Rotational Spectra of Diatomic Molecules

B · Transitions between quantised rotational energy levels

A rotational spectrum arises from transitions between quantised rotational energy levels of a molecule.

B · Microwave and far-infrared

Rotational transitions have the smallest energy gaps and lie mainly in the microwave and far-infrared regions.

A · Two masses joined by a fixed-length rod

The rigid rotor model treats a diatomic molecule as two masses joined by a fixed-length rod with no stretching or bending.

B · $I = \mu r^2$

The moment of inertia is $I = \mu r^2$, where $\mu$ is the reduced mass and $r$ is the bond length.

B · $E_J = hcB\,J(J+1)$

The rigid rotor energy levels are $E_J = hcB\,J(J+1)$ with $J = 0, 1, 2, \dots$

D · $\Delta J = \pm 1$

For a rigid rotor the selection rule is $\Delta J = \pm 1$, which produces equally spaced lines.

B · $2B$

Allowed transitions give lines at $2B, 4B, 6B, \dots$, so the spacing between adjacent lines is $2B$.

A · True

The spacing between adjacent lines in a rigid-rotor spectrum is $2B$, where $B = h/(8\pi^2 c I)$.

A · N$_2$

N$_2$ is a homonuclear diatomic with no permanent dipole moment, so it is microwave-inactive.

A · It has no permanent dipole moment

A permanent dipole moment is required for a molecule to interact with microwave radiation and show a pure rotational spectrum.

C · $B = \dfrac{h}{8\pi^2 c I}$

The rotational constant is $B = \dfrac{h}{8\pi^2 c I}$, usually expressed in cm$^{-1}$.

A · $2.0\ \text{cm}^{-1}$

Spacing $= 2B$, so $B = 4.0/2 = 2.0\ \text{cm}^{-1}$.

D · Line spacing → $B$ → $I$ → $r$

Measure the spacing to get $B$, then $I = h/(8\pi^2 c B)$, then $r = \sqrt{I/\mu}$.

C · Centrifugal distortion stretches the bond

At high $J$, centrifugal distortion stretches the bond slightly, so the rigid-rotor spacing is no longer exact and lines bunch up.

C · CO

CO has a permanent dipole moment and is microwave-active; H$_2$, N$_2$ and CO$_2$ are not.

C · It is larger than the true value by a factor of $\sqrt{2}$

Using $B_{\text{wrong}} = 2B_{\text{true}}$ gives $I_{\text{wrong}} = I_{\text{true}}/2$, so $r_{\text{wrong}} = r_{\text{true}}/\sqrt{2}$ — actually smaller. Wait: $B = h/(8\pi^2 c I)$, so halving $B$ doubles $I$ and multiplies $r$ by $\sqrt{2}$. The correct option is the one st…

B · $B = 1 ext{cm}^{-1}$

The spacing between adjacent lines in a rigid rotor spectrum is $2B$. Given that the measured spacing is $2B$, we can directly conclude that $B$ is half of that spacing, hence $B = 1 ext{cm}^{-1}$.

C · $r = 1.20 ext{Å}$

First, we find $B = rac{3.0}{2} = 1.5 ext{cm}^{-1}$. Then, using $I = rac{h}{8\pi^2 c B}$, we calculate $I$ and subsequently use $r = \sqrt{rac{I}{\mu}}$. This yields $r \approx 1.20 ext{Å}$.

A · The derived bond length is twice the actual bond length.

Since the researcher incorrectly uses the spacing $B$ instead of $2B$, they will derive a bond length that is twice the actual bond length, as they underestimate the rotational constant $B$.

B · $E_J = hcB\,J(J+1) - DJ^2(J+1)^2$

To account for centrifugal distortion, the energy expression must include a correction term $-DJ^2(J+1)^2$, which reflects the effect of molecular stretching at high rotational quantum numbers $J$.