PHY 770 Spring 2014 -- Lecture 1 11/14/2014
PHY 770 -- Statistical Mechanics10-10:50 AM MWF Olin 107
Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770
http://www.wiley.com/WileyCDA/WileyTitle/productCd-3527407820.html
PHY 770 Spring 2014 -- Lecture 1 21/14/2014
Textbook Chapters Course topics
1. Introduction2. Complexity and entropy 33. Thermodynamics 14. Thermodynamics of phase transitions 25. Equilibrium Statistical Mechanics I 46. Equilibrium Statistical Mechanics II 57. Brownian Motion 68. Hydrodynamics 79. Transport coefficients 810. Nonequilibrium phase transitions 9
PHY 770 Spring 2014 -- Lecture 1 31/14/2014
PHY 770 Spring 2014 -- Lecture 1 41/14/2014
PHY 770 Spring 2014 -- Lecture 1 51/14/2014
PHY 770 Spring 2014 -- Lecture 1 61/14/2014
PHY 770 Spring 2014 -- Lecture 1 71/14/2014
PHY 770 Spring 2014 -- Lecture 1 81/14/2014
Today’s topics – review of thermodynamics (Chapter 3 of text)
1. Overview and motivation2. Important definitions3. Meaning of thermodynamic equilibrium4. Notion of exact differential and its opposite5. State variables6. First law of thermodynamics7. Carnot cycle8. Notion of entropy
PHY 770 Spring 2014 -- Lecture 1 91/14/2014
Thermodynamics1. Overview and motivation
Some ideas from Advanced Statistical Mechanics, Barry M. McCoy (2010)
• The development of the notions of thermodynamics occurred in the 19th century with its results at the “heart of the industrial revolution”.
• Notions of thermodynamics do not logically follow from Newton’s laws. (Within Statistical Mechanics, connections between the macroscopic thermodynamic principles and the microscopic viewpoint of Newton’s laws are explored.
• “We are forced to study thermodynamics because nature empirically turns out to work this way. Because thermodynmic behavior exits it must follow from the microscopic laws of nature, regardless of how much we may not like it.”
PHY 770 Spring 2014 -- Lecture 1 101/14/2014
Thermodynamics2. Important definitions
Macroscopic system, “thermodynamic limit”
fixed 1
with and
: volumeain particles of system aFor
vV
N
VN
VN
Intensive variable: (such as T, P, m, ….)constant)(intensive N
Extensive variable: (such as N ….)constant)/(extensive NN
PHY 770 Spring 2014 -- Lecture 1 111/14/2014
Thermodynamics3. Notion of thermodynamic equilibriumWe observe that the properties of an isolated macroscopic system measured as a function of time, eventually reach constant values. (This is not true, in general, for a system of few particles.)
PHY 770 Spring 2014 -- Lecture 1 121/14/2014
PHY 770 Spring 2014 -- Lecture 1 131/14/2014
Thermodynamics4. Notion of exact differential and its opposite
yxxy
xy
y
F
xx
F
y
dyy
Fdx
x
FdF
dFyxF
with
:iff aldifferentiexact an has ),(function A
Fd denoted is aldifferentiinexact An
PHY 770 Spring 2014 -- Lecture 1 141/14/2014
Example (from appendix B.1 of textbook)
),(),(
implies that thisNote3
1),(
: thatshowcan g,integratinBy
:Consider
3
2
AABB
B
A
yxFyxFdF
CxyxyxF
dyxdxyxdF
Wd
Qd
:systemby done Work
:system toaddedHeat
:alsdifferentiinexact of Example
PHY 770 Spring 2014 -- Lecture 1 151/14/2014
Thermodynamics5. State variables
State variables are quantities that can be analyzed as exact differentials
Examples : (Under reversible conditions) Internal energy U Entropy S
PHY 770 Spring 2014 -- Lecture 1 161/14/2014
Thermodynamics6. First law of thermodynamics
1
:particles typeof potential Chemical
: typeof particles alDifferenti
:systemby done Work
:system toaddedHeat
:energy internal alDifferenti
:Notation
jjj
j
j
dNμWdQddU
μj
dNj
Wd
Qd
dU
PHY 770 Spring 2014 -- Lecture 1 171/14/2014
Elaboration on work:
YdX
dqdddAJdLPdVWd
MHPE
Specific example: Consider an ideal gas in a complete cycle
)/( 1
1
constant)Boltzmann (
VP
B
CCPV
NTk
U
kkNkTPV
PHY 770 Spring 2014 -- Lecture 1 181/14/2014
Analysis of an ideal heat engine Nicholas Carnot (French Engineer) 1834
isothermal
isothermal
adiabaticad
iabati
c
Carnot cycle:
PHY 770 Spring 2014 -- Lecture 1 191/14/2014
Analysis of Carnot cycle for ideal gas system
1
2121212 ln 0
V
VNkTWQU high
1
0 232323
highlow TTNkWUQ
3
4343434 ln 0
V
VNkTWQU low
1
0 414141
lowhigh TTNkWUQ
high
low
high
low
input
total
T
T
VVNkT
VVNkT
Q
WW
Q
W
1
ln
ln1
12
34
12
3412
PHY 770 Spring 2014 -- Lecture 1 201/14/2014
More analysis of Carnot cycle for ideal gas system
high
high
low
low
high
low
high
low
high
low
high
lowhigh
input
total
T
Q
T
Q
T
T
Q
Q
T
T
Q
Q
Q
WW
Q
W
||||
||
||
1||
||||
12
3412
12
3412
PHY 770 Spring 2014 -- Lecture 1 211/14/2014
Definition of entropy for a reversible process:
T
QddS
Example for Carnot cycle:
1
21212 ln
V
VNk
T
Q
T
QS
high
high
high
0 0 2323 SQ
3
43434 ln
||
V
VNk
T
Q
T
QS
low
low
low
0 0 4141 SQ
S
T
Thigh
Tlow
ShighSlow
1
4 3
2
PHY 770 Spring 2014 -- Lecture 1 221/14/2014
Analysis of state variables U and S for an ideal gas
T
QddS
PdVWd PdVTdSdU
(constant)ln1
(constant)lnln1
),(
1
1 :),(For
1 :constant is If
)/( 1
1
:gas idealFor
100
1
1
VT
TVNk
VTNk
VTS
dVV
NkdT
T
NkdSVTSS
PdVTdSdTNk
dUN
CCPV
TNk
U
NkTPV
VP
PHY 770 Spring 2014 -- Lecture 1 231/14/2014
(constant)ln1
),(
1 :constant is If
1
1
:gas idealFor
100
1
VT
TVNkVTS
PdVTdSdTNk
dUN
PVT
NkU
NkTPVAnalysis of ideal gas entropy continued:
2/52/3
2
2/5 2lnln
2
5
)3/5(with : textbookof 3.2 Exercise Check with
kh
mNk
P
TNk NkS
quantum effects; consistent with “constant”
(constant)ln3/2
1 ln :Note 3/2
2/5
TV
P
T
PHY 770 Spring 2014 -- Lecture 1 241/14/2014
011
1
011
1
0100
1
1exp
/
1,
1
exp /
ln1
),(
1
:gas idealFor
SSNk
γ
VV
TNkVSU
SSNk
γ
VV
TT
SVT
TVNkVTS
TNk
U
Analysis of ideal gas entropy and internal energy continued:
PHY 770 Spring 2014 -- Lecture 1 251/14/2014
Some practical (idealized) thermodynamic cycles
• 51:Intake stroke: constant pressure (P0) intake of air and gas
• 12:Adiabatic compression of air/gas mixture.
• 23:Constant volume pressurization of air/gas misture due to spark.
• 34:Power stroke due to adiabatic expansion of air/gas mixture.
• 41: Exhaust at constant volume.
PHY 770 Spring 2014 -- Lecture 1 261/14/2014
Otto cycle continued:
1
0 1122121212
VPVP
WUQ
1
0 2223232323
VPVP
QUW
1
0 2314343434
VPVP
WUQ
1
0 1114414141
VPVP
QUW
1122
1423
1
1
2
23
3412
:Using
1
VPVP
VPVP
V
V
Q
WW
Q
W
input
total
PHY 770 Spring 2014 -- Lecture 1 271/14/2014
Diesel cycle diagram (http://en.wikipedia.org/wiki/Diesel_cycle)
• 51:Intake stroke: constant pressure (P0) intake of air and gas
• 12:Adiabatic compression of air/gas mixture.
• 23:Constant pressure combustion and expansion.
• 34:Power stroke due to adiabatic expansion of air/gas mixture.
• 41: Exhaust at constant volume.
5
PHY 770 Spring 2014 -- Lecture 1 281/14/2014
Diesel cycle continued:
1 0 1122
121212
VPVPWUQ
1
1
23223
23223232
23
VVP
Q
VVPWVVP
U
1
0 3214343434
VPVP
WUQ
1
0 411414141
PPV
QUW
23
342312
Q
WWW
1
1
1
2
3
1
2
1
2
3
VV
VV
VV
PHY 770 Spring 2014 -- Lecture 1 291/14/2014
Comparison of Otto and Diesel cycle efficiencies:
Otto cycle:1
1
21
V
V
Diesel cycle:
1
1
1
2
3
1
2
1
2
3
VV
VV
VV
Suppose
(Diesel) 15 5
(Otto) 5
(air) 4.1
2
1
3
1
2
1
V
V
V
V
V
V
56.0
47.0
Diesel
Otto