This question concerns the following exercise from an old exam:
The vibrational motion of a linear diatomic molecule can be approximated as simple harmonic motion.
A CO molecule has a bond with force constant $k = 1900 \, Nm^{-1}$. What frequency of radiation would excite transitions between the different vibrational enery levels? (C and O have masses of $12 \, m_u$ and $16\, m_u$ respecitvely, where $m_u$ is the atomic mass unit.)
Explain why electromagnetic radiation can not excite vibrational transitions in an $O_2$ or $N_2$ molecule. How is this important for the heating of the earth's atmosphere by the sun (the greenhouse effect)?
For the first part I know the energy levels of a harmonic oscillator (in center of mass reference frame)
$$E_n = \hbar\omega(n+1/2),\quad \text{where }\omega = \sqrt{\frac{k}{\mu}}, \quad \mu = \frac{m_Om_C}{m_O + m_C}$$
and the energy of a photon of frequency $\nu$ is given by $E_{ph} = h\nu$. The photon energy must be the energy difference between two states:
$E_{ph} = E_n – E_m = \hbar \omega (n-m)$. This gives me the frequency
$$\nu = \frac{(n-m)}{2\pi}\sqrt{\frac{k}{\mu}}$$
Or minimum frequency $\nu_0 = \frac1{2\pi}\sqrt{\frac{k}{\mu}}$, right?
However, now to my actual question: I don't know why we cannot do the exact same thing for the other two molecules? What is different for $O_2$ and $N_2$?
I'd appreciate some help, thanks! =)
Best Answer
The molecules $O_2$ and $N_2$ are symmetric and have no dipole momentum. That's why they can not interact with EMW (at least within dipole approximation).
One can say that transitions between the oscillator levels in these molecules are forbidden by symmetry in electrodipole approximation.
The molecule $CO$ consists of two different atoms. The average positions of positive and negative charges are not the same. This molecule is polar.