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Five identifiable particles are distributed in three nondegenerate levels with energies 0, E and 2E. The most probable distributions for total energy, 3E corresponds to what combination of options given below?
Select one or more:
As the levels are nondegenerate, there is only one state for each energy.
Let the number of particles occupying the 3 energy states be N_{1}, N_{2, }and N_{3 }respectively
where
The particles are identifiable, the number of ways of choosing the particles is
The energy of system is
Now, the most probable distribution is the one in which W is a maximum, subject to constraint given by equation (1)
Thus if
and
If N_{2} = 3, N_{3} = 0
and N_{1} = 5 – (3 + 0) = 2
No other distribution are possible
For
So must probable distribution is
The correct answers are:
In relation to statistical mechanics choose the correct statements.
Select one or more:
For a given kind, all particles are exactly the same and are thus indistinguishable.
For each degree of freedom, there is one position coordinate and one canonical momentum,
∴ for n degrees, 2n will be the dimension.
Photons follow Bose Einstein statistics.
The correct answers are: All particles of given kind are treated as mutually indistinguishable., The phase space for n degrees of freedom will have 2n dimensions and it unit cell volume in n^{n}.
Degeneracy in perfect gases is the departure of their properties from those of ordinary gas and becomes significant at :
Select one or more:
At very low temperature/high energy densities, the concentration of particles with similar momentum (low) would violate the Pauli’s exclusion principle.
The correct answers are: Very low temperature, High density
Bosons are particles that carry force, there is an attractive force between bosons (except for photons carrier of electromagnetic force, they can be repulsive also.)
Fermions are particles that make up the matter, they due to Pauli’s exclusion principle exert repulsive force on each other.
It is due to these respective attractions and repulsions that at given temperature.
Bose gas has lower pressure as compared to both Fermi gas and classical gas and Fermi gas has higher pressure.
The correct answers are: A fermion gas will exert greatest pressure, A boson gas will exert the least pressure.
In a grand canonical ensemble, a system A fixed volume is contact with a large reservoir B. Then
Select one or more:
A grand canonical ensemble is such that the system can exchange energy and particles with a reservoir so that various possible states of the system can differ in both then total energy and total number of particles.
The correct answers are: A can exchange energy with B, A can exchange particles with B
Which of the following statements are true :
Select one or more:
For system following FD statistics, there is a limit on the number of particles occupying a certain energy level. This comes due to the fact that they follow Pauli’s exclusion principle, hence the particles, are distinguishable. Pauli’s exclusion principle on the other hand is not followed by system with BE statistics and MB statistics, hence the particles here are considered to be indistinguishable.
The correct answers are: particles following FD statistics are distinguishable, Particles following BE statistics are indistinguishable, Particles following Maxwell Boltzmann statistics are indistinguishable.
In a canonical ensemble, system can exchange only energy with environment.
Number of particles remains constant, also volume of the system remains constant. Therefore, temperature and density of system are constant.
The correct answers are: temperature is constant, density is constant
Liouville’s theorem is all about the conservation of density of points ρ is phase space.
i.e.
⇒ ρ is conserved.
Also we know that in microcanonical ensemble, the system is isolated in the sense that it cannot exchange energy or particles with its environment.
Hence Liouville’s theorem holds only for microcanical ensemble.
The correct answer is: micro canonical ensemble
The ensemble average of a physical quantity is equal to its time average. This is true for :
Select one or more:
The ensemble average of any physical quantity is equal to its time average and holds for all the ensembles whether microcanonical, canonical or grand canonical ensemble.
The correct answers are: a microcanonical ensemble, canonical ensemble, grand canonical ensemble
Electronic contribution of specific heat varies linearly with the temperature for :
Select one or more:
Electronic contribution of specific heat is completely independent of the dimension of the system and is always linear with the temperature.
The correct answers are: 3dimensional electron gas, 2dimensional electron gas, 1dimensional electron gas
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