PHY 341/641 Thermodynamics and Statistical Physics Lecture 36

4/27/2012 PHY 341/641 Spring 2012 -- Lecture 36 1

PHY 341/641 Thermodynamics and Statistical Physics

Lecture 36

Review and examples

• Review of general principles • Review of specific examples


4/30– Laurence, Zac, Eric 5/2 -- Kristen, Audrey, Griffin


Fundamental principles of thermodynamics and statistical mechanics

• This course focused on “thermostatics” and statistical mechanics of systems in equilibrium

• Thermal equilibrium fixed temperature • First law of thermodynamics

WQEEE +=∆≡− 12

0 :)(expansion system theBY doneFor work 0 :on)(contracti system theTO doneFor work

0 :system theFROMrawn heat withdFor 0 :system theTO addedheat For

<>

<>

WW

QQ


∫−=

−=−=−=<⇒

→2

1

),(

:0 expands system system; theON work -- your text in conventionSign

21

V

VdVVTPW

PdVPAdxFdxdWW

Thermodynamic process -- WORK various sign conventions !!!#$#!!!


First law of thermodynamics

system theON done work system theTO addedheat

12

≡≡

+=∆≡−

WQ

WQEEE

System of study

Controlling medium

W Q


Fundamental principles of thermodynamics and statistical mechanics -- continued

TdQdS =

• From the analysis of the Carnot cycle, a new state variable – S = entropy was analyzed. For a “reversible” process this is defined by:

• In terms of S, the differential form of the first law of thermodynamics becomes

PdVTdSdWdQdE −=+=


Second law of thermodynamics • Kelvin-Planck: It is impossible to construct an

engine which, operation in a cycle, will produce no other effect than the extraction of energy from a reservoir and the performance of an equivalent amount of work.

• Clausius: No process is possible whose sole result is cooling a colder body and heating a hotter body.

• Gould-Tobochnik:There exists an additive function of state known as the entropy S that can never decrease in an isolated system.



Variables and functions:

µ potential Chemical particles ofNumber

eTemperatur Volume Pressure

Entropy energy Internal

NTVPSE



VE

NE

NST

VSTP

,

,

:defined potential Chemical

:defined pressure micThermodyna

∂∂

−≡

∂∂

≡

µ


dNT

dVTPdE

TdS

dNNSdV

VSdE

ESdSNVESS

VENENV

µ−+=

∂∂

+

∂∂

+

∂∂

=⇒=

1 :lawfirst fromDirectly

),,(

:lawfirst todue ipsrelationshEntropy

,,,


VSNSNV

SVNSNV

NE

VEP

SET

dNNEdV

VEdS

SEdE

NVSEEdNPdVTdSdE

,,,

,,,

),,( suppose -- icsThermodyam of LawFirst of form aldifferenti of Analysis

∂∂

=

∂∂

−=

∂∂

=⇒

∂∂

+

∂∂

+

∂∂

=

=⇒+−=

µ

µ




Maxwell relation


PT

TPPT

TP

TP

TV

PS

PG

TTG

P

PGV

TGS

dPPGdT

TGVdPSdTVdPPdVdFdG

PVFPTGG

∂∂

−=

∂∂

⇒

∂∂

∂∂

=

∂∂

∂∂

∂∂

=

∂∂

−=⇒

∂∂

+

∂∂

=+−=++=

+==

),( :energy free Gibbs

From the mathematical properties of these functions, we can derive the “Maxwell relations”. For example, simplifying to fixed N:


Summary of Maxwell’s relations for a fixed number of particles

PT

VT

PS

VS

TV

PS

TP

VS

SV

PT

SP

VT

∂∂

−=

∂∂

∂∂

=

∂∂

∂∂

=

∂∂

∂∂

−=

∂∂


( )( ) TSE

NENS

dNNEdV

VEdS

SEdNPdVTdSdE

NE

NST

VN

VS

UV

VSNSNV

SV

VE

µ

µ

µ

µ

−=

∂∂∂∂−

=

∂∂

∂∂

+

∂∂

+

∂∂

=+−=

∂∂

=

∂∂

−=

,

,

,

,,,

,

,

//

???consistent thisIs

:energy internal From

:entropy From

:yconsistencCheck

Properties of extended Maxwell’s relations


Categories of functions and variables Extensive depends on system size

Intensive independent of system size

Extensive Intensive

Number of particles N Temperature T

Volume V Pressure P

Entropy S(E,V,N) Density ρ

Internal energy E(S,V,N) Chemical potential µ

Enthalpy H(S,P,N)

Helmholz Free energy F(T,V,N)

Gibbs Free energy G(T,P,N)


vdPsdTdPNVdT

NSddg

dPPgdT

Tgddg

PTgN

NPTGPTgNG

PTNgNPTGNPTG

TP

PT

+−≡+−==

∂∂

+

∂∂

==

=⇒==

∂∂

=

µ

µ

µµ

µ

:equation Duhem-Gibbs--function) (intensive ),(density energy free Gibbs of In terms

),,( ),(

),(),,( : thatArgue:),,( to of iprelationsh Special

,


Some materials parameters based on thermodynamic variables and functions

VVV T

STTEC

V

∂∂

=

∂∂

=

:constant at capacity Heat

In fact, this should be easy, but as we have seen the “natural” variables of E are E=E(S,V,N) and S=S(E,V,N).

( )( )

( )VV

VV

VVV

SESE

SETC

SE

ST

SET

2222

2

2

//

/

∂∂∂∂

=∂∂

=⇒

∂∂

=

∂∂

∂∂

=


Other useful thermodynamic derivatives

NP

NT

TV

V

PV

V

,

,

1:expansion Isobaric

1:ilitycompressib Isothermal

∂∂

=

∂∂

−=

α

κ



The connection between the macroscopic viewpoint of thermodynamics and the microscopic viewpoint of statistical mechanics was made by Boltzmann using the statistical properties of large systems.

tot

tot

ln:entropy toiprelationsh its postulated and

smicrostate ofnumber theofnotion theintroducedBoltzmann system, a analyzingIn

Ω=

Ω

kStot


Other examples of microstate analysis for both classical and quantum systems. For N particles moving according to the classical mechanical (Newton’s) laws of physics in d-dimensional space (d=1,2,3), Liouville’s theorem shows that phase space ddNr ddNp spans all possibilites. In order to count the number of microstates, it is useful to define:

SΩ k Γk N

EdEdEg

dEEE

pdrdE

E

EEnergy

dNdN

=≈

Ω→Γ

=

+

∝Γ ∫≤

lnln , largefor that Note

)()(

: and between energy with smicrostate ofnumber The

)(

: toequalor than lessenergy with smicrostate ofnumber The



Extending Boltzmann’s analysis of “microcanonical” ensemble where E is controlled to “canonical” ensemble where T is controlled

Canonical ensemble:

Eb

Es


Canonical ensemble (continued)

( )( )

( )

( ) ( )+

∂Ω∂

−Ω+≈

−Ω+=

−Ω−Ω

=

<<+=

∑

totE

bstotb

stotbs

sstotb

stotbs

bsbstot

EEEEC

EECP

EEEEP

sEEEEE

lnln

lnln

: microstatein is systemy that Probabilit

''


Canonical ensemble (continued)

( )

( ) ( )+

∂Ω∂

−Ω+≈

−Ω+=

totE

bstotb

stotbs

EEEEC

EECPlnln

lnln

( )bNV

b

E

b

TEES

EEk

tot

1)(ln

,

=

∂∂

≈

∂Ω∂

( )kTE

s

stotbs

seCPkT

EECP

/'

1lnln

−=⇒

+

−Ω+≈


function"partition " :where

1

'

'

/

/

/

'∑ −

−

−

≡

=

=

s

kTE

kTE

kTEs

s

s

s

eZ

eZ

eCPCanonical ensemble:

kTs

E

s

kTE βeeZ ss 1

''

/ where

:functionpartition theusing nsCalculatio'' ==≡ ∑∑ −− β


Canonical ensemble continued – average energy of system:

ββ

β

∂∂

−=∂∂

−=

== ∑∑ −−

ZZZ

eEZ

eEZ

Es

Es

s

kTEss

ss

ln1

11'

''

/'

''

Heat capacity for canonical ensemble:

( )222

2

2

2

22

2

2

2

1

111ln1

1

ss

ssV

EEkT

ZZ

ZZkT

ZkT

EkTT

EC

−=

∂∂

−∂∂

−=∂

∂−=

∂∂

=∂

∂=

βββ

β



β∂∂

−=ZEs

ln:functionpartition canonical

of in terms functions amic thermodynof sEvaluation

( )NVTZkTNVTF ,,ln ),,( −=⇒

Energy Free Helmholz ),( VTFTSEs =−


( )

( ) ZkTNFVT

eZ

GLandau

s'

NEG

ss

ln,,

:ensemble canonical Grand''

−=−=Ω

≡ ∑ −−

µµ

µβ

Partition function Thermodynamic potential

Microcanonical Ω(E,V,N) S(E,V,N)=k ln(Ω)

Canonical Z(T,V,N) F(T,V,N)=-kT ln(Z) F=E-TS

Grand canonical ZG(T,V,µ) ΩLandau(T,V,µ)=-kT ln(ZG) Ωlandau=F-µN



Statistics of non-interacting quantum particles

k

k

n :numbers occupation particle Single :states particle Single ε

0,1 :spin)integer ( particles Fermi 0,1,2,3, :spin)(integer particles Bose

21 =

=

k

k

nn

( ) seTZs

NEG

ss smicrostate allover summing ),(

:systems for thesefunction partition Grand

∑ −−= µβµ



( ) seTZs

NEG

ss smicrostate allover summing ),(

:systems for thesefunction partition Grand

∑ −−= µβµ

( )

( )∑

∏

∏ ∑

∑ ∑

−−

−−

≡

≡

=

==

s

nnkG

kkG

k s

nnG

k k

sksk

sks

skk

sk

skk

sk

eTZ

TZ

eTZ

nNnE

µεβ

µεβ

µ

µ

µ

ε

),( where

),(

),(

,

,


Fermi particle case : ( )

( )µεβ

µεβµ

−−

−−

+=

≡ ∑k

skk

sk

e

eTZs

nnkG

1

),(,

1,0=skn

( )( )

( ) 11

:numbersoccupancy Mean 1lnln

:case for this potentialLandau

k

,k

+=

∂Ω∂

−=

+−=−=Ω

−

−−

µεβ

µεβ

µ k

k

en

ekTZkT

k

kG

µ defines :conditiony consistencSelf ⇒=− ∑k

knN


Bose particle case : ( )

( )( )µεβ

µεβ

µεβµ

−−

∞

=

−−

−−

−==

≡

∑

∑

ksk

skk

skk

sk

ee

eTZ

n

n

s

nnkG

11

),(

0

,

,4,3,2,1,0=skn

( )( )

( ) 11

:numbersoccupancy Mean 1lnln

:case for this potentialLandau

k

,k

−=

∂Ω∂

−=

−=−=Ω

−

−−

µεβ

µεβ

µ k

k

en

ekTZkT

k

kG

0 defines :cases someIn ≤⇒= ∑ µk

knN


Thermodynamic description of the equilibrium between two forms “phases” of a material under conditions of constant T and P

),(),(

),,(:energy Free sGibb' of Review

NV

Pg

NS

Tg

PTgNGPT

dNVdPSdTdGPVFPVTSENPTGG

TP

=

∂∂

−=

∂∂

≡==

++−=+=+−≡=

µµ

µ



Example of phase diagram :


( ) ( )( ) ( )

( )( )NV

NSdTdP

dPNV

NVdT

NS

NS

dPPg

PgdT

Tg

Tg

PTdgPTdgPTgPTg

B

B

A

A

B

B

A

A

T

B

T

A

P

B

P

A

BA

BA

//

0

0

,,,,

EquationClapeyron -Clausius

∆∆

=⇒

=

−+

−−

=

∂∂

−

∂∂

+

∂∂

−

∂∂

==


The van der Waals equation of state -- More realistic than the ideal gas law; contains some of the correct attributes for liquid-gas phase transitions.

( )

parametersdependent -material are here

:state ofequation der Waalsvan

:state ofequation gas Ideal

2

2

a, b

NkTbNVVNaP

NkTPV

=−

+

=

2

2

~3~3

~~8~ :state ofequation der Waalsvan

3~ 827~ 27~

: variablesessDimensionl

ρρ

ρ

ρ

−−

=

≡

≡

≡

TP

VNbkT

abTP

abP

Examples of systems studied using STP principles


Summary of results for classical fluid with pair potential:

:density lowat expansion virialof in terms state ofEquation

)()( 3

21)()( 321

:functionn correlatiopair of in terms state ofEquation

33 rgdr

rdurdrrgdr

rdurdrkTV

NNkTPV

∫∫ −=−=πβρπ

( )

( )ru

ru

erg

edr

rdurdrTB

(T)ρBNkTPV

β

βπβ

−

−

≈⇒

−=

++≈

∫)( :limit At this

)( 3

2)(

1

32

2

Documents

PHY 341/641 Thermodynamics and Statistical Physics Lecture 36