PHY 770 Spring 2014 -- Lecture 24 14/22/2014

PHY 770 -- Statistical Mechanics12:00* - 1:45 PM TR Olin 107

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

Lecture 24

Review and perspective; multicomponent systems (Chapter 3 in 3rd edition of Reichl)

Affinity; degree of reaction Equilibrium relationships Rates of reaction Course assessment forms

*Partial make-up lecture -- early start time

PHY 770 Spring 2014 -- Lecture 244/22/2014 2

PHY 770 Spring 2014 -- Lecture 24 34/22/2014

General comments about presentations:

This exercise is designed to allow you to study a topic (related to statistical and thermal physics) of your choosing in some detail. Please plan your presentation for 10-15 minutes and allow

at least 5 minutes for questions After your presentation, please hand in or email: Your presentation Any additional notes, computer programs, maple or

mathematica sheets A list of references including a copy of any seminal

references Extra points will be awarded for audience questions and

discussion

PHY 770 Spring 2014 -- Lecture 24 44/22/2014

Time Name Title

11:00-11:20 AM David Montgomery Chemical Reactions

11:25-11:45 AM

11:50-12:10 PM Zach Lamport Steam Engines - The Rankine Cycle

12:15-12:35 PM Xiaohua Liu Osmosis

12:40-1:00  PM Junwei Negative temperature state

1:05-1:25   PM Jiajie Xiao Physics in DNA-protein binding prediction

Presentations on Thursday 4/24/2014 in Olin 107

Hyunsu Lee???

PHY 770 Spring 2014 -- Lecture 24 54/22/2014

Time Name Title

11:00-11:20 AM Sam Flynn The ledenfrost effect

11:25-11:45 AM Calvin Arter “Statistical mechanics and the interaction potential”

11:50-12:10 PM Evan Welchman Monte Carlo

12:15-12:35 PM Ahmad Al-Qawasmeh

Linear Response Theory and Dielectric properties

12:40-1:00  PM Chaochao Dun Mo/CZTS interface instability

1:05-1:25   PM Eric Chapman Analysis of the Stirling Engine

Presentations on Tuesday 4/29/2014 in Olin 107

PHY 770 Spring 2014 -- Lecture 24 64/22/2014

PHY 770 Spring 2014 -- Lecture 24 74/22/2014

PHY 770 Spring 2014 -- Lecture 24 84/22/2014

Properties of the Gibb’s free energy -- multicomponent ideal gas

i ii

G N

2 2 2

2 2 21

For example, consider a reaction at fixed and :

2H O 2H O

2H O+2H +O 0 General notation: 0i

n

ii

T P

X

1 1

, 1

Change in Gibbs free energy with fixed and :

affinity

n n

i ii i

i i

i

T

iiP

n

T P

dG dN d

G

A

such that the change

in number of particles o

Define "

f type

degree of re

is

action

given

y

"

b i ii dN d

PHY 770 Spring 2014 -- Lecture 24 94/22/2014

Properties of the Gibb’s free energy -- multicomponent ideal gas

1 1

1

Change in Gibbs free energy with fixed and :

At equilibrium: 0 0

Note that if 0, there is a "driving force" for the reaction.

n n

i ii i

n

iii

i i

T P

dG dN d

dG

A

A

PHY 770 Spring 2014 -- Lecture 24 104/22/2014

Multicomponent ideal gas and possible chemical reactions -- continued

1

Equilibrium condition for Gibbs free energy with fixed and :

, 0

Estimation of the chemical potentials:

For each , assume independent ideal gas particles with internal

energies determin

i

n

ii

T P

T P

i

electron internal kinetic

kineti

ic

1/

3c

22

ed by electronic, internal and kinetic energies:

1 with thermal wavelength

! 2

Canonical partition function for syst

i

i i i i

N

i ii i i

Z Z Z

ZV

T

Z

m kN

1

em: n

ii

Z Z

PHY 770 Spring 2014 -- Lecture 24 114/22/2014

Estimation of chemical potential continued

0

1

Note that if we can show that , , ln where

Then the equilibrium condition , 0 has the following ana

,

lysis

i i

n

ii i

iii

T P T P kT c c

T P

N

N

1

0

1

1

1

1 1

,

1

0

0

0

, ln 0

, ln 0

,Define: ln ( , )

ln ln ln ( , )

( , ) e

n

i i

i i

i i

i i i

i ii i

i i

n

n

i

n

i

n

i

i i

T

n

i i

nP

k

ii

T

T P kT c

T Pc

kT

T PK T P

kT

c c K T P

c K T P

PHY 770 Spring 2014 -- Lecture 24 124/22/2014

Multicomponent ideal gas and possible chemical reactions -- continued

1

( , )i

n

ii

c K T P

2 2 2

Example:

2H O+2H +O 0

: 2 2

1i

1

2

2 22

2

H O( , )

H Oi

n

ii

c K T P

PHY 770 Spring 2014 -- Lecture 24 134/22/2014

Multicomponent ideal gas and possible chemical reactions -- continued

1

2

2 22

2

H O( , )

H Oi

n

ii

c K T P

2 2 2

2

2

2

2H O 2H O

Suppose H O 1

H

O / 2

x

x

x

1

0 ,3

2

( , ) e2(1 )

i in

i

T P

kTxK T P

x

PHY 770 Spring 2014 -- Lecture 24 144/22/2014

0Want to show that , , ln where ,i ii

i iT P T P kT c cN

N

Estimation of chemical potential continued

electroni

e internal kinetic

kinetic

intern

lectronic

1/22

3

electronic

al

1 with thermal wavelength

! 2

(represents vibrations, rotations, ( , )

i

i

ci

i

i i i i

N

i ii i i

Ni

N

i i

Z Z Z

Zm kT

Z

Z e

Z T P

V

N

1

etc.)

Canonical partition function for system: in

i

Z Z

PHY 770 Spring 2014 -- Lecture 24 154/22/2014

13

3

, ,

1

1

Helmholz free energy for this system

ln ln ln 1 ln

ln ln

For an ideal gas:

( , )

( ,

h r

)

w e e

j

eli i i i

i i

n n

i

eli i i

i i iT V N

i

n

ii

VA kT Z kT Z kT kT

N

VkT kT

N

PV NkT

N T P

AT P

N

N

3

0

03

/, ,Let ln ln

, , , ln

where , ln ln

( , )

( , )

elii i i i i

i i

i i i i

eli i i

i

kT PT P c kT kT

c

T P c T P kT c

kTT P kT kT

N

Nc T P

N

T PP

Estimation of chemical potential continued

PHY 770 Spring 2014 -- Lecture 24 164/22/2014

0

1

,

1

Law of "mass action"

( , ) e

n

i

i i

i

T P

kT

i

n

ic K T P

Multicomponent systems -- continued

0 0 0

, , ,

Other relationships:

Gibbs Free Energy: ( , , )

Enthalpy:

For

At equilibrium:

0

i

i i

T P T P T P

G T P N U TS PV

H U PV G TS

d d

G ST

N

H

0

, , , ,

= iP N T P T P P

GS

H GT

T T

PHY 770 Spring 2014 -- Lecture 24 174/22/2014

Example:2 4N O 0

2 4 2

Consider the gas phase reaction at constant and with initially = :

N O 2NO

T P N N

2 4 2N O NO

0

03

, , , ln

and , ln ln

1

w

In this ca

here ( , )

2se:

1 1

i i i i

elii i i i

i

Nc T P

T P c T P kT c

kTT P k

N PT k

c c

T

0

2

0

,

, , , ln

2, ln

1 1

( , , )i i i i i i ii i

i iiT P

T P c d TdG T P N

G

P kT c d

T P kT

PHY 770 Spring 2014 -- Lecture 24 184/22/2014

Example continued:

2 4

2

2

N O

NO

0

,

0

0

2, ln

1 1

In this case at standard and : 23.49kcal/mol

12.39kcal/mol

Solving for equilibrium

i iiT P

T P kT

T P

G

2 2 4NO N O0 02

2

value of

2

at standard and

exp

:

40.166

1 eqk

T P

T

2 4 2N O NO

Equilibrium concentrations:

1=0.715

2 0.284

1 1c c

PHY 770 Spring 2014 -- Lecture 24 194/22/2014

Estimation of chemical potential continued

0 0

0

Extension to other (non ideal gas) systems:

, , , ln ,

"standard state"

"activity coefficie

, , ln

nt"

,

i i i i i i i i i

i

i

T P c T P kT c T P c T P kT c

T P

PHY 770 Spring 2014 -- Lecture 24 204/22/2014

Extension of analysis to irreversible reactions

1

Affinity has previously been defined:

Note that if 0, there is a "driving force" for the reaction.

In this case, there is typically an irreversible contribution to the entropy:

n

ii

i

irrdS dT

A

A

A0

2Example: Cl(g)+H (g) HCl(g)+H(g)

2

Typically it is possible to determine the reaction rate:

1Cl H HCl H

Here the forward and reverse reaction rates are estimated from

the nearby equilibrium values of the Gibbs Free Energies.

rf

dk k

V dt