6.
You have a system of non-interacting particles that can exist in three possible states, with
energy 0, E, 2E (E > 0). At a temperature T = E/kB (where kB = Boltzmann’s constant)
the energy states are occupied in approximately the ratio
a. 1 : 1 : 1
b. 1 : 0.5 : 0.25
c. 1 : 2.72 : 7.39
d. 1 : 0.37 : 0.14
Boltzmann Distribution Basics
Non-interacting particles follow the Boltzmann distribution, where occupation probability for state i is proportional to e-Ei / kBT. Here, energies are E0 = 0, E1 = E, E2 = 2E, and T = E / kB, so βE = 1 where β = 1 / kBT.
Normalized occupations are n0 : n1 : n2 = e0 : e-1 : e-2 = 1 : 0.37 : 0.14.
Option Analysis
- a. 1 : 1 : 1 assumes equal populations, valid only at infinite temperature (
kBT ≫ E) where Boltzmann factors approach 1. - b. 1 : 0.5 : 0.25 implies linear energy dependence, not exponential as required by Boltzmann statistics.
- c. 1 : 2.72 : 7.39 reverses the trend (higher energy, more occupation), impossible since
e-E/kBT < 1forE > 0. - d. 1 : 0.37 : 0.14 matches
e-1 ≈ 0.37ande-2 ≈ 0.14, confirmed by partition functionZ = 1 + e-1 + e-2 ≈ 1.50.
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Introduction to Non-Interacting Particles Three Energy States Occupation Ratio
In statistical mechanics for CSIR NET Life Sciences, understanding the non-interacting particles three energy states occupation ratio at T = E/kB is crucial for Boltzmann distribution applications. This MCQ tests equilibrium populations across energies 0, E, 2E (E > 0). The correct ratio follows exponential decay, vital for thermodynamics and biophysical chemistry.
Detailed Boltzmann Calculation
Particles occupy states per ni ∝ gie-Ei / kBT, assuming non-degenerate levels (gi = 1). At T = E/kB:
n0 ∝ 1n1 ∝ e-1 ≈ 0.3679n2 ∝ e-2 ≈ 0.1353
Ratio: 1 : 0.37 : 0.14. Partition function Z ≈ 1.503 confirms normalization.
Why Other Options Fail in CSIR NET Context
| Option | Ratio | Reason for Incorrectness |
|---|---|---|
| a | 1:1:1 | High-T limit only (kBT >> 2E) |
| b | 1:0.5:0.25 | Linear, ignores exponential Boltzmann factor |
| c | 1:2.72:7.39 | Inverted (e+E/kBT), violates energy minimization |
| d | 1:0.37:0.14 | Matches e-E/kBT exactly |
CSIR NET Exam Tips
Practice similar questions on canonical ensemble and partition functions. For non-interacting particles, always compute ratios relative to ground state. This concept links to protein folding energetics and enzyme kinetics in life sciences.
