1/21/2014PHY 770 Spring 2014 -- Lecture 3 & 41 PHY 770 -- Statistical Mechanics 12:30-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course

PHY 770 Spring 2014 -- Lecture 3 & 4 11/21/2014

PHY 770 -- Statistical Mechanics12:30-1:45 PM TR Olin 107

Instructor: Natalie Holzwarth (Olin 300)Course Webpage: http://www.wfu.edu/~natalie/s14phy770

Lecture 3-4* -- Chapter 3Review of Thermodynamics – continued

1. Properties of thermodynamic potentials2. Response functions3. Thermodynamic stability

*Special double lecture, starting at 11 AM on 1/21/2014

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





Summary of thermodynamic potentials

Potential Variables Total Diff Fund. Eq.

U S,X,Ni

H S,Y,Ni

A T,X,Ni

G T,Y,Ni

W T,X,mi

i

iidNYdXTdSdU i

iiNYXTSU

i

iidNXdYTdSdH

i

iidNYdXSdTdA

i

iidNXdYSdTdG

i

iidNYdXSdTd

YXUH

TSUA

YXTSUG

i

iiNTSU


Derivative relationships of thermodynamic potentials

jii NXSiNSNX

i

N

U

X

UY

S

UT

NXSU

,,

i,,

:),,(energy Internal

jii NYSiNSNY

i

N

H

Y

HX

S

HT

NYSH

,,

i,,

:),,( Enthalpy

jii NXTiNTNX

i

N

A

X

AY

T

AS

NXTA

,,

i,,

:),,(energy free Helmholz

jii NYTiNTNY

i

N

G

Y

GX

T

GS

NYTG

,,

i,,

:),,(energy free sGibb'


Properties of thermodynamic potentials“Potential” in the sense that energy can be stored and retrieved through thermodynamic work

Equalities reversible processesInequalities general processes

EntropySd

T

QddS i

Entropy production due to irreversible processes 0SdiReversible entropy

contribution

T

QddS


Inequalities associated with thermodynamic potentials – continued

Entropy

0

0 :system isolated thermallyaFor

SddS

Qd

SdT

QddS

i

i

At equilibrium when dS=0 , S is a maximum



Internal energy

0

and ,, fixed with system isolatedan For

Since

dU

NXS

dNYdXTdSdU

dNYdXdUQdTdS

i

iii

iii

At equilibrium when dU=0 , U is a minimum (for fixed S, X, and {Ni})



Enthalpy

0


dH

NYS

dNXdYTdSdH

i

iii

At equilibrium when dH=0 , H is a minimum (for fixed S, Y, and {Ni})



Helmholz free energy

0


dA

NXT

dNYdXSdTdA

i

iii

At equilibrium when dA=0 , A is a minimum (for fixed T, X, and {Ni})



Gibbs free energy

0


dG

NYT

dNXdYSdTdG

i

iii

At equilibrium when dG=0 , G is a minimum (for fixed T, Y, and {Ni})



Grand potential

0


d

XT

dNYdXSdTd

i

iii

At equilibrium when d =0W , W is a minimum (for fixed T, X, and {mi})


Example:Consider a system having a constant electrostatic potential f at fixed T and P containing a fixed number of particles {Ni} for i=1,…n-1. Find the change in the Gibbs free energy when dNn particles, each with charge q n are reversibly added to the system.

dNqdG

NPT

dNdNqdNVdPSdTdG

i

iii

:, ,constant For

1

1


Thermodynamic response functions -- heat capacity

i

i

jii

NXNX

i

iii

NXTiNTNX

i

iii

T

UC

NX

dNN

UdXY

X

UdT

T

UQd

NXTUU

dNYdXdUQd

dT

QdC

,,

,,,,

:, constant at capacity heat For

:),,( Assuming


Thermodynamic response functions -- heat capacity -- continued

ii

ii

iii

i

jii

jii

NYNTNXNY

NYNTNX

NYi

ii

NYTiNTNY

i

iii

NXTiNTNX

i

iii

T

XY

X

UCC

dTT

XY

X

UdT

T

UQd

dTT

XdXNY

dNN

XdY

Y

XdT

T

XdX

NY

dNN

UdXY

X

UdT

T

UQd

NXTUU

dNYdXdUQd

dT

QdC

,,,,

,,,

,

,,,,

,,,,

:, constant At

: thatNote


:),,( Assuming


Example: Heat capacity

i i

iNV

NVNV

i

kN

C

T

UC

NVX

i

i

i

1

, constant at capacity heat For

,

,,

ii

i i

i

kTNPV

kTN

U

1 :gas ideal of

i i

ii

ii

i i

i

NPNTNVNP

i

kN

kNkN

T

VP

V

UCC

NPY

ii

ii

11


,,,,


Thermodynamic response functions -- heat capacity -- continued

ii

i

i

jii

NXNXNX

NX

i

ii

NXTiNTNX

i

T

AT

T

STC

dTT

STQd

NX

dNN

UdX

X

SdT

T

STQd

NXTSS

TdSQdS

,

2

2

,,

,

,,,,


:),,( Assuming

: of in terms analysis Alternate

ii

i

NYNYNY

i

T

GT

T

STC

NY

,

2

2

,,

-)- algebra someAfter (



Thermodynamic response functions -- mechanical response Results stated here without derivation:

ii

i

NTNTNT Y

G

Y

X

,

2

2

,, :itysusceptibl Isothermal

ii

i

NSNSNS Y

H

Y

X

,

2

2

,, :itysusceptibl Adiabatic

i

i

NYNY T

X

,, :yexpansivit Thermal

i

i

i

i

NX

NY

NS

NT

C

C

,

,

,

, :functions responsebetween ipsRelationsh

2,,,,

2,,,,

iiii

iiii

NYNTNTNY

NYNXNYNT

TC

TCC


Thermodynamic response functions -- mechanical response Typical parameters:

Isothermal susceptiblity isothermal compressibility

ii

i

NTNTNT P

G

VP

V

V,

2

2

,,

11

Adiabatic susceptiblity adiabatic compressibility

ii

i

NSNSNS P

H

VP

V

V,

2

2

,,

11

Thermal expansivity Thermal expansivity

i

i

NPNP T

V

V ,,

1


Example: mechanical responses for an ideal gas

NkTPV

kTN

U

1 :gas idealFor

PP

V

ViNT

T

11 :ilitycompressib Isothermal

,

PVPPV

P

V

V

S

SS

1 : thatNote

1 :ilitycompressib Adiabatic

00

TT

V

V PP

11 :yexpansivit Thermal


Stability analysis of the equilibrium state

iAiA

AAA

N

TVP

,

,,,

iBiB

BBB

N

TVP

,

,,,

Assume that the total system is isolated, but that there can be exchange of variables A and B.

iBiA

BA

BA

NN

VV

UU


Stability analysis of the equilibrium state

iAiA

AAA

N

TVP

,

,,,

iBiB

BBB

N

TVP

,

,,,

iBiA

BA

BA

NN

VV

UU

:assumecan weIf

iiA

B

iB

A

iAA

B

B

A

AA

BA

BA ii

itotal

NTT

VT

P

T

PU

TT

NN

SV

V

SU

U

SS

11

:nsfluctuatioentropy of Analysis

,

, , :mequilibriuAt iBiABABA PPTT

Note that this result does not hold for non-porous partition.


Stability analysis of the equilibrium state – continuedNote that the equilibrium condition can be a stable or unstable equilibrium.

Stable function(convex)

Unstable function(concave)

f

x

f

x

02

2

dx

fd0

2

2

dx

fd


Stability analysis of the equilibrium state -- continued

ii

itotal N

N

SV

V

SU

U

SS

2

1

sorder term second leading analyzing

:system dpartitione-multi ain nsfluctuatioentropy of analysis theExtending

i

i

i

NUN

SV

UV

SU

U

S

U

S

22

2

2

iiitotal

iii

ΔNΔΔVΔPΔSΔTT

ΔS

NVPUST

2

1

:becomesexpansion order second the

:identitiesother andrelation lawfirst theUsing

Documents

1/21/2014PHY 770 Spring 2014 -- Lecture 3 & 41 PHY 770 -- Statistical Mechanics 12:30-1:45 PM TR Olin 107 Instructor: Natalie Holzwarth (Olin 300) Course