chapter 3 classical statistics of maxwell-boltzmann

Chapter 3

Classical Statistics of Maxwell-Boltzmann

1- Boltzmann Statistics

- Goal:

Find the occupation number of each energy level (i.e. find(N1,N2,…,Nn)) when the thermodynamic probability is a maximum.

- Constraints: )1(1

i

iNN )2(1

i

iiNU

N and U are fixed

- Consider the first energy level, i=1. The number of ways of selectingN1 particles from a total of N to be placed in the first level is

)!(!

!

111 NNN

NN

N


We ask:

In how many ways can these N1 particles be arranged in the firstlevel such that in this level there are g1 quantum states?

For each particle there are g1 choices. That is, there are

possibilities in all.

Thus the number of ways to put N1 particles into a level containingg1 quantum states is

1

1Ng

!!

!

11

11

NNN

gN N


For the second energy level, the situation is the same, except thatthere are only (N-N1) particles remaining to deal with:

!!

!

212

212

NNNN

gNN N

Continuing the process, we obtain the Boltzmann distribution ωB as:

3213

321

212

21

11

121

!

!

!!

!

!!

!)...,(

3

21

NNNNN

gNNN

NNNN

gNN

NNN

gNNNN

N

NN

nB


!!!

!),(321

32121

321

NNN

gggNNNN

NNN

nB

)3(!

!),(1

21

n

i i

Ni

nB N

gNNNN

i

2- The Boltzmann Distribution

Now our task is to maximize ωB of Equation (3)

At maximum, . Hence we can

choose to maximize ln(ωB ) instead of ωB itself, this turns theproducts into sums in Equation (3).

0)(ln0 B

BBB

ddd

Since the logarithm is a monotonic function of its argument, themaxima of ωB and ln(ωB) occur at the same point.

From Equation (3), we have:

1

)!ln()ln()!ln()ln(i

iiiB NgNN


1

)!ln()ln()!ln()ln(i

iiiB NgNN

iiii NNNN )ln()!ln(Applying Stirling’s law:

1

)ln()ln()!ln()ln(i

iiiiiB NNNgNN

)ln()ln(11

)ln()ln()ln(

iii

iiii

B NgN

NNgN


Now, we introduce the constraints

Introducing Lagrange multipliers (see chapter 2 in ClassicalMechanics (2)):

0)ln( 21

iii

B

NNN

0)ln()ln( iii Ng

iii

ii

iii

NNU

NNN

2

12

1

11

0

10


ii

ii

i

i

g

N

N

g

lnln

ii eegNeg

Nii

i

i ??e

11 i

ii

iieegNN

11

iii

i i

i

eg

NeegeN

)4(

1

ii

ii i

i

eg

eNgN

We will prove later that TkB

1


(5)on)distributi (Boltzmann

iii

ii i

i

eg

Ne

g

Nf

and hence

The sum in the denominator is called the partition function for asingle particle (N=1) or sum-over-states, and is represented by thesymbol Zsp:

)6(1

n

iisp

iegZ

where fi is the probability of occupation of a single state belongingto the ith energy level.


or )7(1

s

spseZ

Ω = total number of microstates of the system,s = the index of the state (microstate) that the system can occupy,εs = the total energy of the system when it is in microstate s.

Example:

A system possesses two identical and distinguishable particles (N=2),and three energy levels (ε1=0, ε2=ε, and ε3=2ε) with g1=2, andg2=g3=1. Calculate Zsp by using Eqs.(6) and (7)


Macrostate Nb. microstates

(N1,N2,N3) ωB εs

(1,0,0) 2 0

(0,1,0) 1 ε

(0,0,1) 1 2ε

2

)2()0(

321

3

1

2

2

321

ee

eee

egegegegZi

ispi

2

)1,0,0(

)2(

)0,1,0(

)(

)0,0,1(

)0(

)0,0,1(

)0(4

1

2

ee

eeeeeZs

sps


If the energy levels are crowded together very closely, as they arefor a gaseous system:

whereg(ε)dε: number of states in the energy range from ε to ε+dε,N(ε)dε: number of particles in the range ε to ε+dε.

We then obtain the continuous distribution function:

dNN

dggbyreplaced

i

byreplacedi

)(

)(

deg

Ne

g

Nf

)()(

)()(

3- Dilute gases and the Maxwell-Boltzmann Distribution

The word “dilute” means Ni << gi, for all i.

This condition holds for real gases except at very low temperatures.

The Maxwell-Boltzmann statistics can be written in this case as:

n

i i

Ni

MB N

g i

1 !


The Maxwell-Boltzmann distribution corresponding to ωMB,max:

n

i i

Ni

MB N

g i

1 !

n

i i

Ni

B N

gN

i

1 !!

- ωMB and ωB differ only by a constant – the factor N!,

- Since maximizing ω involves taking derivatives and the derivativeof a constant is zero, so we get precisely the Boltzmann distribution:

on)distributiBoltzmann -(Maxwellspi

ii Z

Ne

g

Nf

i


- Boltzmann statistics assumes distinguishable (localizable) particlesand therefore has limited application, largely solids and some liquids.

- Maxwell-Boltzmann statistics is a very useful approximation forthe special case of a dilute gas, which is a good model for a realgas under most conditions.

4- Thermodynamic Properties from the Partition Function

In this section, we will state the relationships between the partitionfunction and the various thermodynamic parameters of the system.

1ii

sp iegZ

1iii

sp iegZ

NT

i

ii

NTii

NT

sp

Vegeg

VV

Zii

,1,1,

NT

i

ii

NT

sp

Veg

V

Zi

,1,


11

1,

1

iii

spspii

iii

i

i

egZ

uZ

eNgNandN

Nu

- Calculation of average energy per particle:

sp

spii

sp

Z

Zeg

Zu i

11

1

)ln( spZu

NuU

- Calculation of internal energy of the system:

)ln( spZNU


- Calculation of Entropy for Maxwell-Boltzmann statistics:

i

i

eN

Z

N

g

Z

eNgN sp

i

i

spii

1

max !i i

Ni

N

g i

1

)!ln()ln()ln(i

iii NgN

1

)ln()ln()ln(i

iiiii NNNgN

11

ln)ln(i

ii i

ii N

N

gN

1

ln)ln(i

spi Ne

N

ZN i

with


NNN

ZN

iii

spi

1

)(ln)ln(

NNNN

Z

U

iii

N

ii

sp

11

ln)ln(

NUNNZN sp )ln()ln()ln(

UNZN sp )1)ln()(ln()ln( )ln( BkS

UkNZNkS BspB 1)ln()ln(


- Calculation of β

VU

S

TpdVdUTdS

1

UkZNkS BspB )ln(

VBB

V

spB

V UUkk

U

ZNk

U

S

)ln(

Tk

UUkk

U

ZNk

U

SB

VBB

V

NU

BV

1)ln(

/

TkB

1


- Calculation of the Helmholtz free energy:

U

BspB UTkNZTNkUATSUA

1)ln()ln(

1)ln()ln( NZTNkA spB

- Calculation of the pressure:

dVV

AdT

T

ApdVSdTdA

TV

TV

Ap

T

spB V

ZTNkp

)ln(

5- Partition Function for a Gas

1i

Tkisp

BiegZ

The definition of the partition function is

For a sample of gas in a container of macroscopic size, the energylevels are very closely spaced.

Consequences:

- The energy levels can be regarded as a continuum.- We can use the result for the density of states derived in Chapter 2:

dmh

Vdg s

21233

24)(

5- Partition Function for a Gasγs = 1 since the gas is composed of molecules rather than spin 1/2particles. Thus

dmh

Vdg 2123

3

24)(

Then

0

2123

03

24)( dem

h

VdegZ TkTk

spBB

The integral can be found in tables and is

TkTk

de BBTk B

20

21

23

2

2

h

TmkVZ B

sp

Partition function depends on both thevolume V and the temperature T.

6- Properties of a Monatomic Ideal Gas23

2

2),(

h

TmkVTVZ B

sp

2

2ln

2

3)ln(

2

3)ln()ln(

h

mkTVZ B

sp

- Calculation of pressure:

V

TNkp

V

ZTNkp B

T

spB

)ln(

RN

RN

nknRNknRTpV

ABB

1 Since

where NA is the Avogadro’s number and n the number of moles.

AB N

Rk

6- Properties of a Monatomic Ideal Gas

- Calculation of internal energy:

T

T

ZNUZNU sp

sp

)ln()ln(

22

11Tk

T

TkTTk BBB

TT

Z

h

kmTVZ spB

sp

1

2

3)ln(2ln

2

3)ln(

2

3)ln()ln(

2

TTNkU B 2

32

TNkU B2

3

6- Properties of a Monatomic Ideal Gas- Calculation of heat capacity at constant volume:

BVBNV

V NkCTNkUT

UC

2

3

2

3and

,

CV is constant and independent of temperature in an ideal gas.

BAV knNC2

3 nRCV 2

3

- Calculation of entropy T

UNZNkS spB 1)ln()ln(

T

TNkN

h

mkTVNkS

BB

B2

3

1)ln(2

ln2

3)ln(

2

3)ln( 2

3

23)2(ln

2

5)ln(

2

3ln

h

mkNkT

N

VNkS B

BB

chapter 3 classical statistics of maxwell-boltzmann

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