Question: The interatomic potential energy in a diatomic molecule (Figure 10.16) has many features: a minimum energy, an equilibrium separation a curvature and so on. (a) Upon what features do rotational energy levels depend? (b) Upon what features do the vibration levels depend?

Local minimum is at x = a.

The expression for rotational energy is, ${\mathit{E}}_{\mathbf{r}\mathbf{o}\mathbf{t}}\mathbf{=}\frac{{\mathbf{\hslash }}^{\mathbf{2}}\mathbf{l}\mathbf{(}\mathbf{l}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{2}\mathbf{\mu }{\mathbf{a}}^{\mathbf{2}}}$.

Where,$\hslash \to$reduced Planck's constant,$I\to$rotational quantum number,$\mu \to$reduced mass and$a\to$bond length in meters.

The reduced mass $\mathit{\mu }$ is, $\mathit{\mu }\mathbf{=}\frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{m}}_{\mathbf{2}}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{3}}}$.

The expression for vibration energy is, ${\mathit{E}}_{\mathbf{\text{vib}}}\mathbf{=}\left(n+\frac{1}{2}\right)\mathit{\hslash }\sqrt{\frac{\mathbf{k}}{\mathbf{\mu }}}$.

Where, $\hslash \to$ Reduced Planck's constant, $n\to$ vibration quantum number and $\mu \to$ Reduced mass.

(a)

The reduced mass$\mu$is,$\mu =\frac{m_1{m}_2}{m_1+m_3}$.

The rotational energy is expressed as$E_{rot}=\frac{{\hslash }^{2}l(l+1)}{2\mu a^2}$ .

In the above expression$\hslash ,l$are constant, anddepends on the mass.

Therefore, the quantitywill affect the rotational energy$E_{rot}$.

The separation of the two atoms is equivalent to the orbital radius in a planetary system, which determines the rotational energy.

Therefore,Will affect the rotational energy$E_{rot}$.