QUANTUM STATISTICS

 

There are three types of statistics —

 

i) Maxwell – Boltzmann statistics or MB statistics (classical statistics)

ii) Bose – Einstein statistics or BE statistics (quantum statistics)

it) Fermi-Dirac statistics on FD statistics (quantum Statistics) 

 

Maxwell – Boltzmann Statistics

 Basic postulates of MB-statistics : 

 i) The particles are distinguishable and identifiable.

ii) The particles of the system are spinless.

iii) There is no restriction imposed on the number of particles having same energy (i.e. no priory restriction).

iv) The particles do not obey Pauli’s exclusion principle.

v) Heisenberg’s uncertainty principle is not applicable for the particles .

 

 Further, if the system is isolated

      a) Total number of particles is constant i.e.

N = ΣNi = Constant.

𝛿N = Σ𝛿Ni = 0 —————– a

            Ni being the no of particles in ith energy stage.

     b) Total energy of ine system is constant,if the particles are non interacting.

     i.e. E = ΣNiEi = constant. 

𝛿E = ΣEi𝛿Ni = 0 —————– b

 

Note : The particles which obey MB statistics, called Maxwellian.

          Example : gas molecules.

 

Derivation of MB – distribution function

 

To determine MB distribution function we divide the problem into two parts –

 i) Calculate G

ii) Calculate G’

Thermodynamic probability  w = GG’

 

Derivation of MB distribution function

 

(a) Thermodynamic probability

 

Let N be the total number of distinguishable particles in the assembly ;  N1 , N2 , N3 , ……. Nn be the number of particles with energies E1 , E2 , E3 ,…….. En respectively and for the sake of generality gi be the no. of Q states for energy level Ei To determine the total number of ways G in which the total no of particle could be distributed among the quantum states.

i) The number of ways in which the group of particles N1 , N2 , N3 , ——- Ni could be chosen from N particles is, from the rule of permutation,

ii) Since there is no restriction on the number of particles with same energy, each of Ni particles can be distributed among gi sub shells (states) in gi ways.

Thus total number of ways of distributing Ni particles in gi  sub shells will be , 

 

Considering all groups , Hence total number of ways of distributing all the N particle in different states ; i.e. Thermodynamic probability

w = GG’

(b) Most probable distribution function

Taking logarithm on the both sides of equation 3 ,

According to Stirling’s approximation ;

In (N!) = NInN – N 

 For most probable distribution ; entropy of the system s = klnw must be maximum i.e. 

𝛿s = 0

According to Lagrange’s method of undetermined multiplier , multiplying equation (a) by -α and equation (b) by -β and adding with equation 5 , we get , 

 Since 𝛿Ni is independent, and above equation holds for any value of i , thus

 The ratio of the number of particle Ni distributed in gi stales to the number of states gi, is called the distribution function .

So , MB distribution function ,

Evaluation of e^α :

Total number of particles ,

where ,

is called partition function or sum of states .

 

Evaluation of β :

From equation 4 ,

So , Entropy

 Introducing temperature T from thermodynamics ,

Again ,

MB distribution function :-

 

Calculation of G in details :

Let N be the total number of distinguishable particles in the assembly ; N1 , N2, N3 , ………. be the number of particles with energies E1 , E2 , E3 , ………….. respectively .

For the first assembly ;

number of ways choosing the 1st particle = N

number of ways choosing the 2nd particle = N – 1

and number of ways choosing the 3rd particle = N – 2

& number of ways choosing the N1 th particle = N – N1 +1

Thus , number of ways to forming the group of N1, particles out of N particles 

Since the assembly or groups indistinguishable , hence number of ways , to form the 1st assembly

 

Similarly

  and so on

Hence , number of ways in which the group of particles N1 , N2, N3 , ………. Ni could be chosen from N particles ,

Limitations of MB Statistics

Maxwell–Boltzmann statistics is valid only for the classical limit. 

1)  This statistics is applicable only to an isolated gas- molecular system in equilibrium when the mean potential energy due to mutual interaction between the molecules is negligible compared to their mean kinetic energy.

2) The expression for MB count does not lead to the correct expression for entropy of an ideal gas. It leads to the Gibbs paradox which can be resolved if the expression divided by N

3) When the MB – statistics is applied to “electron gas” a number of discrepancies arise between the theory and the observation.

 4) When the MB -statistics is applied to “photon gas” i.e. a batch of electromagnetic radiant energy, it predicts a continuously increasing number of photons per unit range of frequency, as frequency increases. 

The actual distribution however shown by Planck that shows a maximum , falling off asymptotically on either side.

All these difficulties with MB statistics have been satisfactory resolved by the quantum statistics.

5) If we put T = 0 in expression for entropy of an ideal gas, s becomes a negative quantity which is at variance with the 3rd law of thermodynamics, 

s → 0 at T → 0.

Read more –

STATISTICAL MECHANICS | Bose – Einstein Statistics | Thermodynamic probability | Fermi Dirac Statistics | Comparison among three distribution curve