At higher temperature, which one of the following statement are false?a)Fermi-Dirac statistics approaches Maxwell-Boltzmann statistics and Bose- Einstein statistics approaches to Fermi-Dirac statistics.b)Bose-Einstein approaches Maxwell-Boltzmann statistics and Fermi-Diracstatistics approaches Bose-Einstein statistics.c)Both Fermi-Dirac and Bose-Einstein statistics approach Maxwell-Boltzmannstatistics.d)None of these.Correct answer is option 'A,B,D'. Can you explain this answer?

Question

Answer

∴

for M.B

for B.E

for F.B and

At higher temperature

[For higher T, terms

and higher terms can be neglected.]

[Since, eα vary at higher temperature ]

[factor T is neglected Since

]
∴

Hence,

Thus both Fermi-Dirac and Bose-Einstein statistics approach Maxwell

Answer

Introduction:
The given question is related to statistical mechanics and the behavior of particles at higher temperatures. It asks to identify the false statement among the given options regarding the approach of different statistics at higher temperatures.

Explanation:
At higher temperatures, the behavior of particles is influenced by their statistics, which can be described by Fermi-Dirac, Bose-Einstein, or Maxwell-Boltzmann statistics. Let's analyze each statement to identify the false one:

Statement a:
Fermi-Dirac statistics approaches Maxwell-Boltzmann statistics and Bose-Einstein statistics approaches Fermi-Dirac statistics.

This statement is false. At higher temperatures, Fermi-Dirac statistics tends to approach Maxwell-Boltzmann statistics, not the other way around. Fermi-Dirac statistics describes the behavior of particles with half-integer spin, such as electrons, which obey the Pauli exclusion principle. As the temperature increases, the energy levels become more populated, and the effects of the Pauli exclusion principle become less significant. In the high-temperature limit, Fermi-Dirac statistics approaches Maxwell-Boltzmann statistics.

Similarly, Bose-Einstein statistics approaches Fermi-Dirac statistics at lower temperatures, not at higher temperatures.

Statement b:
Bose-Einstein statistics approaches Maxwell-Boltzmann statistics and Fermi-Dirac statistics approaches Bose-Einstein statistics.

This statement is also false. Bose-Einstein statistics describes the behavior of particles with integer spin, such as photons or bosons. At higher temperatures, Bose-Einstein statistics does not approach Maxwell-Boltzmann statistics. Instead, it remains distinct, as bosons can occupy the same energy levels without any restriction. In the high-temperature limit, Bose-Einstein statistics does not change its nature.

Similarly, Fermi-Dirac statistics does not approach Bose-Einstein statistics at any temperature range.

Statement c:
Both Fermi-Dirac and Bose-Einstein statistics approach Maxwell-Boltzmann statistics.

This statement is true. At higher temperatures, both Fermi-Dirac and Bose-Einstein statistics tend to approach Maxwell-Boltzmann statistics. Maxwell-Boltzmann statistics describes the behavior of classical particles without any restrictions on their occupancy of energy levels. As the temperature increases, the effects of quantum statistics become less significant, and both Fermi-Dirac and Bose-Einstein statistics converge towards Maxwell-Boltzmann statistics.

Statement d:
None of these.

This statement is false because the correct answer is option 'A,B,D'.

Conclusion:
In conclusion, at higher temperatures, the false statements among the given options are A, B, and D. Fermi-Dirac statistics does not approach Maxwell-Boltzmann statistics, Bose-Einstein statistics does not approach Maxwell-Boltzmann statistics, and the correct answer is not 'None of these'.

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