Question

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:

• A. \( q_{{trans}}(T, V), \quad q_{{vib}}(T, V), \quad q_{{rot}}(T, V) \)
• B. \( q_{{trans}}(T, V), \quad q_{{vib}}(T), \quad q_{{rot}}(T) \)
• C. \( q_{{trans}}(T), \quad q_{{vib}}(T, V), \quad q_{{rot}}(T) \)
• D. \( q_{{trans}}(T, V), \quad q_{{vib}}(T), \quad q_{{rot}}(T, V) \)

Hint:

Remember: Translational motion depends on both temperature and volume, but vibrational and rotational motions depend only on temperature for ideal gas models.

Answer 1

The Correct Option is B

Let us analyze the dependence of each molecular partition function:

• Translational partition function \( q_{\text{trans}} \): Depends on both temperature \( T \) and volume \( V \), since the number of available translational states increases with both parameters.
  \[ q_{\text{trans}} \propto V T^{3/2} \]
• Vibrational partition function \( q_{\text{vib}} \): Depends only on temperature \( T \), as it arises from quantized vibrational energy levels which are temperature-dependent but not affected by the system's volume.
  \[ q_{\text{vib}} = \frac{1}{1 - e^{-h\nu/kT}} \]
• Rotational partition function \( q_{\text{rot}} \): Also depends only on temperature \( T \), since the rotational energy levels are functions of temperature and the moment of inertia, which are independent of volume.
  \[ q_{\text{rot}} \propto T \]

Hence, the correct dependency is:

\[ q_{\text{trans}}(T, V), \quad q_{\text{vib}}(T), \quad q_{\text{rot}}(T) \]

\[ \boxed{\text{Correct option is (B)}} \]