Pure Rotational Spectroscopy Study Set

Vocabulary and key concepts regarding Pure Rotational Spectroscopy, including selection rules, the rigid rotor model, population distributions, and centrifugal distortion.

Rotational Spectroscopy

The study of transitions between quantized rotational energy levels of molecules, typically occurring in the microwave region of the electromagnetic spectrum, approximately $1 - 100\,\text{cm}^{-1}$.

Gross Selection Rule (Rotational)

For a molecule to exhibit a pure rotational spectrum, it must possess a permanent electric dipole moment ($\mu \neq 0$).

Microwave Active

Molecules with a permanent electric dipole moment such as heteronuclear diatomics ($HCl$, $CO$) and asymmetric polyatomics ($H_2O$, $NH_3$).

Microwave Inactive

Molecules that do not exhibit a pure rotational spectrum because they lack a permanent dipole, including homonuclear diatomics ($H_2$, $N_2$, $O_2$) and highly symmetric molecules ($CH_4$, $SF_6$).

Rigid Rotor Model

A model where the bond length ($r$) is assumed to be fixed, treating the molecule as two point masses ($m_1$ and $m_2$) rotating around their center of mass.

Reduced Mass ($\mu$)

A mathematical definition used in the rigid rotor model given by the formula $\mu = \frac{m_1 m_2}{m_1 + m_2}$.

Moment of Inertia ($I$)

A measure of an object's resistance to rotational acceleration, defined as $I = \mu r^2$, with units in $\text{kg m}^2$.

Rotational Constant ($B$) in Joules

The constant defined by the expression $B = \frac{h^2}{8\pi^2 I}$.

Rotational Constant ($\bar{B}$) in Wavenumbers

The constant used to express rotational energy levels in $\text{cm}^{-1}$, defined as $\bar{B} = \frac{h}{8\pi^2 cI}$.

Rotational Energy Levels ($E_J$)

The allowed energy levels quantized by the rotational quantum number $J$ ($J = 0, 1, 2 \dots$), defined as $E_J = \bar{B} J(J + 1)$.

Specific Selection Rule (Rotational Transitions)

Transitions are only allowed between adjacent levels where $\Delta J = \pm 1$.

Line Spacing (Rigid Rotor)

In a rigid rotor spectrum, rotational lines are equally spaced by $2\bar{B}$, with the first line appearing at $2\bar{B}$ and the second at $4\bar{B}$.

Degeneracy ($g_J$)

Defined as $g_J = 2J + 1$, it represents the number of spatial orientations ($m_J$ states) and increases population at higher $J$ levels.

Boltzmann Factor

The term $e^{-E/kT}$ in the Boltzmann distribution that causes population to decrease as rotational energy increases.

$$J_{max}$$

The rotational level with the maximum population, calculated using the formula $J_{max} \approx \sqrt{\frac{kT}{2hc\bar{B}}} - \frac{1}{2}$.

Isotope Substitution Effect

Replacing an atom with a heavier isotope (e.g., $^{12}C$ to $^{13}C$), which increases $\mu$ and $I$ while decreasing $\bar{B}$, causing the spectrum to shift to lower wavenumbers.

Centrifugal Distortion

The phenomenon in non-rigid rotors where bonds stretch as rotational speed increases, causing the moment of inertia ($I$) to increase and the rotational constant ($\bar{B}$) to decrease.

Centrifugal Distortion Constant ($D$)

A correction factor utilized in the energy level formula for a non-rigid rotor: $E_J = \bar{B} J(J + 1) - D [J(J + 1)]^2$.

Spectral Consequence of Non-Rigidity

The spacing between lines ($2\bar{B}$) in the rotational spectrum slowly decreases at very high $J$ levels.