chemical potential (a summary) - berkeley cosmology...

Phys 112 (S2006) 7 Chemical Potential 1

Chemical Potential (a Summary)

Definition and interpretations K&K chap 5.• Thermodynamics definition• Concentration• Normalization• Potential

Law of mass action K&K chap 9

Saha EquationThe density of baryons in the universe


Chemical Potential DefinitionMicrocanonical method

Isolated systems 1 and 2 put in contactEquilibrium of isolated system 1+2= Probability distribution sharply

peaked around configuration of maximum entropy

<=>

Thermodynamics functions• U

• F=U-τσ

Note:

Does not apply to U since entropy changes with the number of particles• G=F+PV

Note that

Proof: divide system into two subsystems. Subsystems are in equilibrium

!"T

!N1

= 0 with N1+ N

2= 0

!"1

!N1 U,V

=!"

2

!N2 U,V

= #µ

$

1 2!1 N1 ! 2 N2

dF = !"d# ! PdV + µdN

µ = F N( ) ! F N !1( ) at constant " ,V

! µ ="F

"N # ,V

dG = !"d# +VdP + µdN! µ =

"G

"N # ,PG = Nµ P,!( )

dU = !d" # PdV + µdN ! µ ="U

"N # ,V

!G1

!N1

= µ P," , N1( ) =!G

2

!N2

= µ P," , N2( ) !"µ

"N= 0

!G

!N= µ P,"( )# G = Nµ P,"( )+ f P,"( ) N = 0!G = 0! f P,"( ) = 0


Chemical Potential <=> ConcentrationCanonical method

Probability

=>

=>

Ideal gas:

increases with temperatureincreases with concentrationNote:

Review examples of attachment to a molecule or trapping by an impurity site (K&K p.141, Noteschap. 3, p.12)

ps =1

Ze!"s

#Z = e

!" s

#

s

$

U = !s pss

" = #2 $ logZ

$#! = " ps log ps

s

# =$ % log Z( )

$%F =U !"# = !" log Z

µ =!F

!N " ,V

= #"! log Z

!N ",V

Z =1

N!nQV( )

N

�

nQ

= exp !"#

$

% &

'

( ) D "( )d"0

*+ =1

4, 2

2m

!2

$

%

& &

'

(

) )

3 / 2

exp !"#

$

% &

'

( ) "d"

0*+ =

m#2,! 2$

% &

'

( )

3/2

µ = !"# log Z#N ",V

= " logN ! log nQV( )( ) = " logn

nQ

$

% &

'

( )

log Z ! " N logN " N( ) + Nlog nQV( )

G = F + PV = N! logn

nQ

"

# $

%

& ' ( 1

"

# $

%

& ' + N! = N! log

n

nQ

"

# $

%

& ' = N! log

P

!nQ

"

# $

%

& ' = Nµ P,!( )

�

! "0

#

$ exp %"&

'

( )

*

+ , d" =

&2

'

(

)

)

*

+

,

,

3 / 2

2x2

0

#$ e

x2

2 dx

with x =2"&

! " # $ #

= 2-&2

'

(

)

)

*

+

,

,

3 / 2


Grand Canonical methodProbability

In order to determine µ, impose that <N> is the given N

Examples: Ideal gases• Fermi Dirac or Bose Einstein. With our convention for the density of states

which does not include volume (≠K&K)

• Classical limit (non relativistic)

same result as in the canonical case!Proof:

Chemical Potential = normalization method

p s,! s,N( ),N( ) =1

Zexp

µN " ! s,N( )#

$%&

'()

Z = expµN ! " s,N( )

#$%&

'()

s,N

*

N = Ns,N

! p s," s,N( ), N( ) = Nsystem

! s,N( ) = N!s

1

exp! " µ

#( ) ± 1

D !( )0

$

% d! = n

expµ !"#

$

% & '

( ) D "( )

0

*

+ d" = n, µ = # log n

nQ

$

% &

'

( )

! expµ

"( ) m"

2#!2$ %

& '

3 /2

= n ! µ = " logn

nQ

$

% (

&

' )

�

exp !"#

$

% &

'

( ) D "( )d"0

*+ =1

4, 2

2m

!2

$

%

& &

'

(

) )

3 / 2

exp !"#

$

% &

'

( ) "d"

0*+ =

m#2,! 2$

% &

'

( )

3/2

= nQ


Chemical Potential as a PotentialRaising the potential energy of a system

Let us consider an isolated system at zero potential energy

Let us then raise it at uniform potential energy per particleThe entropy is not changed by uniform potential (number of states not changed)

Example: Barometric pressure equation

Uo µo =!Uo

!N " ,V

Uo !U = Uo + N"#

µ =!U

!N " ,V

= µointernal

!+ #$

external

!


Let us consider two isolated systems in interaction with each other.If concentration 1 > concentration 2

The only way to maintain the difference of concentration is to give somepotential energy to system 2 with respect to system 1

In practice, if you attempt to prevent an evolution of the concentration, youwill have to generate a difference of potential energy (and vice versa).

At a deeper level the constancy of the total chemical potential in asystem in equilibrium reflects the balance between the drift currentgenerated by the external potential and the diffusion current due tothe random thermal velocity. cf. Chapter 10 of the notes

Difference of Concentration and Potential

µint 1 > µint 2

µint 1 +!1 = µint 2 +!2

µint 1

!1

µint 2

!2

!"

µint 1 ! µint 2 ="2 !"1

!µint = "!#


Examples: Battery (K&K p.129)Consider an electrolyte AB: ions A- B+ In the middle of the cell, equilibrium between positive and

negative ions. But on electrodes, difference of behavior => selective depletion + repulsion

If the two electrodes are not connected

p-n Diode

On n side, the donors give their electrons and positive charges remain behindOn p side, the acceptors capture the electrons, generating fixed negative

charges. The resulting field generate a potential barrier which preventscurrent to flow in one direction

Balance between drift and diffusion

A!+ N " AN + e

!B++ P + e

!" BP

N P

B

+

B

+B

+

B

+

B

+

B

+

B

+

B

+

B

+

B

+

A

!

A

!

A

!

A

!

A

!

A

!

A

!

A

!

++++++++

--------

! ! .! E =

"

##o

!

E

N P

V

P

V = !! E .d! r

x

"N

Stops A- neutralization

p n

eeeeeeeeeeeeee

hhhhhhhhhh

!Fn( )

!Fp( )

p

n

eeeeeeeeeeeeee

hhhhhhhhhh!Fn( )!

Fp( ) -----+++


Calculation Method

Consider 2 species of opposite charge q±: number density n±x( )

Combine µint

+x( ) + q+

n+x( )! x( ) = Constant µ

int

"x( ) + q"

n"x( )! x( ) = Constant

and !#.!E = "#

2! =

q+n+x( ) + q"

n"x( )

$%o

!!!!!& 3 equations for 3 functions n+!n

"!!

($ ,often also written %, is the relative dielectric constant of the medium)

Note that in semiconductor books, the constancy of the chemical potential

is expressed in terms of the sum of the drift and diffusion currents being zero.

This is the same physics expressed in different ways!


Several speciesEquilibrium with several species i

If the two systems are in equilibrium, each kind separately has to be inequilibrium

=>

Conserved quantitiesIn a reaction between species, the number of disappearing particles or

molecules is related to the number of produced particles or molecules

The probability distribution at equilibrium will be sharply peaked around theconfiguration of maximum total entropy :

or

with the constraints

µi 1( ) = µi 2( ) !i21

µi 1( ) µi 2( )

!1A1 +!2A2 "# 3A3 +# 4A4

! "i Ai

i

# $ 0 with "3 = %&3 , "4 = %& 4

!" =#"

#NA1

!NA1+

#"

#NA2

!NA2+

#"

#NA3

!NA3+

#"

#NA4

!NA4= 0

µ1!NA1+ µ2!NA

2+ µ3!NA

3+ µ4!NA

4= 0

!NA1

"1

=!N

A2

"2

=!NA

3

"3

=!N

A4

"4

!NA1

"1

=!N

A2

"2

=!NA

3

"3

=!N

A4

"4

! "iµi

i

# = 0

or "iµi

initial

# = $iµi

Final

#

Conservation of chemical potential


ConsequencesPhotons have µ=0 <= they can disappear by interaction with electronsIf there is no asymmetry particles and antiparticles have opposite

µ’s.


Important Note!

The energies of all the states have to be measuredfrom the same origin.

This is taken automatically into account by including in the internalpartition function the ground state energies (i.e. rest mass)

Important to take into account threshold/energy release effects

The reaction A + B! AB can be exothermic

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!" ground state of AB( ) < " ground state of A( ) + " ground state of B( )

or endothermic " ground state of AB( ) > " ground state of A( ) + " ground state of B( )

e.g. Z = exp !"s

#$%&

'()

s

* = exp !"Ki

#$%&

'()

i

*Kinetic=ZK

! "## $##

exp !"ground state

#$

%&'

()exp !

"int j

#$%&

'()j

*Internal=Zint

! "###### $######


Law of mass actionClassical Ideal Gas

Recall: 1 particle

Kinematic part

Internal e.g. multiplicity g (rotation, spin, binding energy -see below) N particles

=>

Law of Mass ActionConsider the reaction

or

=>

Z = exp !"s#

$

% &

'

( )

s

* = exp !"

Ki

#$

% &

'

( )

i

*Kinetic=Z

K

! " # # $ # #

exp !"int j

#$

% &

'

( )

j

*

Internal =Zint

! " # # $ # #

�

ZK

= nQV

�

Z =1

N!nQV( )

N

Zint

N F = !" log Z# µ =$F$N " ,V

= " logn

nQZint

%

& '

(

) *

!iAii

" # 0

!iµii

" = 0 with µi = # logni

nQiZint i

$

% &

'

( )

ni

nQiZint i

!

" #

$

% &

i

'( i

= 1

ni!i

i

" = K #( ) with K #( ) = nQiZint i( )i

"!i


Kinetic View of Mass ActionDetailed balance

Equilibrium

Does not say anything about reaction time (e.g. independent of catalyst)!

dnAB

dt= C !( )nAnB " D !( )nAB

nAB

nAnB

=C !( )

D !( )= K !( )

A + B! AB" AB # A # B! 0 nAB

nAnB

= K !( )


ExamplespH and the Ionization of Water

Define

=>In normal conditions:

Definition

pH>7 basicpH<7 acidic

Non degenerate semiconductors (classical limit)

H++OH

!" H2O

A[ ] ! concentration of A in mole/liter

H+[ ] OH![ ]H2O[ ]

= K "( )

�

H+

[ ] << H2O[ ] = 56 moles/liters

�

H+[ ] OH ![ ] = 10

!14mole/liter( )

2 independently of H

+[ ]

pH = ! log10 H+[ ]( ) = 14 + log10 OH![ ]( )

ne ! nQe exp "#c " µ$

%&'

()*

with!nQe = 2me

*$2+!2

%&'

()*

3

2

!!

nh ! nQh exp "µ " #v$

%&'

()*

with!nQh = 2mh

*$2+!2

%&'

()*

3

2

!!!!!!!!, nenh = nQenQh exp "#c " #v

$%&'

()*


Ionization of HydrogenSaha Equation:

Where we identify the rest energies of the proton, electron. Similarly for hydrogen

there are 4 spin states!

Taking into account that the masses of the proton and the hydrogen are very close, themass action law can be written:

where I is the ionization energy of atomic hydrogen (if we neglectexcited states)

If the concentrations of protons and electrons are the same:

Not a Boltzmann factor!

e + p! H

nQeZ

int e= 2

me!

2"!2#$%

&'(3/2

exp )*e

!#$%

&'(

nQpZint p = 2mp!2"!2

#$%

&'(

3/2

exp )* p!

#$%

&'(

nQHZint H = 4

mH!

2"!2

#$%

&'(

3/2

exp )*H

!#$%

&'(

nenp= n

H

me!

2"!2#$%

&'(3/2

exp )*p+ *

e) *

H

!#$%

&'(= n

H

me!

2"!2#$%

&'(3/2

exp )I

!#$%

&'(

ne= n

p= n

H

me!

2"!2#$%

&'(3/4

exp )I

2!#$%

&'(

�

nenp

nH

=nQeZ int enQpZ int pe

nQHZ

int H

I = !p+ !

e" !

H


How to build Nuclei?

Nuclear attraction

Requires emission of a photon!Need low enough temperature for n and p to stay bound.Otherwise

More generally equilibrium function of temperature

Coulomb repulsion e.g. 4He

Need s enough energy to “penetrate” barrier=> needs high enough temperature(but not too high lest its dissociates)

Large amount of He in the universe: hot Big Bang

Pote

ntia

l

rn-p

B

“Potential well”

p + n!2H +"

p + n!2 H + "

p + n!2H +"

Epot + Ekin = constant

Deuterium =2H

B= 2.22 MeV

p

n

γ

Pote

ntia

l

rDD

potential barrier <- Coulomb repulsion

�

2H

++

2H

+!

4He

++ + "

γBinding energy (28MeV)


Reaction Rates and ExpansionStrong dependence of reaction rate on temperature

cf. ordinary cooking Breaking of bounds (tenderness)Pressure cooker, refrigerator

Establishing new bounds (e.g. custard)

Dependence on densityThe particles have to find each other!

In an expanding universeThe reaction rate has to be greater than the expansion rate

otherwise nuclei are diluted away before having time of reacting!Freeze-out

3 temperature regimes• At high temperature, nucleus cannot exist <- dissociation• At low temperature: nucleus is stable but not enough energy to be

formed The reactions proceed too slowly :Freeze-out• Intermediate: nucleus can be formed and is stable enough to survive

�

rate!"1"2


Need protons and neutronsBut do it fast enough as neutron will decay(half life time 10.6

minutes)

2 body reactionsDensity not large enough for 3 body

+ a number of other possible reactions!

Process stopped by Freeze-OutTemperature and density become too smallNo element heavier than Lithium

Building Nuclei by Fusion

p+n

HydrogenDeuterium=2H

3He

4He

7Li

+

+3He

2H


Primordial Nucleosynthesis

2H bottleneck

Freeze out

n depleted by low T

• 4He very much more bound than 2H => higher equilibrium concentration at low temperatureBut cannot be reached because we have to go through 2-body reactions: deuterium bottleneck!Really starts at 0.150MeV ≈1 minute

• everything is over in 5 minutes: stops below carbonHigher A elements will have to be produced in stars and supernovaeDependent on expansion rate and density of p&n = Ωb

Same formalism asSaha equation butexpanding universe


Non BaryonicDark Matter

Big Bang Nucleosynthesis

Self consistent

Confirmedby Cosmic Microwave

Background

Density of baryons ≈ 5%

4He

7Li

D

chemical potential (a summary) - berkeley cosmology...

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