PHY 770 Spring 2014 -- Lecture 2 11/16/2014

PHY 770 -- Statistical Mechanics10-10:50 AM MWF Olin 107

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

Lecture 2 -- Chapter 3Review of Thermodynamics – continued

1. Some empirically obtained equations of state2. Some properties of entropy3. Thermodynamic potentials

PHY 770 Spring 2014 -- Lecture 2 21/16/2014

Equations of state

Variable UnitsT (temperature) oK

P (pressure) PaV (volume) m3

n (moles) n=N/NA

N (particles) N=nNA

Avogadro’s number: NA=6.022 141 29 x 1023 mol-1

Boltzmann constant: k=1.380 6488 x 10-23 J K-1

Molar gas contant: R=NAk=8.314 4621 J mol-1 K-1

http://physics.nist.gov/cuu/Constants/index.html

PHY 770 Spring 2014 -- Lecture 2 31/16/2014

Equations of State -- examples

NkTnRTPV Law Gas Ideal

Range of validity

dilute limit; ignor particle interactions

includes effects of higher density in terms of virial coefficients Bi

approximates gases and liquids in terms of an excluded volume nb and a cohesion parameter a

TB

V

nTB

V

n

V

nRTP 32

2

21

expansion Virial

nRTnbVV

anP

2

2

state ofequation der WaalsVan

PHY 770 Spring 2014 -- Lecture 2 41/16/2014

Special properties of entropy

For a reversible process:T

QddS

Thermo “laws” involving entropy2. Heat flows spontaneously from high temperature to

low temperature3. It is not possible to reach the coldest temperature

using a finite set of reversible steps

These relationships, together with the notion that entropy is an extensive and additive property leads to the

Fundamental equation of thermodynamics:

i

ii NXYUTS

PHY 770 Spring 2014 -- Lecture 2 51/16/2014

Fundamental equation of thermodynamics

i

ii NXYUTS

internal energy

generalized displacement (V)

generalized force (-P)

chemical potential

number of particles

Derivation of fundamental equation of thermodynamics:

ii

i

iii

dNT

dXT

YdU

TdS

dNYdXTdSdU

1

:process reversible afor amics thermodynof lawFirst

PHY 770 Spring 2014 -- Lecture 2 61/16/2014

Derivation of fundamental equation of thermodynamics -- continued:

TN

S

T

Y

X

S

TU

S

dNN

SdX

X

SdU

U

SdS

dNT

dXT

YdU

TdS

i

NXUiNUNX

ii

NXUiNUNX

ii

i

jii

jii

,,,,

,,,,

1

:also

1

ii

i

NXUSNXUS

NXUSS

,,,,

:iprelationsh scaling following infer thecan we

extensive are quantities theseof all because note Also

,,

PHY 770 Spring 2014 -- Lecture 2 71/16/2014

Derivation of fundamental equation of thermodynamics -- continued:

iii

ii

i

ii

NXUiNUNX

i

i

NXUiNUNX

ii

NYXUTS

NT

XT

YU

TS

NN

SX

X

SU

U

SS

d

Nd

N

S

d

Xd

X

S

d

Ud

U

S

d

Sd

NXUSNXUS

jii

jii

1

,,,,

,,,,

,,,,

PHY 770 Spring 2014 -- Lecture 2 81/16/2014

Fundamental equation of thermodynamics

i

ii NXYUTS

Some consequences:

0

:equation Duhem-Gibbs

:amics thermodynof lawfirst From

iii

iii

iii

dNXdYSdT

dNXYUdTSd

dNYdXTdSdU

PHY 770 Spring 2014 -- Lecture 2 91/16/2014

Thermodynamic potentials

iii

iii

i

NYXTSU

dNYdXTdSdU

NXSUU

:equation lFundamenta

:alDifferenti

,, :energy Internal

i

NXSiNSNX

ii

NXSiNSNX

jii

jii

N

UY

X

UT

S

U

dNN

UdX

X

UdS

S

UdU

,,,,

,,,,

:state of Equations

:ipsrelationshFurther

PHY 770 Spring 2014 -- Lecture 2 101/16/2014

Analysis of internal energy continued:

iii

i

jii

NXNSNSX,N

i

NXSiNSNX

S

Y

X

T

S

U

X

N

UY

X

UT

S

U

,,,

,,,,

:relations sMaxwell'

:state of Equations

:ipsrelationshFurther

PHY 770 Spring 2014 -- Lecture 2 111/16/2014

???),( ),( zyxyxz

dzz

xdy

y

xdxzyx

dyy

zdx

x

zdzyxz

yz

xy

),(

),(

y

x

zxz

yz

y

x

/

/ :But

Mathematical transformations for continuous functions of several variables & Legendre transforms:

PHY 770 Spring 2014 -- Lecture 2 121/16/2014

Mathematical transformations for continuous functions of several variables & Legendre transforms continued:

and Let

),(

xy

xy

y

zv

x

zu

dyy

zdx

x

zdzyxz

vy

z

y

wx

u

wvdyxdudw

xduudxvdyudxxduudxdzdwuxzw

dyy

wdu

u

wdwyuw

xuy

uy

,For

),(

function new Define

PHY 770 Spring 2014 -- Lecture 2 131/16/2014

SP

SP

SV

SV

P

HV

S

HT

dPP

HdS

S

HVdPTdSVdPPdVdUdH

PVUPSHH

V

UP

S

UT

dVV

UdS

S

UdU

PdVTdSdU

VSUU

),( :Enthalpy

),( :energy Internal

For thermodynamic functions:

PHY 770 Spring 2014 -- Lecture 2 141/16/2014

Thermodynamic potentials – Internal Energy

PHY 770 Spring 2014 -- Lecture 2 151/16/2014

Thermodynamic potentials – Enthalpy

PHY 770 Spring 2014 -- Lecture 2 161/16/2014

Thermodynamic potentials – Helmholz Free Energy

PHY 770 Spring 2014 -- Lecture 2 171/16/2014

Thermodynamic potentials – Gibbs Free Energy

PHY 770 Spring 2014 -- Lecture 2 181/16/2014

Thermodynamic potentials – Grand Potential

PHY 770 Spring 2014 -- Lecture 2 191/16/2014

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

ii NYXTSU

i

iidNXdYTdSdH

i

iidNYdXSdTdA

i

iidNXdYSdTdG

i

iidNYdXSdTd

YXUH

TSUA

YXTSUG

i

ii NTSU

PHY 770 Spring 2014 -- Lecture 2 201/16/2014

TNk

H

PVUXYUH

PVT

NkU

NkTPV

CCkk VPB

1

1

1

/ constant;Boltzmann

Some examples Ideal gas with N particles of the same type and with X V Y-P

PHY 770 Spring 2014 -- Lecture 2 211/16/2014

SVT

TVNkVTS

TSUA

PVT

NkU

NkTPV

CCkk VPB

0100

1

ln1

),(

:shown have we,Previously

1

1

/ constant;Boltzmann

Some examples -- continued Ideal gas with N particles of the same type and with X V Y-P

PHY 770 Spring 2014 -- Lecture 2 221/16/2014

TNk

SVT

VTNkT

TSVT

TVT

NkA

SVT

TVNkVTS

TSUA TNk

U

1ln1

ln11

ln1

),(with

1

0

01

1

0

1

1

0100

1

0100

1

Some examples -- continued Ideal gas with N particles of the same type and with X V Y-P