Dr. Neelabh Srivastava(Assistant Professor)

Department of PhysicsMahatma Gandhi Central University

Motihari-845401, BiharE-mail: neelabh@mgcub.ac.in

Programme: M.Sc. Physics

Semester: 2nd

Method of Ensembles: Partition Functions

Lecture NotesPart-1 (Unit - IV)

PHYS4006: Thermal and Statistical Physics

2

In earlier lectures, we have discussed the concept of

Ensembles viz. Microcanonical, Canonical and Grand-

canonical.

Herein, we will use that concept in deriving the

thermodynamical functions of a thermodynamic system.

Before we proceed, let us recall the concept of partition

function .

3

The Maxwell-Boltzmann distribution function is given by,

Occupation index,

This equation gives the number of gas molecules in the ith cell and

known as Maxwell-Boltzmann law of energy distribution.

simply,

where,

),.....3,2,1( kieegn i

ii

i

iiegZandeA

i

i

i

i

i

i ZAegeeegnN ii .

This Boltzmann distribution applies

to systems which have

distinguishable particles and N, V

and U are fixed. The Maxwell-

Boltzmann Distribution is

applicable only to dilute gases.

ieZ

N

g

nf

i

ii

4

The quantity Z represents the sum of the Boltzmann factor

over all the accessible states and is called the partition function (derived from German term

Zustandssummae). The quantity Z indicates how the gas molecules of an assembly are

distributed or partitioned among the various energy levels.

The energy term in the expression for partition function does not

mean only the translational component but also may contain the

components corresponding to other degrees of freedom too e.g.

rotational, vibrational and electronic too.

i

iiegZ

kTii eeZ/

i

i

iii

i

i

ii

i

i

i

i

eg

eg

N

nPobability

eg

eNgn

Z

NA

,Pr

• This partition function can be used for calculating the

various thermodynamical properties of ensembles having

independent systems (obeying classical laws) irrespective

of whether the ensembles have distinguishable or

indistinguishable independent systems.

• Consider an assembly of classical gas where the

distribution of energy states is considered to be

continuous. So, the number of energy levels in the

momentum range p and (p+dp) is given by -

5

dph

Vpdppg

3

24)(

Number of energy levels between energy range ε and

(ε+dε) is given by –

Since, the distribution of energy states is continuous,

therefore

6

d

h

Vdg

m3

2/12/3

22)(

0

2/1

3

2/3

0

22

)(

deh

VZ

edgegZ

kT

kT

i

kT

i

m

i

After solving, we get

This gives the translational partition function for a

gas molecule.

7

mkT

kTm

h

VZ

h

VZ

2

2

2/3

3

2/3

3

2/3

2

2

Partition Function and its relation

Thermodynamic Quantities

1. with Entropy (S):

Consider an assembly of ideal gas molecules obeying

M-B distribution law and according to Boltzmann’s

entropy relation –

The maximum thermodynamic probability is given by -

8

).........(lnln ikWkS

).........(!

! iin

gNW

i i

n

ii

Taking logarithms and apply Stirling’s approximation,

we get

According to M-B distribution law

Now, from eqn. (iii)

Putting,

9

i

iiii iiinngnNNW )........(lnlnlnln

ii Aegeegn iii

i i i

iiii

i

iii nAngngnNNW lnlnlnlnln

EnandNn i

i

i

i

i

EZNW

EA

NNEANNNW

lnln

lnlnlnln

but for an ideal gas,

from (iv) and (v),

10

)..(..........ln

lnlnln

ivT

EZNkS

kT

kEZNkEZNkWkS

)..(..........2

3vNkTE

)..(..........2

3ln viNkZNkS

2. with Helmholtz Free Energy (F):

3. with Total Energy (E):

Average energy of a system of N particles is given by,

11

)..(..........ln

ln

viiZNkTF

T

EZNkTETSEF

)(.......... viiiZ

eg

E

eAg

eAg

n

n

N

EE

i

ii

i

ii

i

ii

i

i

i

ii

i

i

i

Since, partition function

For isothermal-isochoric transformation,

Total energy,

12

T

Z

T

Z

T

Z

Vi

kT

ii

V

i

kT

ii

V

i

kT

i

i

i

Z

kTEEZegkT

egkT

egegZ

i

i

ii

2/2

/

2

/

1

).........(log22

ixNkTZ

NkTEENE Z

TT

Z

VV

4. with Enthalpy (H):

Enthalpy is given by,

H = E+PV = E+RT (for an ideal gas, PV=RT)

5. with Gibb’s Potential (G):

putting the value of S from eqn. (iv) in above, we get

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).........(log2 xRTNkTH ZT

V

TSRTNkTTSHG ZT

V

log2

putting the value of E from eqn. (ix), we get

14

EZNkTRTNkTG

T

EZNkTRTNkTG

ZT

ZT

V

V

ln

ln

log

log

2

2

Z

TZ

TVV

NkTZNkTRTNkTG loglog 22 ln

)(..........ln xiZNkTRTG

6. with Pressure (P) of the gas:

7. with Specific heat at constant volume, (CV):

15

)(..........2

22 )(log)(log2

log2

xiiiNkC

NkTT

C

ZT

TZT

T

ZTT

E

V

V

VV

V

)(..........log xiiNkTP ZVV

F

VV

References: Further Readings

1. Statistical Mechanics by R.K. Pathria

2. Statistical Mechanics by K. Huang

3. Statistical Mechanics by B.K. Agrawal and M. Eisner

4. Statistical Mechanics by Satya Prakash

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For any questions/doubts/suggestions and submission of

assignment

write at E-mail: neelabh@mgcub.ac.in