CHAPTER 7: STATISTICAL THERMODYNAMICS 7.1 BACKGROUND Statistical thermodynamics was developed by Boltzmann in Germany and concurrently by Gibbs in USA. The theory was modified to some extent later on by S. N. Bose (India, 1894-1974), Einstein ( Germany, 1879-1955), Enrico Fermi ( Italy, 1901-1954) and Paul A. M. Dirac (UK, 1902-1984). It is applicable to molecules, photons, elastic waves of solids, abstract entities such as wave function, electrons, boson, meson, etc. Classical mechanics applies to large number of particle collections (order of Avogadro’s number ). Molecules being large in number, average properties are calculated without knowing about the individual molecule. Further, classical thermodynamics assumes the flow field to be continuum and conservative. But it fails to predict the value of properties like specific heat. In case of kinetic theory, number of molecules per cubic centimeter is in a chamber. To know velocity vector (

346.02626 10A = ×

2010, ,x y z ) direction with particles, we require

to solve second order equations. Each molecule is to be considered. Thus it is a formidable task to solve such huge number of equations.

N203 10N ×

Statistical thermodynamics deal with the evaluation thermodynamics properties in terms of entropy. All the models or relations developed in statistical thermodynamics aim at finding the entropy. Study of relationships among the thermodynamic properties alone is generally the topic of classical thermodynamics. On the other hand, establishing relationships between non thermodynamic and thermodynamic properties of matter in equilibrium states is the task of statistical thermodynamics. Thermodynamic and non-thermodynamic properties have been discussed in the following section. 7.2 THERMODYNAMIC (MACROSCOPIC) AND NON-THERMODYNAMIC (MICROSCOPIC) PROPERTIES A property of matter is any characteristic, which can distinguish a given quantity of a matter from another. These distinguishing characteristics can be classified in several different ways, but for the convenience, it can be divided into thermodynamic and non-thermodynamic properties. The non-thermodynamic properties describe the characteristics of the "ultimate particles" of matter. An ultimate particle, from a thermodynamic viewpoint, is the smallest subdivision of a quantity of matter, which does not undergo any net internal changes during a selected set of processes, which alter properties of the entire quantity. The ultimate particles are generally considered to be molecules or atoms, or in some cases groups of atoms within a molecule. Because it has no

150

internal changes, an ultimate particle can always be regarded as a rigid mass. Its only alterable distinguishing characteristics which could possibly be detected, if some experimental procedure could do so, are its position and its motion. As a result, the fundamental properties of this particle, which cannot be calculated or derived from any others, consist only of its mass and shape plus the vectors or coordinates needed to describe its position and motion. It is convenient to combine the mass and motion characteristics and represent them as a momentum property. These fundamental characteristics, mass, position, and momentum, are called "microstate" properties and as a group they give a complete description of the actual behavior of an ultimate particle. In reality, the molecules are not the inert rigid masses. The forces of attraction and repulsion which we ascribe to them are in reality the consequence of variations in the quantum states of a deformable electron cloud which fills practically all the space occupied by a molecule so that when we represent it as a rigid mass we are constructing a model which allows us to apply classical mechanics to relate its energy changes to changes in its microstate properties. For example, an effective model for a complex molecule is to regard it as a group of rigid spheres of various size and mass held together by flexible springs. The only justification for this model is that calculations of its energy, when properly averaged, give good agreement with values of energy per molecule obtained from experimental measurements using bulk quantities of the substance. Constructing models is important in all aspects of thermodynamics, not only for individual molecules, but also in describing the behavior of bulk matter. Values, which can be calculated from the microstate properties of an individual particle or of a cluster containing only a few particles, represent another group of non-thermodynamic properties. We will refer to these derived values as "molecular" properties. Examples are the translational, vibrational, or rotational energies of an individual molecule, and also the calculated potential energy at various separation distances in a pair of molecules or between other small groups of near neighbors. In some cases we wish to calculate special functions of the potential energy within a group composed of a few neighbors. An important feature of all of these combinations of fundamental microstate properties is that they can produce the same value of a calculated molecular property. For example, assigning values to the microstate properties of a molecule determines its energy but specifying the energy of a molecule does not specify any one particular set of values for its microstate properties. Whereas the non thermodynamic properties pertain to a single or to only a few ultimate particles, the characteristics of matter which are called thermodynamic properties or macrostates are those which result from the collective behavior of a very large number of its ultimate particles. Instead of only one or a few particles, this number is typically on the order of Avogadro's number. In a manner analogous to the way in which molecular properties can be calculated from the fundamental microstate properties of an individual or small group of particles, the various thermodynamic properties likewise depend upon the vastly greater

151

number of all the microstate properties of the very large group. Furthermore, an even larger number of different sets of microstate properties can produce the same overall thermodynamic property value. In contrast to non-thermodynamic properties, thermodynamic properties can always be measured experimentally or calculated from such measurements. 7.3 BOHR-SOMMERFIELD RULE Atoms can exist in a number of energy states. In case of gas like hydrogen, when atom is heated, the electrons jump to higher orbits and when cooled, the electrons come down to lower orbits. Each orbit represents a certain energy value of electron. Further, all the orbits are not available for electrons. Electrons are allowed with those orbits which satisfy the action integral

.p dq nh=∫ (7.1)

where p is the momentum is the coordinate of the electron ( q .xp dx or .p dθ θ )

n is an integer ( 1 ) , 2,3,..........Integrating for one cycle of close path

. .2 .I n hϖ π = (7.2)

2. .2 .vmv n h

rπ = (7.3)

or,

.2nhmvr Iϖπ

= = (7.4)

This is known as Bohr-Sommerfield rule. It states that the only possible circular orbit of the electrons are those in which the angular momentum ( .Iϖ ) of the

electron is a multiple of 2hπ

.

7.4 WAVE PARTICLE DUALITY – dE BROGLIE EQUATION hpλ

=

de Broglie equation relates the momentum of the photon ( or electron) with its wave length, representing the wave-particle duality. The total energy ε and mass of a moving particle are related by Einstein’s famous equation, m

2mcε = (7.5) where is the velocity of light ( m/s). For a photon ( photon does not have mass but has momentum) of electromagnetic radiation, Planck’s equation gives,

c 83 10×

152

hε ν= (7.6)

where is the Planck’s constant and h ν is the frequency of the wave. Now momentum p of the photon is

2 / h hp mc mc cc cε ν

λ= = = = = (7.7)

When this equation is applied to a heavy particle, the wavelength is so small that the modification made on rectilinear motion by the wave can be neglected, thus reverting back to classical mechanics. 7.5 HEISENBERG’S UNCERTAINITY PRINCIPLE (The quantitative limit on the product of the uncertainities in position and momentum of a particle.) Suppose we want to determine the position and momentum of a particle, say an electron. The position of the electron can be determined by illuminating it with a light source of wavelength λ and observing through a microscope. The uncertainity x∆ of the position in x-direction is known to be a function of λ and the aperature of the lens, given by the angle θ , or

sinx λ

θ∆ = (7.8)

The uncertainiy x∆ can be made small by selecting radiation of short wave length, such as X-rays and γ − rays. But the photon of short wave length or high frequency striking the electron yields extensive scattering. The change of momentum of the electron can be found from the angle of scattering of the photon. The angle must be within the visible range of the lens. The uncertainity of x-momentum is given by

sinxhp θλ

∆ = (7.9)

where is the Planck’s constant. The product of uncertainities in position and momentum is

h

. xx p h∆ ∆ = (7.10)

This is known as Heisenberge’s uncertainity principle.

153

Similarly,

. yy p h∆ ∆ = (7.11)and

. zz p h∆ ∆ = (7.12)

Attempt to improve the accuracy or to decrease the uncertainity for determination of momentum or position results in a corresponding decrease in the accuracy of the conjugate variable. 7.6 PHASE SPACE If the position and momentum of each atom of a gas in an enclosure is to be determined, we require six quantities, e.g., , , , ,x yx y z p p and zp . A six dimensional hyperspace with three location and three momentum coordinates is known as a phase space. Each particle in space can be represented as a point in a phase space and it is termed as phase point. Let us subdivide the phase space into small elements of volume, called cells, and volume of one cell, , is: H

. . . . .x yH dx dy dz dp dp dp= z (7.13)

This volume is small compared to the dimensions of the system, but large enough to contain many atoms. Every atom of the gas must be in a cell. The cells are numbered 1, 2, 3, ….., I, … and the number of particles in the cells are

with . The basic problem of statistical mechanics is to determine how the particles distribute themselves in the cells of phase space.

1 2 3 ,, , ,......., ...........iN N N N 1iN >

According to the Heisenberg’s uncertainity principle, . xx p h∆ ∆ = , , . The total uncertainity in locating a particle in phase space is

. yy p h∆ ∆ =

. zz p h∆ ∆ =

3. . . . .x y zx y z p p p h∆ ∆ ∆ ∆ ∆ ∆ = (7.14)

The particle lies somewhere within an element of phase space of volume . This is known as a compartment. The number of compartment per cell is

3h

3

Hgh

= (7.15)

where and has the dimension of volume in phase space. 1g >> 3h

154

( ) ( )3 33h length momentum= × (7.16)

A quantum state corresponds to a volume in space, and an energy level corresponds to a cell of volume , so is nothing but the degeneracy of the energy level.

3hH g

7.7 ENERGY STATES AND ENERGY LEVELS As discussed in the previous section, classical mechanics does not apply at microscopic scale. For such analysis, quantum mechanics (wave mechanics) is applied. In quantum theory, to each energy level there correspond one or more quantum states described by a wave function Ψ . For so-called stationary states, will be a function dependent on the position coordinates and time. When there are several quantum states that have the same energy, the states are said to be degenerate. The quantum state associated with the lowest energy level is called the ground state of the system; those that correspond to higher energies are called excited states. The energy levels can be thought of as a set of shelves at different heights, while quantum states correspond to a set of boxes on each shelf. For each energy level

Ψ

iε the number of quantum states is given by the

degeneracy . igGeneral method of attacking a problem of quantum mechanics/wave mechanics is to set up and solve an equation known as Schrodinger’s equation. In many problems, this equation is exactly analogous to the wave equation describing the propagation of transverse in a stretched string, fixed at both ends. As is well known, the string can vibrate in a steady state in any one of a number of stationary waves (Fig. 7.1).

N NA

N NA A

N

N NA A

N

AN

L Fig. 7.1 Stationary waves in a stretched string fixed at both end

155

There may be a node N at each end and an anti node A at the center, or there may be a node at the center as well as at the ends, with anti nodes midway between the nodes, and so on. There is always an integral number of antinodes in the steady state modes of vibration (Fig. 7.1). Distance between nodes (or anti nodes) is one half of the wave length, so if is the length of the string, the wave length

Lλ of the possible stationary waves** are

1 2Lλ = (7.17)

21 22

Lλ = (7.18)

31 23

Lλ = (7.19)

1 2jj

Ln

λ = (7.20)

where is an integer equal to the number of anti nodes and can have some one of the values

jn

1,2,3,......jn =

** Erwin Schrodinger, Austrian Physicist (1887-1961), won Nobel Prize in physics in 1933 shared with Dirac. *** A stationary wave is equivalent to two traveling waves propagating in opposite directions, the waves being reflected and re-reflected at the ends of the string. This is analogous to the motion of a particle moving freely back and forth along a straight line and making elastic collisions at two points separated by the distance

. L

156

According to quantum mechanics, a stationary Schrodinger wave is in fact completely equivalent to such a particle, and the wavelength λ of the stationary wave is related to the momentum p of the particle through the relation

hpλ

= (7.21)

where = Planck’s constant. The momentum of the particle for different nodes/antinodes are

346.6262 10 h x J−= s−

1 2 3; 2. ; 3. ;........; 2 2 2 nh h hp p p p nL L L 2

hL

∴ = = = = (7.22)

It can be observed that the momentum of the particle is not continuous rather it is discrete. We can write down the momentum of a particle in the form

2j jhp nL

= (7.23)

If the particle is free to move in any direction within a cubical box of side length whose sides are parallel to the

L,x y and axes of a rectangular co-ordinate

system, the z

,x y and components of its momentum are permitted to have only the values

z

2

2

2

x x

y y

Z z

hp nL

hP nLhP nL

=

=

=

(7.24)

where and are integers called the quantum numbers, each of which can have some one of the values 1, 2, 3, etc. Each set of quantum numbers therefore corresponds to a certain direction of momentum.

,x yn n zn

If jp is the resultant momentum corresponding to some set of quantum numbers

, , &x y zn n n

( )2

2 2 2 2 2 2 224j x y z x y z

hp p p p n n nL

= + + = + + (7.25)

157

If we let , ( )2 2 2x y zn n n n+ + = 2

j

2

2 224j j

hp nL

= (7.26)

7.8. WAVE EQUATION Suppose, the motion of the string discussed in Section 7.7, is restricted to the x-y plane only (Fig. 7.2). Considering motion of the element PQ, net force, in -direction is

yF y

y

x x + dx x

ds

Q

p

T

Tθ 1θ

θ 2

Fig. 7.2 Forces acting on an elemental length of a string

2 1sin sinyF T Tθ θ= − (7.27)

where is the tension in the string. Further, T θ being small

2 2sin tanx dx

yx

θ θ+

∂⎡ ⎤= = ⎢ ⎥∂⎣ ⎦

(7.28)

158

and,

1 1sin tanx

yx

θ θ ∂⎡ ⎤= = ⎢ ⎥∂⎣ ⎦

(7.29)

From Eqs. (7.27-7.29), we get

yx dx x

y yF T Tx x+

∂ ∂⎡ ⎤ ⎡= − ⎤⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦

x x

y yT T dx Tx

yx x x x∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤ ⎡= + − ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x

yT dx x∂ ∂ x⎡ ⎤= ⎢ ⎥∂ ∂⎣ ⎦

(7.30)

By Newton’s law of motion,

2

2x

y yT dx mdxx x t∂ ∂ ∂⎡ ⎤⎡ ⎤ = ⎢ ⎥⎢ ⎥∂ ∂ ∂⎣ ⎦ ⎣ ⎦

(7.31)

where is the mass of the string per unit length. m

2 2

2 2

y m yx T t∂ ∂

=∂ ∂

2 2

2 2

1

p

y y2x V t

∂ ∂=

∂ ∂

(7.32)

where 1 Tp m= = phase velocity

amplitude of vibration at time t y = at distance x from fixed end.

Use method of separation of variable to solve

( ) ( )y f x g t= (7.33)

159

2 2

2 2

y fgx x∂ ∂

=∂ ∂

(7.34)

And,

2 2

2 2

y gft t

∂ ∂=

∂ ∂

(7.35)

substituting in Eq. (7.32), we get

2 2

2 2

1

p2

f gg ft v t

∂ ∂=

∂ ∂

(7.36)

or,

2 22

2 2 2

1 1 ( )p

f g sayf x gv t

α∂ ∂= = −

∂ ∂

(7.37)

where 2α is the separation constant. Since, each side of the Eq. (7.37) is a function of a single variable,

22

2 2

1

p

ggv t

α∂= −

∂

(7.38)

or, 2

2 22 0p

g v gt

α∂+ =

∂

(7.39)

The characteristic equation is

2 2 21 0pm vα+ = (7.40)

or, 1 pm i vα= ± (7.41)

So, the general solution is

1 2cos( ) sin( )p pg C v t C v tα α= + (7.42)

Let,

2pvα ω π= = (7.43)

160

where ω is the angular velocity and ν is the frequency.

∵ pv νλ= (7.44)

2pvα ω π= = (7.45)

2 2πν πανλ λ

= = (7.46)

Again from Eq. (7.37) ⇒

22

2

1 ff x

α∂= −

∂

(7.47)

22

2 0d f fdx

α∴ + = (7.48)

or,

2 2

2 2

4 0d f fdx

πλ

+ = (7.49)

The function of represents the amplitude of the standing wave at any position x for a vibration of associated wavelength. 7.8.1 SCHRODINGER WAVE EQUATION (Applied to Matter wave of de-Broglie) Using the symbol xψ for the x −direction amplitude or wave function f ,

2 2

2 2

4xx

d odxψ π ψ

λ+ =

(7.50)

From de -Broglie Equation,

x

h hp mv

λ = = (7.51)

161

2 2 2

2 2

4 ( ) 0x xx

d mvdx hψ π ψ∴ + =

(7.52)

Total Energy,

21( ) ( )

2x x x xKE PE mv xε φ= + = +

(7.53)

or,

2 2 ( )x xmv m xε φ= − (7.54)

where, x PEφ = Substituting Eq. (7.54) in Eq. (7.52), we get the 1 D− wave equation

[ ]2 2

2 2

8 0xx x

d mdx hψ π ε φ+ − =

(7.55)

In order to generalize the equation to 3 ,D− a wave function ψ is defined as

x y zψ ψ ψ ψ= (7.56)

and the molecular internal energy ( KE PE+ ) as

x y zε ε ε= + + ε 2 2 21 ( )2 x y zm v v v= + +

2 2 212 x y zp p pm⎡ ⎤= + +⎣ ⎦

(7.57)

and x y zφ φ φ φ= + + (7.58)

Time independent wave Equation becomes

22

2

8 ( )mhπψ ε φ ψ∇ + − = 0

(7.59)

162

7.8.2 APPLICATION OF SCHRODINGER WAVE EQUATI ON - PARTICLE IN A BOX Molecules have only translational K.E. For simplification, let us consider that the particles move only in x −direction. The schrodinger equation applied to a single particle is

[ ]2 2

2 2

8 0xx x x

d mdx hψ π ε φ ψ+ − =

(7.60)

Since, the particle must remain in the box, When

0, 0xx ψ= =

, 0xx a ψ= =

(7.61)

( )x amplitudeψ →

For an ideal gas , 0xφ = inside the box. Therefore, Eq.(7.60) becomes,

2 2

2 2

8 0xx x

d mdx hψ π ε ψ+ =

(7.62)

Characteristic equation is,

221 2

8 0x

mmhπ ε+ =

(7.63)

or,

1/ 22

1 2

8 xmm ih

π ε⎡ ⎤= ± ⎢ ⎥⎣ ⎦

(7.64)

The general solution to Eq.(7.62) is

1/ 2 1/ 22

1 22 2

8 8sin cosx xx

mC x C xh h

π ε π ε2mψ⎡ ⎤ ⎡⎧ ⎫ ⎧ ⎫= +⎢ ⎥ ⎢ ⎥⎨ ⎬ ⎨ ⎬⎩ ⎭ ⎩ ⎭⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎤

(7.65)

163

where, are constants 1 &C C2

where, 1 , 0xx oψ= =

2 0C∴ = 1/ 22

1 2

8sin xx

mC xh

π εψ⎡ ⎤⎧ ⎫∴ = ⎢ ⎥⎨ ⎬⎩ ⎭⎢ ⎥⎣ ⎦

(7.66)

Using , 0xx a ψ= = 1/ 22

1 2

80 sin xmC ah

π ε⎡ ⎤⎧ ⎫= ⎢ ⎥⎨ ⎬⎩ ⎭⎢ ⎥⎣ ⎦

(7.67)

1 0C ≠∵

1/ 22

2

8sin 0xmah

π ε⎡ ⎤⎧ ⎫ =⎢ ⎥⎨ ⎬⎩ ⎭⎢ ⎥⎣ ⎦

(7.68)

or, 1/ 22

2

8 xx

ma nh

π ε π⎡ ⎤ =⎢ ⎥⎣ ⎦

(7.69)

where , quantum numbers 1,2,3,.................xn = = For, result is trivial giving 0,xn = 0xψ = for all x which is not permissible.

22

28nx x

hnma

ε = (7.70)

Similarly, 2

228ny y

hnmb

ε = (7.71)

And 2

228nz z

hnmc

ε = (7.72)

For each energy level nxε , for a particular quantum number, ,there is a corresponding wave function

xnnxψ as given below

2 22

1 2 2

8sin8nx x

m hC x nh mπ

aψ

⎡ ⎤= ⎢ ⎥

⎢ ⎥⎣ ⎦

(7.73)

164

1 sin x

xC n xa

⎛ ⎞= ⎜ ⎟⎝ ⎠

The constant can be evaluated by normalization. 1C2

0

1a

m dxψ =∫ (7.74)

or,

2 21

0

sin 1a

x

xC n dxa

π⎛ ⎞ =⎜ ⎟⎝ ⎠

∫ (7.75)

or,

21

0

1 cos21

2

a x

xnaC d

π−x =∫

(7.76)

or, 1/ 22

11

212

C a Ca⎡ ⎤= ⇒ = ⎢ ⎥⎣ ⎦

(7.77)

or, 1/ 22 sinnx x

xna a

ψ π⎡ ⎤ ⎡= ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(7.78)

Similarly, 1/ 22 sinny y

ynb b

ψ π⎡ ⎤ ⎡= ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(7.79)

1/ 22 sinnz z

znc c

ψ π⎡ ⎤ ⎡= ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

(7.80)

1/ 2

, ,

8 sin sin sinx y zn n n x y z

x y zn n nabc a b c

ψ π π π⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤∴ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎥⎦

(7.81)

x y zε ε ε= + + ε22 2 2

2 2 28yx z

nh n nm a b c⎡ ⎤

= + +⎢ ⎥⎣ ⎦

(7.82)

If a b c , L= = =

22 2 2

28 x y z

hC n nmL

n⎡ ⎤∴ = + +⎣ ⎦ (7.83)

165

Specification of an integer for each is the specification of a quantum

state (or energy state) of a particle. All states characterized by the values of s such that constant will have same energy.

, &x y zn n n' 'n

2 2 2x y zn n n+ + =

=

2 2 2x y zn n n+ + Constant (7.84)

If, , energy levels and energy states can be presented as per Table 7.1.

2 2 2 66x y zn n n+ + =

Table 7.1. Various energy states and levels for 2 2 2 66x y zn n n+ + =

Energy Level

1 2 3 4 5 6 7 8 9 10 11 12

xn 8 1 1 7 1 4 7 4 1 5 5 4

yn 1 8 1 4 7 1 1 7 4 5 4 5

zn 1 1 8 1 4 7 4 1 7 4 5 5

It is clear that for the same energy (2

2 /36.68

hmV

ε = ) there are 12 possible

quantum states. Thus there are 12 degeneracy levels. 7.9 EQUILIBRIUM DISTRIBUTION In case of an ideal gas, there are many quantum states corresponding to the same energy level and degeneracy of each level is much larger than the number of particles which would be found at any one level at any one time. In a container of volume V with number of particles and an energy state U we may have the following distribution

N

1N particles in energy level 1ε with degeneracy 1g

2N particles in energy level 2ε with degeneracy 2g

3N particles in energy level 3ε with degeneracy 3g

….. …… …… …….

…. ….. ….. …….

iN particles in energy level iε with degeneracy ig….. …… …… …….

166

This is an example of a microstate of the gas. The number of ways Ω in which

this microstate may be achieved is given by the product of the terms !

iNi

i

gN

1 21 2

1 2

........! !

N Ng gN N

Ω = (7.85)

The quantity is called the thermodynamic probability (also called number of microstates and number of complexity) of the particular microstate. The larger

is, the greater the probability of finding the system of particles in this state.

Ω

Ω N 7.10 MODELS FOR STUDY OF STATISTICAL THERMODYNAMICS There are three different statistical models viz, Bose-Einstein, Fermi-Dirac and Maxwell-Boltzmann models to find the thermodynamic probability of microstates in a thermodynamic system. Distinguishing features of these models are presented in Table 7.2.

Table 7.2 Dishtinguishing features of various statistical models

B-E F-D M-B 1.Identical particles 2.Indistinguishable particles. 3.Wave function is symmetrical 4. State of the system is determined if the number of particles in a given quantum state is known 5. No restriction on the occupancy of the particles in a given 6. .Application to α particles, light quanta (Photons), deuterium, all atoms, all molecules composed of an even number of elementary particles like 12 ,

, 16 , etc

C14 N O 2CO6. Symmetric eigenfuctions -- Bosons

1.System of identical particles 2.Indistinguishable particles 3.Satisfy Pauli’s exclusion principle(Postulates that there is never more than one particle in a given quantum state ). 4. Applicable to all elementary particles electrons, protons, neutrons, positrons, anti-protons, neutrinos. ------ by all atoms or molecules having odd number of elementary particles.

5. Spin = 12

or odd integral

multiple of 12

6. Antisymmetric eigenfuction- Fermions

1. Distinguishable and Non- distinguishable particles

167

7.10.1 MAXWELL-BOLTZMANN STATISTICS A description of the statistical distribution of the energies of the molecules of a classical gas was first set forth by the Scottish physicist James Clerk Maxwell in 1859, on the basis of probabilistic arguments, and gave the distribution of velocities among the molecules of a gas. Maxwell's (1871) finding was generalized by a German physicist, Ludwig Boltzmann. Let us consider distinguishable particles to be distributed in various cells of phase space. Let, are the number of particles in cells

respectively. The number of ways of putting particle (out of ) in cell 1 =

N1 2 3, , ,........N N N

1,2,3,........ 1N N

1

NNc

Number of ways of putting from the remaining ( 2N 1N N− ) particles in cell

2=( )

1

2 1

!! !

N NN N N N

−− − 2

2

Number of ways of putting from the remaining ( 3N 1N N N− − ) particles in cell

3=( )

1 2

3 1 2

!! !

N N NN N N N N

− −− − − 3

, and so on.

Therefore, the total number of ways of arranging the particles in the cells are

i iW Wπ=

( ) ( ) ( )1 1 2

1 1 2 1 2 3 1 2 3

! !! . . .....! ! ! ! ! !

N N N N NNN N N N N N N N N N N N

− − −=

− − − − − −

1 2 3

!! !N

N N N=

!

(7.86)

! !i i

NWNπ

∴ = (7.87)

Let us take the simple case of two cells and b and six distinguishable particles. Possible macrostates and corresponding thermodynamic probabilities are given in Table-7.3.

a

168

Table 7.3 Distribution of 2 cells with 6 distinguishable particles

Macrostate 1 2 3 4 5 6 7

aN 6 5 4 3 2 1 0

bN 0 1 2 3 4 5 6

W =64 1 6 15 20 15 6 1 Number of microstates is estimated from W ( = number of ways of arranging the particles in the cells). There are 7 (seven) macrostates. If the particles are continually shifting around from cell to cell, so that one microstate after the other turns up with equal frequency, the first and the seventh macrostate will each be

observed 164

of the time, the second and sixth seventh macrostate will each be

observed 6 364 32

= of the time, the third and fifth each 1564

of the time and the

fourth macrostate will be observed 20 5

1664= of the time. The fourth macrostate

is having maximum probability (20 564 16

= ) with uniform distribution of particles in

the 2 cells. In an actual situation, however, for a particular macrostate, W will be very large compared to other macrostates. This macrostate will correspond to the most probable distribution of particles. This is referred to as thermodynamic equilibrium state. Other macrostates also do occur, but their occurances are much less frequent than this macrostate. The distribution of particles in cells of phase space means distribution at different energy levels. Now we consider how the particles occupy the different

quantum states (each quantum state represents a compartment in the phase

space) of an energy level (cell)

iNig

iε . Since, there are choices for the location

of each particle, the total number of ways in which dishtinguishable particles

could be distributed among quantum states of an energy level, would be:

igiN

ig........ iN

i i i ig g g g= (7.88)

Hence, the thermodynamic probability of the macrostate is the product of this factor over all energy levels and the factor for interchanges between the levels.

169

!!

iNi i

i i

NW gN

ππ

= (7.89)

or,

!!

iNi

i

i

gW NN

π= (7.90)

This is the thermodynamic probability of a molecular model based on Maxwell-Boltzmann statistics for distinguishable particles with no limit on the number of particles per quantum state. Three distinguishable particles and can occupy quantum states 3, and

in six different ways, as given below: ,a b c 8

12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 a b c a c b b a c b c a c a b c b a In case of indistinguishable particles, there will be only one way to occupy the quantum states. For the example above, number of ways of placing the

indistinguishable particles will be 6 13!= . Similarly from Eq. (7.90), for

indistinguishable particles,

N

!

iNi

i

i

gWN

π= (7.91)

This is the thermodynamic probability of Maxwell-Boltzmann statistics for indistingshable particles. 7.10.1.1 STIRLING APPROXIMATION The factorial can be written as x

! 1.2.3........x x= (7.92) or,

ln ! ln 2 ln3 ...... lnx = + + + x (7.93)

170

This is exactly equal to the area under the stepped curve (Fig. 7.3) between

and . 1x = x x=

ln 2ln 3ln 4ln 5ln 6

1 2 3 4 5 6

ln 2

ln 3

ln 3 ln 4ln 4 ln 5 ln 6

0x

y =ln x

Fig. 7.3 Stirling approximation This area is approximately equal to the area under the smooth curve when is large. Therefore,

x

1

ln ! ln x

x x= ∫ dx (7.94)

We know that

d = u v uv v du−∫ ∫ (7.95)

Putting and v x , lnu x= =1ln ln . . lnx dx x x x dx x xx

= − =∫ ∫ x−

+

x

(7.96)

1

ln ! ln ln 1x

x x dx x x x= = −∫ (7.97)

Since, is large, 1 can be neglected, So, x

ln ! lnx x x= − (7.98) Equation (7.98) is known as Stirling’s approximation.

171

7.10.1.2 MAXWELL-BOLTZMANN DISTRIBUTION FUNCTION Considering the probability of M-B statistics for indistinguishable particles

!

iNi

i

i

gWN

π= (7.99)

Taking logarithm of both sides of Eq. And using Stirling’s approximation,

[ ]ln ln lni i i iW N g N N N= −∑ i+ (7.100)

As particles in the cells of phase space shift around, they change from one energy level to another, and ’s will change. If the system is in thermodynamic

equilibrium with maximum W , a small change in W represented by iN

lnWδ will be zero. Since the value of is assumed to be constant, ig

1ln ln ln 0i i i i i i i

i

W g N N N N N NN

δ δ δ δ δ⎡ ⎤

= − − +⎢ ⎥⎣ ⎦

∑ = (7.101)

or,

ln 0ii

i

g NNδ =∑

(7.102)

where is the number of particles in the iiN th− cell ( or i th− energy level) in

thermodynamic equilibrium. However, iNδ ’s are not independent, but are subject to equation of constraints,

iN = ∑N =constant, 0iNδ =∑ (7.103)

iU iNε= ∑ =constant, 0i iNε δ =∑ (7.104)

Multiplying Eq. (7.103) by and Eq(7.104) by ln B− β− and adding to Eq (7.102),

ln ln 0ii i

i

q B NN

βε δ⎡ ⎤

− − =⎢ ⎥⎣ ⎦

∑ (7.105)

Since, iNδ ’s are now independent,

172

ln ln 0ii

i

q BN

βε− − = (7.106)

or,

[ ]1/ ii

i

N Beg

βε= (7.107)

Eq. (7.107) is known as M-B distribution function of particles among cells in phase space at equilibrium. 7.10.2 BOSE-EINSTEIN STATISTICS In this model, the particles are indistinguishable and the energy levels quantized. We have an energy level with a degeneracy and we must distribute within this

level indistinguishable particles, sometimes called bosons. Let this energy level be represented by the cell i in phase space, and let this cell be divided into compartments numbered 1 , each representing a quantum state. Let

the particles be temporarily numbered as shown in Fig. ( ). For a particular arrangement. If the numbers and letters are arranged in all possible sequences, each sequence will represent a microstate, provided that the sequence start with a number. The sequence can begin in ways, and in each

sequence the remaining

igiN

,2,3,....... ig

)

, , ,......., ia b c N

ig( 1i iN g+ − letters and numbers can be arranged in

any order. Therefore, the number of different sequences begin with a number is ( )1 !i i ig N g+ −

Since, both and are indistinguishable, it is necessary to divide the above number by the permutations of these subsets. Then the number of possible arrangements (or microstates) in the

ig iN

i th− energy level is

( )1 !! !

i i ii

i i

g g NW

g N+ −

= (7.108)

or,

( )( )

1 !1 ! !

i ii

i i

g NW

g N+ −

=−

(7.109)

173

The total number of microstates including all the cells, or the thermodynamic probability W is

(7.110)( )( )

1 !1 ! !

i ii i i

i i

g NW W

g Nπ π

+ −= =

− Taking the logarithm and using the Stirling approximation,

( ) ( ) ( ) ( ) ( )( )

ln 1 ln 1 1 1 ln 1 ]

1 lni i i i i i i i

i i i i

W g N g N g N g g

g N N N

= + − + − − + − − − −⎡ ⎤⎣ ⎦+ − − +

∑ (7.111)

7.10.3 FERMI-DIRAC STATISTICS Fermi-Dirac statistics is based on the Pauli’s exclusion principle. According to this principle, no two electrons having same spin can occupy the same energy level. If we allow not more than one particle per quantum state, then it is necessary that

and particles governed by this restriction are called Fermions. Of the quantum states in an energy level

ig N≥ i igiε , states are occupied and

states are empty. The problem thus consists of counting the number of ways in which quantum states can be divided into two groups, with occupied states in one group and the empty one in the other.

iN ( )i ig N−

ig

∴ Thermodynamic probability of energy level iε is

iW i

i i i

gN g N

=−

(7.112)

For all energy levels,

W = = i iWΠ i

ii i i

gN g N

Π−

(7.113)

174

7.11 ENTROPY AND PROBABILITY s = Entropy additive W = probability multiplicative At equilibrium: s, W maximum Let us consider two system of probabilities or disorder number and and entropies

1W 2W1s and 2s .

1s = 1( )f W (7.114)

2s = 2( )f W (7.115) For a single system of the two together

1 2s s s= + (7.116)

1 2.W W W= (7.117)∴ Finally,

1 2 1( ) ( . ) 2s f W f W W s s= = = + (7.118) or,

1 2 1 2( . ) ( ) ( )f W W f W f W= + (7.119)

Differentiating w.r.t. W W , 1 2 and 1 2 2 1( ) ( )f WW W f W′ ′= (7.120)

1 2 1 2( ) ( )f WW W f W′ ′= (7.121)

Dividing Eq. (7.120) by Eq. (7.121), we get

2 1

1 2

( )( )

W f WW f W

′=

′

(7.122)

or, 1 1 2 2( ) ( ) ( )W f W W f W K say′ ′= = (7.123)

∵each side is a function of one variables, hence

( )Wf W K′ = (7.124)

175

or,

( )df WW KdW

= (7.125)

or,

( ( )) dWd f W KW

= (7.126)

or,

( ) lnf W K W C= + (7.127)where C is a constant Hence,

lnS K W C= + (7.128)when 1, 0 0W s C= = ⇒ =

lnS K W∴ = (7.129) Greater the disorder or probability, greater is the entropy.

176