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Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

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Q.1. Compare the rate of fall of temperature of two solid sphere of the same material and similar surfaces, where the radius of one surface is four times and the Kelvin temperature of the large sphere is twice that of the small one. (assume that the temperature of the sphere is so high that absorption from the surrounding may be ignored).

P = 4πR2σT4

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

P = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics where s is specific heat 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 4


Q.2. Three identical Boson particles are to be distributed in three Quantum states. (a) Make a detailed pictorial representation showing a stage by a horizontal line and a particle by a dot for each possible way of distribution. (b) How many possible arrangements are there? (c) Use the method of permuting vertical lines and dots to find the number of possible arrangements. (d) What is the probability that all three particles occupy the same

quantum state?

(a) 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics all three together 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics two in one state and one in other

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics all in different states 

(b) 10

c) There are three vertical lines for three quantum states, and three dots for the particles. The first vertical line remains undisturbed. The rest five symbols can be permuted in 5! ways. The two vertical lines can be permuted 2! ways and the three dots can be permuted 3! ways. These permutations are not be counted separately. So the number of possible arrangements is

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 10

Look at the solution of part (a), there are 3 ways. out of a total of 10 , in which all the particles can occupy the same state. Assuming all arrangements to be equally probable, the probability of all three occupying the same state is 3 / 10 .


Q.3. Find the wavelength λm corresponding to the maximum energy density for blackbody radiation at (a) 300K (b) 2000K and (c) 5000K .

From Wien's displacements law,

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

or λm = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(a) T = 300K . So, λm = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 9670nm

(b) T = 2000K . So, λm = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 1450nm

(c) T = 5000K . So, λm = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 580nm


Q.4. Estimate the number photons with wavelength in the range 400 to 405nm in a cavity of volume 100cc at (a) 300K , (b) 1000K (c) 3000 K .

The energy of the radiation in the wavelength  λ to  λ +  dλ is given by

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

The energy of a photon with wavelength λ is ε = hc /λ . As dλ is small, the number of photons with wavelength λ to  λ +  dλ is

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

As you know, hc = 1240 eV nm and k = 8.62´10-5 eV / K

So N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(a) T = 300K, N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) T = 1000K, N =Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(c) T = 3000K, N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.5. Estimate the number of photons with wavelength in the range 800and 805nm in a cavity of volume 100 cc at (a) 300K , (b) 1000K (c) 3000K

N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(a) T = 300K, N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) T = 1000K, N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(c) T = 3000K, N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.6. Three identical Fermions particles are to be distributed in two energy levels 1 E and 2 E . Each of the two energy levels is 5 -fold degenerate. (a) What are the possible values of the total energy of three particles? (b) for each such possible total energy, find the number of ways in which the particles can be distributed. (c) what is the total number of ways in which the particles can be distributed?

(a) Since each of the energy level is 5 -fold degenerate, any number of particle, out of three given, can be put in any of the energy level. We show below the differential ways of distributing these three particles in the two energy levels and the total energy corresponding to each such distribution.

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) In case (i), the three particles are to be distributed in 5 quantum states. This can be done in 5C= 510 ways.

In case (ii) one particle is to be put in one of the 5 quantum states of energy E1 , and two particles are to be put in the 5 quantum states of energy of E2 . One particle can be put in 5 state in 5C1 = 5 ways and two particles in 5 state can be put in 5C2 = 10 ways. So the distribution in case (ii) cab be done in 5C1 x 5C1= 50 ways.

(c) The total number of ways in which particles can be arranged is 10 + 50 + 50 +10 = 120

You can also get it directly from the fact that we have distributed 3 Fermions in 10 quantum states and that can be done in 10C3 = 120 .


Q.7. The Fermi level for a system of identical Fermions is μ . Find the energy at which the distribution function f (E) is (a) 0.90 (b) 0.10

The distribution function is f(E) = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(a) let the energy be E1 where value of f (E ) is 0.90 . Then,

0.9 = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

or,Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

or, Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

or, Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) Let the energy be E2 when this probability is 0.10 , then,

0.1 = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics 

This gives, Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.8. Copper contains 8.5 x 1028 free electrons per cubic meter. Assuming free electron model, calculate the Fermi energy for these electrons.

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

= Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

= Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.9. Consider a one-dimensional harmonic oscillator of angular frequencyε . If 5 identical particles occupy the energy levels of this oscillator at zero temperature. Answer the following questions.

(a) If the particles are electrons what is ground state energy of system?

(b) If the particles are fermions of spin 3/2, what is ground state energy of system?

(c) If the particles are spin-less fermions, what is ground state energy of system?

(d) If the particles are bosons what is ground state energy of system?

(a) If particles are electrons then spin is 1/2 then ground state energy 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) If particles are fermions with spin is 3/2 then ground state energy 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(c) If the particles are spin-less fermions, then energy is 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(d)  If the particles are bosons E0= Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.10. Let A and B represent two types of non-interacting spin-1/2 particles. If 3 particles of type A and 4 particles of type B are in a simple harmonic oscillator potential characterized by ω, then what is ground state energy of the system.

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.11. Consider a system of 3 particle which can occupy any of the 4 available energy states with equal probability. Then Find the entropy of the system.

(a) If particles are fermions without spin

(b) If particle are fermions with spin 1/2

(c) Particles are Bosons

(d) If particle are classical distinguishable

(a)  n= 3, gi =  4

Number of ways that 3 fermions will adjust  in 4 available energy  isIdentical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics So entropy is S = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) In case of fermions with spin s = 1/2, ni = 3, so multiplicity is 2s +1= 2 then g= 4 x2=8

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Number of ways that 3 fermions will adjust  in 8 available energy  is Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 56

so entropy is S = kBInW = kBIn56

(c) In case of boson n=3, gi =  4

W = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 20

S = kBInW = kBIn20

(d) In case of distinguishable particle 

N = 3 , n=3, gi =  4

W = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics ⇒ Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = 43 = 64

S = kBInW = kBIn64


Q.12. If fermions are confine in Volume V at equilibrium temperature T = 0K .if U is internal energy of system prove that Bulk modulus Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics is given by Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics.

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

U = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

dU = TdS- PdV ∵ Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Hence Bulk modulus, B = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics, Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.13. If fermions of rest mass m are moving with relativistic speed such that pc = mc2 in volume V at equilibrium temperature T = 0K , then Find the average energy .

Fermions at temperature T = 0K Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

For three dimensional case g ( E ) ∝ E2 ⇒ g ( E ) = CE2 relativistic

(U) = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.14. (a) In case of photon find density of quantum state between v and dv in two

dimensional area A

(b) If number of photon is proportional to Tα then find value of α .

(c) If energy of photon is proportional to T β then find value of β

(a) Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

If  polarization is included  then g(v) dv = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(b) f(v) = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

N = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

= Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

(c) U = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

= Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.15. Calculate the probability that an allowed state is occupied if it is lied above Fermi level by kT .

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

It is given E - EF= kT

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics


Q.16. Calculate the Fermi wavelength and Fermi energy for 4.2x1021 electrons in box of volume 1cm3 .

From Fermi Energy: 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

On raising the power of both sides by 3 / 2 , we can write 

Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

Hence, Fermi wavelength λF = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

On substituting the given values, we get

λF = Identical Particle: Assignment Notes | Study Kinetic Theory & Thermodynamics - Physics

= 1.52 x 10-19 J

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