Question

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?

Short answer

Answer

(a) Will affect the rotational energy E_rot .

(b) Will affect the rotational energy E_vib .

Step-by-step solution

Given data 

Local minimum is at x = a.

Concept of rotational and vibration energy

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.

 Step 3: Determine the factor on which rotational energy depends

(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$E_{rot}$.