If q t and Q t,m are the molecular and molar translational partition functions of X 2, respectively, then (Q t,m)= (N is the Avogadro number)

GATE 2024 CY – Question 47

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 · Multiple choice

If $q_t$ and $Q_{t,m}$ are the molecular and molar translational partition functions of $\mathrm{X_2}$, respectively, then $\ln(Q_{t,m})=$ ($N$ is the Avogadro number)

$N\ln q_t-N\ln N$
$N\ln q_t-\ln N$
$N\ln q_t+N\ln N+N$
$N\ln q_t-N\ln N+N$

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Correct answer: (D) $N\ln q_t-N\ln N+N$

Explanation

For $N$ identical, indistinguishable, non-interacting molecules, the molar translational partition function is
$$Q_{t,m}=\frac{q_t^{\,N}}{N!}.$$

**Take the natural logarithm** and use Stirling's approximation, $\ln N!\approx N\ln N-N$:
$$\ln Q_{t,m}=N\ln q_t-\ln N!=N\ln q_t-N\ln N+N .$$

Answer **D**.