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GATE 2022 CY – Question 64

Physical Chemistry · Statistical Thermodynamics: Micro-canonical, Canonical and Grand canonical ensembles, Boltzmann distribution, Partition functions and thermodynamic properties. Statistical mechanics of non-interacting systems, ideal monoatomic, diatomic gases, translational, rotational, vibrational and electronic partition functions. · 2 marks · Numerical answer

Each molecule has two nondegenerate levels separated by 5000 cm$^{-1}$. Ground-state population is ten times upper-state population. The measurement temperature is ______ K (nearest integer). Boltzmann constant is 0.695 cm$^{-1}$ K$^{-1}$.

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Correct answer: 3100 to 3150

Explanation

For two nondegenerate levels, the Boltzmann population ratio is Nground/Nexcited=exp(ΔE/kBT). The given ratio is 10, so taking natural logarithms gives ΔE/(kBT)=ln(10). Expressing both energy quantities as wavenumbers uses kB/(hc)=0.695 cm⁻¹ K⁻¹.

Therefore T=5000/[0.695 ln(10)]=3124.42 K, giving **3124 K**. No statistical-weight correction is needed because both levels are nondegenerate.