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
Six distinguishable particles are distributed over 3 non‐degenerate levels, of energies 0, ε and 2ε. The most probable value for the total energy is
The partition function for a gas is given by
Q(N, V, T) = \(\frac{1}{N!}\left(\frac{2\pi m}{h^2\beta}\right)^{3N/2}\) (v - Nb)Ne \(\frac{\beta aN^2}{V}\)
The internal energy of the gas is
A three-state system with energies E = −ε0, 0, +ε0 is in a thermal equilibrium at a temperature T. If β ε0 = x, the probability of finding the system with energy E = 0 is [recall, cosh x = \(\frac{1}{2}\)(ex + e−x)]
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)- At temperature T, the canonical partition function of a harmonic oscillator with fundamental frequency ($\nu$) is given by
$q_{vib} (T) = \frac{e^{-h\nu/2k_BT}}{1-e^{-h\nu/k_BT}}$
For $\frac{h\nu}{k_BT} = 3$, the probability of finding the harmonic oscillator in its ground vibrational state is ____________ (Up to two decimal places)
