AIIMS 2007 Physics Free Electron Theory Assertion Reason Question

To analyze the question, we need to evaluate both the assertion and the reason, and determine their validity in the context of solid-state physics, specifically regarding the behavior of free electrons in metals.

Assertion: "In a metal, all the free electrons have the same energy."

This assertion is false. In a metallic solid, free electrons are not uniformly distributed in terms of energy. According to the free electron model, electrons occupy a range of energy levels, which can be described by the Fermi-Dirac statistics. At absolute zero temperature, electrons fill the available energy states up to a certain maximum energy known as the Fermi energy ($E_F$). However, these electrons have different energies due to thermal excitations and the quantization of energy levels, meaning not all electrons can have the same energy.

Reason: "Electrons do not obey Pauli’s exclusion principle."

This reason is also false. The Pauli exclusion principle states that no two fermions (particles with half-integer spin, such as electrons) can occupy the same quantum state simultaneously. In metals, free electrons are indeed subject to this principle. As a result, each energy level can only be occupied by two electrons (one with spin up and one with spin down). This principle is crucial in explaining why electrons fill up energy levels up to the Fermi energy, rather than all being at the same energy level.

Summary of Evaluation

• Assertion is false: Not all free electrons in a metal have the same energy level.
• Reason is false: Electrons do obey the Pauli exclusion principle.

Conclusion

Given that both the assertion and the reason are false, the correct answer is D) Both Assertion and Reason are false.

Clarification of Other Options

• Option A: Both Assertion and Reason are true and Reason is the correct explanation of Assertion.

  • Incorrect: Both the assertion and the reason are false.

• Option B: Both Assertion and Reason are true but Reason is not the correct explanation of Assertion.

  • Incorrect: Both statements are false, not true.

• Option C: Assertion is true but Reason is false.

  • Incorrect: The assertion is false; thus, this option cannot be correct.

Additional Concepts

To further understand the distribution of energy levels in metals, we can look at the Fermi-Dirac distribution function given by:

$f(E) = \frac{1}{e^{(E - E_F)/kT} + 1}$

where:

• $E$ is the energy level,
• $E_F$ is the Fermi energy,
• $k$ is the Boltzmann constant,
• $T$ is the absolute temperature.

At temperatures above absolute zero, this function describes the occupancy of energy states by electrons, indicating that many different energy levels are occupied rather than all electrons having the same energy.

In conclusion, the correct answer to the assertion-reason question is D) Both Assertion and Reason are false.