My book, Pavia's Introduction to Spectroscopy, tells me this about the infrared spectrum of alkanes:
$\ce{C-C}$ : Stretch not interpretatively useful; many weak peaks.
This is very interesting. What makes the C–C bond so reluctant to absorb infrared wavelengths?
I have been doing a bit of thinking lately. Bonds absorb infrared radiation that match with the harmonic vibrational frequency of the bond. I believe the extent of absorption is based on the tendency of the bond to get vibrated. For some reason, the $\ce{C-C}$ bond is very reluctant to absorb the radiation, and I believe it's because the bond is too "rigid".
If a bond is too rigid, it must show large change in intermolecular force per change in bond distance. In other words, I believe, the value of
$$\frac{\mathrm dF}{\mathrm dr}$$
is quite high. In such a case, the bond wouldn't have a tendency to compress and expand, since the bond has such a high tendency to oppose such changes. Assuming the bond to act as a spring, I have,
$$F=kr$$
where $k$ is the spring constant of the bond-spring. Differentiating this w.r.t $r$, we get:
$$\frac{\mathrm dF}{\mathrm dr}=k$$
So, this makes me conclude that a higher value of $k$, implies a more rigid bond. I also know that
$$U=\frac{1}{2}kr^2$$
Plugging the values of $U$ ($\pu{=346000J}$) and $r$ ($\pu{=1.54 \times 10^{-10}m}$) for the $\ce{C-C}$ bond, I get,
$$k_{\ce{C-C}}\pu{= 2.92Nm^-1}$$
If I compare this with the value of $k$ for the $\ce{C=O}$ bond (one of the most prominent absorptions), I get,
$$k_{\ce{C=O}}\pu{= 1.11Nm^-1}$$
The $k_{\ce{C=O}}$ is just 3 times smaller than $k_{\ce{C-C}}$, but the existence of $\ce{C-C}$ absorption is rather non-existent.
I can conclude one thing for sure, that all that I have thought before is completely wrong.
So here's my question. Why does the C–C bond absorb extremely weakly? And what's wrong with what I have thought about above?