausius’s statement of the 2nd Law does not allow this !
K realistic  
Entropy
Recall from a Carnot Cycle, we have derived the following relationship:
QC
T
 C
QH TH
QH  QC

0
TH
TC
Formally, we can rewrite this as,
Q
0

cycle T
(We have absorbed the explicit
sign back into the Q variable.)
where Q represents the heat absorbed/released along the isotherm at temp T.
Entropy
Any reversible cycles can be approximated as a series
of Carnot cycles !
Entropy
This suggests that the following generalization to be true for any reversible
processes,
dQr
 T  0
cycle
where,
dQr is the infinitesimal heat absorbed/released by the system at an
infinitesimal reversible step at temp T.

cycle
denotes the integration evaluated over one complete cycle.