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
Partition Function Dependencies Analysis
The question asks for the thermodynamic variables upon which the translational ($q_{trans}$), vibrational ($q_{vib}$), and rotational ($q_{rot}$) molecular partition functions depend for ideal diatomic gas molecules in the canonical ensemble (N, V, T).
Translational Partition Function ($q_{trans}$)
The translational partition function depends on the available volume ($V$) and temperature ($T$). It describes the motion of the center of mass of the molecule. Its dependence is given by $q_{trans} \propto V T^{3/2}$. Therefore, $q_{trans}$ depends on both T and V.
Vibrational Partition Function ($q_{vib}$)
The vibrational partition function depends on the vibrational frequency of the molecule (a molecular property) and the temperature ($T$). For ideal gases, vibrational energy levels are generally considered independent of the volume ($V$). Thus, $q_{vib}$ depends only on T.
Rotational Partition Function ($q_{rot}$)
The rotational partition function depends on the rotational constants (related to the moment of inertia, a molecular property) and the temperature ($T$). For ideal diatomic gases, rotational energy levels are also independent of the volume ($V$). Hence, $q_{rot}$ depends only on T.
Conclusion on Dependencies
Based on the analysis:
- $q_{trans}$ depends on (T, V)
- $q_{vib}$ depends on (T)
- $q_{rot}$ depends on (T)
This matches the dependencies stated in Option B.
Important Questions from Partition Functions and Their Relation
- If the translational partition function for H$_2$ confined in a 1 L vessel at 300 K is $y \times 10^{27}$, then the value of $y$ is ______ (rounded off to one decimal place).
(Given: Atomic mass (in amu): H = 1.008; 1 amu = $1.661 \times 10^{-27}$ kg; $h = 6.626 \times 10^{-34}$ J s; $k = 1.381 \times 10^{-23}$ J K$^{-1}$) If $q_t$ and $Q_{t,m}$ are the molecular and molar translational partition functions of $X_2$, respectively, then $ln(Q_{t,m})$ =
(N is the Avogadro number)- The rotational partition function of a diatomic molecule with energy levels corresponding to J = 0 and 1, is (where, $\epsilon$ is a constant)
- The molecular partition function of a system is given by
$q(T) = (\frac{k_B T}{hc})^{\frac{3}{2}} (\frac{8\pi^2 mk_B T}{h^2})^{\frac{3}{2}}$, where the symbols have their usual meanings.
The heat capacity at constant volume for this system is - The molecular partition function for a system in which the energy levels are equispaced by $ε$, is
