14. A system consists of molecules that can be in one of two energy levels, E1 and E2>E1.
At a temperature T, what fraction of the molecules are in energy level E2?
a. E2/(E1+E2)
b. exp(-E2/kT)/exp(-(E1+E2)/kT)
c. exp(-(E2-E1)/kT)
d. exp(-E2/kT)/(exp(-E1/kT)+exp(-E2/kT))

The fraction of molecules in energy level E2 follows the Boltzmann distribution in a two-level system, crucial for CSIR NET Life Sciences statistical mechanics questions. At temperature T, lower energies dominate due to thermal equilibrium probabilities.

Correct Answer

The correct answer is option d: exp(-E2/kT) / [exp(-E1/kT) + exp(-E2/kT)]

Detailed Derivation

In statistical mechanics, molecules distribute across energy levels according to the Boltzmann distribution at thermal equilibrium. The probability p_i of occupying state i with energy ε_i is p_i = exp(-ε_i/kT) / Z, where Z is the partition function Z = Σ_j exp(-ε_j/kT) and k is Boltzmann’s constant.

For this two-level system, the states are E1 (ground) and E2 > E1 (excited), assuming non-degenerate levels. Thus, Z = exp(-E1/kT) + exp(-E2/kT). The fraction in E2 is the population ratio N2/N = p2 = exp(-E2/kT) / Z, matching option d exactly.

This form accounts for arbitrary E1, E2; often E1 = 0 simplifies it to exp(-(E2-E1)/kT) / [1 + exp(-(E2-E1)/kT)], but the general expression is d.

Option Analysis

• a. E2/(E1+E2): Ignores temperature and exponential probability; represents energy-weighted average, not population fraction.
• b. exp(-E2/kT)/exp(-(E1+E2)/kT): Simplifies to exp((E1)/kT), independent of E2 and unnormalized, yielding incorrect probabilities exceeding 1.
• c. exp(-(E2-E1)/kT): Boltzmann factor exp(-ΔE/kT) gives N2/N1 ratio, not absolute fraction N2/N.
• d. Correct: Normalized probability for excited state population.

Boltzmann Distribution Formula

Probability p2 = exp(-E2/kT) / [exp(-E1/kT) + exp(-E2/kT)] normalizes total probability to 1. As T rises, excited state fraction increases since exp(-ΔE/kT) approaches 1 for small ΔE = E2 – E1.

CSIR NET Exam Relevance

This MCQ tests partition function understanding in molecular energy distributions, appearing in physical chemistry and biophysics sections. Practice reinforces concepts like population ratios in spectroscopy and enzyme kinetics.