1/14/2014PHY 770 Spring 2014 -- Lecture 11 PHY 770 -- Statistical Mechanics 10-10:50 AM MWF Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage:

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

http://www.wfu.edu/~natalie/s14phy770

http://www.wiley.com/WileyCDA/WileyTitle/productCd-3527407820.html


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







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


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.”


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


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.)



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


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


Thermodynamics5. State variables

State variables are quantities that can be analyzed as exact differentials

Examples : (Under reversible conditions) Internal energy U Entropy S


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


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


Analysis of an ideal heat engine Nicholas Carnot (French Engineer) 1834

isothermal

isothermal

adiabaticad

iabati

c

Carnot cycle:


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


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

QQ

Q

QQ

Q

WW

Q

W

||||

||

||

1||

||||

12

3412

12

3412


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


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


(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


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:


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.


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


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

http://en.wikipedia.org/wiki/Diesel_cycle




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


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

Documents

1/14/2014PHY 770 Spring 2014 -- Lecture 11 PHY 770 -- Statistical Mechanics 10-10:50 AM MWF Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course Webpage: