$q_{vib} (T) = \frac{e^{-h\nu/2k_BT}}{1-e^{-h\nu/k_BT}}$
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)
Given
Canonical partition function of harmonic oscillator:
$q_{vib}(T) = \dfrac{e^{-h\nu/2k_BT}}{1 - e^{-h\nu/k_BT}}$
Given ratio: $\dfrac{h\nu}{k_BT} = 3$
Step 1: Write expression for ground state probability
The probability of finding the oscillator in the ground state is
$P_0 = \dfrac{e^{-h\nu/2k_BT}}{q_{vib}}$
Step 2: Substitute $q_{vib}$ into $P_0$
$P_0 = \dfrac{e^{-h\nu/2k_BT}}{\dfrac{e^{-h\nu/2k_BT}}{1 - e^{-h\nu/k_BT}}}$
This simplifies to
$P_0 = 1 - e^{-h\nu/k_BT}$
Step 3: Substitute numerical value
Given $\dfrac{h\nu}{k_BT} = 3$
$P_0 = 1 - e^{-3}$
$P_0 = 1 - 0.0498 = 0.9502$
Final Answer
$\boxed{0.95}$
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
