GATE 2025 CY – Question 20
The translational, vibrational, and rotational molecular partition functions for a system containing ideal diatomic gas molecules in the canonical ensemble $(N,V,T)$ are written as $q_{trans},q_{vib}$, and $q_{rot}$, respectively. The option that correctly defines their thermodynamic variable(s) dependency is
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Correct answer: (B) $q_{trans}(T,V),q_{vib}(T),q_{rot}(T)$
Explanation
The molecular partition function is the product of the contributions from each kind of motion: $q=q_{trans}\,q_{vib}\,q_{rot}\,q_{elec}$.
- **Translation:**
$$q_{trans}=V\left(\frac{2\pi mk_BT}{h^2}\right)^{3/2},$$
which depends on both $T$ and $V$ (the energy levels of a particle in a box depend on the box size).
- **Vibration:** $q_{vib}=\dfrac{1}{1-e^{-h\nu/k_BT}}$ (or with the zero-point factor). The vibrational frequency does not depend on the volume for an ideal gas, so $q_{vib}$ depends on $T$ only.
- **Rotation:** $q_{rot}=\dfrac{T}{\sigma\Theta_{rot}}$ (high-temperature form). The rotational constant depends only on the bond length, so $q_{rot}$ depends on $T$ only.
So $q_{trans}(T,V)$, $q_{vib}(T)$, $q_{rot}(T)$: option **B**.