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)
$N ln q_t - N ln N + N$
Translational Partition Function Relationship
The question asks for the relationship between the molar translational partition function ($Q_{t,m}$) and the molecular translational partition function ($q_t$) for a gas like $X_2$. We use the standard formula connecting molar and molecular partition functions:
$ Q_{t,m} = \frac{(q_t)^N}{N!} $
where $N$ is the Avogadro number.
Deriving Logarithm of Molar Partition Function
To find the expression for $ln(Q_{t,m})$, we take the natural logarithm of both sides of the equation:
$ ln(Q_{t,m}) = ln\left(\frac{(q_t)^N}{N!}\right) $
Using the properties of logarithms ($ln(a/b) = ln(a) - ln(b)$ and $ln(a^b) = b \cdot ln(a)$):
$ ln(Q_{t,m}) = ln((q_t)^N) - ln(N!) $
$ ln(Q_{t,m}) = N \cdot ln(q_t) - ln(N!) $
Applying Stirling's Approximation
For a large number like Avogadro's number (N), Stirling's approximation for $ln(N!)$ is used:
$ ln(N!) \approx N \cdot ln(N) - N $
Substitute this approximation back into the equation for $ln(Q_{t,m})$:
$ ln(Q_{t,m}) = N \cdot ln(q_t) - (N \cdot ln(N) - N) $
$ ln(Q_{t,m}) = N \cdot ln(q_t) - N \cdot ln(N) + N $
Final Result
This derived expression matches Option D, which represents the natural logarithm of the molar translational partition function using Stirling's approximation.
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)]
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
- 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)
